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MATHEMATICS OF COMPUTATION Volume 75, Number 253, Pages 183–207 S 0025-5718(05)01772-2 Article electronically published on September 16, 2005

GLOBAL OPTIMIZATION OF EXPLICIT STRONG-STABILITY-PRESERVING RUNGE-KUTTA METHODS

STEVEN J. RUUTH

Abstract. Strong-stability-preserving Runge-Kutta (SSPRK) methods are a type of time discretization method that are widely used, especially for the time evolution of hyperbolic partial di?erential equations (PDEs). Under a suitable stepsize restriction, these methods share a desirable nonlinear stability property with the underlying PDE; e.g., positivity or stability with respect to total variation. This is of particular interest when the solution exhibits shock-like or other nonsmooth behaviour. A variety of optimality results have been proven for simple SSPRK methods. However, the scope of these results has been limited to low-order methods due to the detailed nature of the proofs. In this article, global optimization software, BARON, is applied to an appropriate mathematical formulation to obtain optimality results for general explicit SSPRK methods up to ?fth-order and explicit low-storage SSPRK methods up to fourth-order. Throughout, our studies allow for the possibility of negative coe?cients which correspond to downwind-biased spatial discretizations. Guarantees of optimality are obtained for a variety of third- and fourth-order schemes. Where optimality is impractical to guarantee (speci?cally, for ?fthorder methods and certain low-storage methods), extensive numerical optimizations are carried out to derive numerically optimal schemes. As a part of these studies, several new schemes arise which have theoretically improved time-stepping restrictions over schemes appearing in the recent literature.

1. Introduction Popular time-stepping schemes are typically based on linear stability analysis. Such analysis is often very e?ective on problems having smooth solutions. However, these schemes often perform poorly on problems having discontinuous or shocklike solutions. This poor performance can manifest itself in the form of spurious oscillations, overshoots, or loss of positivity. On the other hand, strong-stabilitypreserving (SSP) time discretization methods [21, 22, 8, 24] are designed to preserve the nonlinear stability properties that arise when the forward Euler time-stepping scheme is applied to the spatially discretized system. The ability to preserve this underlying nonlinear stability makes SSP time-stepping methods particularly suitable for the simulation of partial di?erential equations with nonsmooth solutions.

Received by the editor July 2, 2003 and, in revised form, August 31, 2004. 2000 Mathematics Subject Classi?cation. Primary 65L06, 65M20. Key words and phrases. Strong-stability-preserving, total-variation-diminishing, Runge-Kutta methods, high-order accuracy, time discretization, downwinding, low-storage. This work was partially supported by a grant from NSERC Canada.

c 2005 American Mathematical Society

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In this paper, particular attention will be paid to the development of guaranteed optimal1 explicit SSP Runge-Kutta (SSPRK) time-stepping methods for hyperbolic PDEs; e.g., (1.1) ut + f (u)x = 0, subject to appropriate initial conditions. Solutions to these and other PDEs are often approximated by sequentially discretizing the temporal and spatial derivatives. For example, in the method of lines, a discretization of the spatial derivatives of the PDE is carried out to produce a large set of coupled time-dependent ordinary di?erential equations (ODEs) ˙ (1.2) U = F (U ). These ODEs can then be treated by suitable time-stepping techniques such as linear multi-step or Runge-Kutta methods. For hyperbolic conservation laws, papers by Shu [21], Shu and Osher [22], and subsequent work [7, 8, 20] have shown that improved nonlinear stability can sometimes be found by appropriately selecting nonlinearly stable upwind-biased (F (U )) ? and downwind-biased (F (U )) spatial discretizations according to the coe?cients of the time-stepping method. Thus, this paper considers optimality over the broad class of explicit SSPRK schemes that includes both upwind-and downwind-biased spatial discretizations. We remark, however, that most of the optimal schemes that we present involve only nonnegative coe?cients. Nonnegative coe?cient schemes are appropriate to apply to general problems (including dissipative PDEs) since ? they do not involve F (·) Optimal explicit SSPRK schemes with nonnegative coe?cients and where the number of stages s is equal to the order p for s = p = 1, 2, and 3 have been known for some time [21, 22, 7]. Gottlieb and Shu [7] showed that no such method exists with nonnegative coe?cients when s = p = 4. In [24, 25], Spiteri and Ruuth studied the general class of explicit nonnegative coe?cient SSPRK methods with s > p. They gave optimal SSPRK schemes with s stages and orders 1 and 2 (see also [21, 7]), as well as speci?c schemes for p = 3, s = 4, 5 and p = 4, s = 5, 6, 7, 8. The advantage a?orded by these high-stage schemes is that the increase in the CFL coe?cient allows for a large enough increase in the stable time step to more than o?set the increase in computational cost per step. However, in [19] they showed that it was impossible to have an explicit SSPRK method with order greater than 4 with nonnegative coe?cients. Ruuth and Spiteri [20] later gave a uni?ed treatment of explicit SSPRK schemes with unrestricted coe?cients and found that many of the optimal explicit nonnegative coe?cient SSPRK methods are also optimal in terms of e?ective CFL coe?cient when the sign of the coe?cients is not restricted. However, optimality proofs for third-order schemes were limited to the simplest cases (s = 3, 4), and proofs were not attempted for higher orders. Indeed, even ?nding e?cient ?fth-order SSPRK schemes proved challenging, and motivated the development of this study. To date, most of the practical interest in SSPRK schemes has been for explicit schemes, and, unless otherwise stated, all studies appearing in this report are for explicit SSPRK schemes. Implicit SSPRK schemes have been investigated by a number of authors [8, 4, 10, 9]. It is noteworthy that implicit SSPRK schemes with

1 Optimality will be determined with respect to the e?ective CFL coe?cient de?ned in Section 2.

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nonnegative coe?cients cannot be unconditionally SSP [4, 10, 9] and that diagonally implicit schemes with unrestricted coe?cients cannot be unconditionally SSP [8]. The remainder of the paper is organized as follows. In Section 2 we review some relevant results on SSP schemes and de?ne key concepts such as the e?ective CFL coe?cient. In Section 3 we develop a mathematical formulation of the problem suitable for global optimization and provide details on our optimization procedures. Section 4 uses these techniques to achieve the ?rst guarantees of optimality of several third- and fourth-order SSPRK schemes. New ?fth-order methods are also derived which exceed the theoretical e?ciency of previously known methods. In Section 5 our approach is used to guarantee the optimality of a variety of low-storage SSPRK schemes appearing in the literature. Some improvements for recently derived lowstorage schemes are also found. Finally Section 6 concludes by summarizing the main ?ndings of the paper. 2. Background on SSP schemes In this section we give some theoretical background on SSPRK schemes. We begin by recalling the de?nition of strong stability: De?nition 1. A sequence {U n } is said to be strongly stable in a given semi-norm || · || if ||U n+1 || ≤ ||U n || for all n ≥ 0. Now assume that upwind-biased spatial discretizations are appropriate and express an s-stage, explicit Runge-Kutta method using an α ? β (or Shu-Osher [3]) representation (2.1a) (2.1b) (2.1c) U (0) U (i) U n+1 = U n,

i?1

= = U

(αik U (k) + ?tβik F (U (k) )),

k=0 (s)

i = 1, 2, . . . , s,

,

where all the αik ≥ 0 [21]. A given Runge-Kutta scheme will typically have many di?erent representations of this type; see [24] for a discussion and an illustrative example. Throughout this paper, we construct representations that maximize the CFL coe?cient, as de?ned in Theorem 1 or 2 below. We call an optimal representation of this type an optimal α ? β representation and note that such representations have been constructed both numerically (e.g., [22, 8, 24]) and via a contractivitybased approach [3, 9]. i?1 By consistency we must have that k=0 αik = 1, i = 1, 2, . . . , s. Assuming that both sets of coe?cients αik , βik are nonnegative, it is clear that (2.1) is a convex βik combination of forward Euler steps with various step sizes αik ?t. The strong stability property follows easily, and we conclude [22, 8]: Theorem 1. If the forward Euler method is strongly stable under the CFL restriction ?t ≤ ?tF E , then the Runge-Kutta method (2.1) with βik ≥ 0 is SSP provided ?t ≤ C?tF E , where C is the CFL coe?cient C ≡ min {cik : 0 ≤ k < i ≤ s} , where cik = ∞

αik βik

if βik = 0, otherwise.

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The CFL coe?cient for strong stability has an interesting relationship with the time-stepping restriction associated with the more classical concept of contractiv? ity, where the di?erence ||U n+1 ? U n+1 || between any two sequences is required to be nonincreasing with increasing n (see, e.g., [23, 14, 15]). It turns out that many of the optimal SSP schemes found in [24] agree with optimal contractive schemes in [15]. In fact, for a given Runge-Kutta method involving only upwindbiased spatial discretrizations (2.1), Ferracina and Spijker [3, 4] have proven that the CFL coe?cient C corresponding to an optimal α ? β representation equals the related quantity R(A, b) [15] arising in contractivity studies, provided C ≥ 0. (If C < 0 for an optimal α ? β representation, then R(A, b) = 0 [3].) See also the recent work of Higueras [9] for some interesting relationships between contractive and strong-stability-preserving Runge-Kutta methods when both upwind- and downwind-biased operators arise. SSPRK schemes with negative coe?cients βik are also accommodated by modifying the spatial discretization. Following the procedure ?rst suggested in papers ? by Shu [21] and Shu and Osher [22], whenever βik < 0, the operator F (·) is used in? (·) approximates the same derivatives as F (·) but is assumed stead of F (·), where F to be strongly stable for Euler’s method solved backwards in time under a suitable time-step restriction. In practice, this corresponds to di?erencing in the downwind direction. Remark 1. Suppose that an appropriate spatial discretization has been derived ? for computing F for the hyperbolic conservation law (1.1). Then ?F is normally found by applying this same spatial discretization technique to the backwards-intime variant ut + (?f (u)x ) = 0. To explicitly illustrate, consider discretizing the linear, variable coe?cient problem (2.2) ut + a(x)ux = 0

using a uniform mesh (step size h) and ?rst-order one-sided di?erencing. Then F can be found by applying ?rst-order upwinding to (2.2) to give Fj = ? a(xj )(Uj ? Uj?1 )/h if a(xj ) > 0, a(xj )(Uj+1 ? Uj )/h otherwise.

Similarly, ?rst-order upwinding can also be applied to the backwards-in-time version of (2.2) to give ? ?Fj = a(xj )(Uj+1 ? Uj )/h if a(xj ) > 0, a(xj )(Uj ? Uj?1 )/h otherwise,

? ? from which F is trivially obtained. Clearly F approximates the same quantity ? is based on ?rst-order downwind di?erencing instead of ?rstas F , however, F order upwinding. See [21] for further discussion on constructing downwind-biased discretizations. As shown by Shu [21] and Shu and Osher [22], an interesting generalization of Theorem 1 is obtained by using both upwind- and downwind-biased operators: Theorem 2. Let Euler’s method applied forward in time combined with the spatial discretization F (·) be strongly stable under the CFL restriction ?t ≤ ?tF E . Let Euler’s method applied backward in time combined with the spatial discretization

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? F (·) also be strongly stable under the same CFL restriction ?t ≤ ?tF E . Then the Runge-Kutta method (2.1) is SSP provided ?t ≤ C?tF E , where C is the CFL coe?cient (2.3) C ≡ min {cik : 0 ≤ k < i ≤ s} , where cik =

αik |βik |

∞

if βik = 0, otherwise,

? and βik F (·) is replaced by βik F (·) whenever βik is negative. We note that the assumptions on strong stability of Euler’s method applied forward and backward in time restricts the theoretical advantages of this result to non-dissipative equations such as (1.1). If every coe?cient βik is nonnegative, then the number of stages is trivially equal to the number of function evaluations for an irreducible explicit Runge-Kutta ? method. However, if both F (U (k) ) and F (U (k) ) are required for some k, the RungeKutta method (2.1) has more function evaluations2 than stages. As discussed by Ruuth and Spiteri [20], the ?rst step in deriving optimal schemes is to create a fair comparison of the computational cost of a given Runge-Kutta method by considering general methods that allow precisely one (new) function evaluation per stage. This can be achieved by insisting that the nonzero coe?cients βik for a given k are all of the same sign [20]. For the remainder of the paper, we will tacitly assume that the schemes under consideration are of this form, and we remark that schemes that are written combining nonnegative and negative coe?cients βik within a given level k can be augmented with additional stages to be of this form [20]. Thus, without loss of generality, we have that the total number of evaluations of F (·) and ? F (·) is identically equal to the number of stages of the method. We emphasize that with this formulation it is quite natural to search for the optimal scheme for a given order and a given number of stages (function evaluations) by maximizing the CFL coe?cient. In earlier formulations the e?ective ? CFL coe?cient could vary with the number of F (·) evaluations, complicating the development of optimal schemes. Another advantage to this formulation is that schemes can be represented in Butcher array form via

i?1

(2.4) aik

= βi?1,k?1 +

j=k+1 s

αi?1,j?1 ajk , k = 1, 2, . . . , i ? 1, i = 2, 3, . . . , s, αs,j?1 ajk , k = 1, 2, . . . , s,

bk

= βs,k?1 +

j=k+1

2 The only di?erence between F (·) and F (·) is a change in the upwind direction, so F (·) can ? ? clearly be computed with the same cost as F (·) [8]. Recent studies make the assumption that ? if both F (U (k) ) and F (U (k) ) must be computed for some k, the cost as well as the storage requirements for that k doubles [7, 8, 24, 20].

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? since di?erences of the form F (U (i) ) ? F (U (i) ) do not arise. Thus the method can be implemented as ? ? F U n + ?t Ki = ? ? F U + ?t

n s i?1 j=1 i?1 j=1

aij Kj aij Kj

if bi ≥ 0, otherwise,

i = 1, 2, . . . , s,

U n+1 = U n + ?t

i=1

bi Ki .

Butcher array form may be desirable for a variety of reasons such as to simplify optimality proofs, facilitate optimization (see Section 3), or (in some instances) reduce storage requirements. Note that by construction, if the βij , j +1 ≤ i ≤ s, are of one particular sign, then the corresponding ai,j+1 , j + 2 ≤ i ≤ s, and bj+1 values ? must be of that same sign. We further remark that the di?erences F (U (i) )? F (U (i) ) contribute to arti?cial dissipation and smearing [20]. For example, this di?erence is proportional to the discrete Laplacian when ?rst-order upwinding is applied to the linear advection equation. A natural consequence of this formulation is that during optimization these dissipative di?erences do not arise, leading to schemes with smaller errors and less smearing than would otherwise occur [20]. In order to make a fair comparison of the relative e?ciencies of these methods and to derive optimal schemes, we make the following de?nition. De?nition 2. The e?ective CFL coe?cient Ce? of a SSPRK method is C/s, where C is the CFL coe?cient of the method and s is the number of stages (function evaluations) required for one step of the method. As conjectured in Shu and Osher [22] and subsequently proven in Gottlieb and Shu [7], the optimal two-stage, order-two SSPRK scheme with nonnegative coe?cients is the modi?ed Euler scheme. It has a CFL restriction ?t ≤ ?tF E , which implies a CFL coe?cient of 1. Henceforth, we will refer to this scheme as SSP(2,2). In general, we adopt the convention of referring to a numerically optimal s-stage, order-p SSPRK scheme as SSP(s,p). This scheme is the ?rst member of a class of s-stage, order-two SSPRK schemes [5, 24] with a CFL coe?cient of s ? 1. For nonnegative coe?cients, the optimality of these schemes was proven in [24]. Optimality for the case of s general function evaluations was shown in [20]. Shu and Osher [22] also conjectured that the optimal three-stage, order-three SSPRK scheme is U (1) U (2) U n+1 = U n + ?tF (U n ), 3 n 1 (1) U + U + = 4 4 1 n 2 (2) U + U + = 3 3

1 ?tF (U (1) ), 4 2 ?tF (U (2) ), 3

which has a CFL coe?cient of 1 as well. This scheme is commonly called the TVD Runge-Kutta scheme, but we will refer to it as SSP(3,3). Optimality of this scheme was proved for nonnegative coe?cients in [7] and for three general function evaluations in [20].

GLOBAL OPTIMIZATION

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In [19], Ruuth and Spiteri derived a linear bound that is used to prove that the optimal four-stage, order three SSPRK scheme with nonnegative coe?cients is U (1) U (2) U (3) U n+1 1 = U n + ?tF (U n ), 2 1 = U (1) + ?tF (U (1) ), 2 2 n 1 (2) 1 U + U + ?tF (U (2) ), = 3 3 6 1 = U (3) + ?tF (U (3) ), 2

which has a CFL coe?cient of 2. Following [24] we will refer to this scheme as SSP(4,3). Optimality in the unrestricted case is shown in [20]. Moving on to nonnegative coe?cient SSPRK schemes with ?ve stages and order three gives a numerically optimized scheme, SSP(5,3), with a CFL coe?cient of approximately 2.65 [24]. This scheme has a CFL coe?cient that agrees with the contractivity bound R(A, b) for linear constant-coe?cient problems [14]. Because these restrictions are equivalent [3] and the time-stepping restriction for nonlinear problems cannot exceed that for linear problems, we conclude that SSP(5,3) is also an optimal ?ve-stage, third-order nonnegative coe?cient SSPRK scheme. Similarly, it was noted in [20] that third-order schemes with up to nine stages can be constructed that have CFL coe?cients that agree with the contractivity bound R(A, b) for linear constant-coe?cient problems. Unfortunately, proving optimality for any of these schemes (s ≥ 5) when s general function evaluations are involved is much more complicated than for the three- or four-stage cases. In Section 4.1 we use global optimization to directly guarantee the optimality of SSP(5,3) and SSP(6,3) over this broader class of schemes. We also give third-order schemes with seven and eight function evaluations and guarantee their optimality using global optimization in combination with the linear contractivity bound. The analysis for orders greater than 3 is more complicated still, since even for the nonnegative coe?cient case the linear contractivity bound fails to give a sharp bound for the nonlinear problem. However, by appropriately applying global optimization techniques, we are able to guarantee optimality of the ?ve-stage, fourthorder SSPRK scheme given in [24, 15] in the general setting where downwind-biased spatial discretizations may arise. See Section 4.2. We remark that this scheme is particularly important since it is impossible for a fourth-order SSPRK scheme to use fewer than ?ve (general) function evaluations [7, 20]. In [19] it is shown that explicit SSPRK schemes with only nonnegative coe?cients do not exist with order greater than four. A similar restriction to orders four or less was shown for contractive schemes by Kraaijevanger [15]. This means that the search for schemes of order-?ve and higher must involve evaluations of ? the downwind-biased operator F (·). Fortunately, this still leads to schemes with competitive e?ective CFL coe?cients. In Section 5 we derive a ?fth-order scheme with an e?ective CFL coe?cient of 0.3395, exceeding that of the popular SSP(3,3) scheme. Optimal low-storage SSPRK schemes have also received some attention in the literature [7, 8, 24, 13]. In particular, Gottlieb and Shu [7] ?nd an optimized threestage, third-order scheme of Williamson [32] type and Kennedy, Carpenter and Lewis [13] ?nd several optimized schemes of van der Houwen and Wray type. Using

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global branch-and-bound optimization techniques, we are able to guarantee the optimality of these schemes and others. In the course of this work, improvements to some recently published low-storage schemes are also found. We remark that until now only nonnegative coe?cients have been used to construct low-storage schemes, since it has been assumed that downwind-biased operators destroy the lowstorage property. Section 5 disproves this assumption by exhibiting two examples of low-storage schemes that are more e?cient than the corresponding nonnegative coe?cient schemes. The remainder of the paper commences with a discussion on our optimization techniques. 3. Global optimization of SSPRK schemes Traditional nonlinear programs are guaranteed to converge to the optimal solution only under certain convexity assumptions [28]. On the other hand, deterministic global optimization algorithms of the branch-and-bound type are available for a variety of problems that are guaranteed to provide global optima under fairly general assumptions [28]. In this section we give a mathematical formulation appropriate for BARON [28, 29], and show how to bound the variables and their expressions in the nonlinear programming (NLP) problem. We also e?ectively break the problem into parallel subproblems to reduce the total number of CPU cycles required. Subsequent sections use these ideas to determine and to guarantee the optimality of several third- and fourth-order schemes in both the low-storage and general settings. 3.1. Formulation of the optimization problem. We seek to optimize an sstage, order-p SSPRK scheme by maximizing the CFL coe?cient de?ned in Theorem 2. To achieve this goal, we will transform the problem into a smooth version that has additional constraints, but is more amenable to treat using standard numerical optimization techniques. As a ?rst step, we replace the nonsmooth objective function arising in the original formulation with a new variable, z, that is constrained to be a lower bound on all the ratios {αik /|βik |}. This well-known technique applied to our problem gives the equivalent formulation [6, p. 96, 97] (3.1a) subject to (3.1b) (3.1c) αik βk+1,k , βk+2,k , . . . , βsk or βk+1,k , βk+2,k , . . . , βsk

i?1

max z,

(αik ,βik )

≥ 0, ≥ 0, ≤ 0, = 1,

k = 0, . . . , s ? 1, i = 1, 2, . . . , s, t ∈ Tq , q = 1, 2, . . . , p,

(3.1d)

k=0 s

αik

(3.1e)

j=1

bj Φj (t) =

1 , γ(t)

(3.1f)

αik ? z|βik | ≥ 0,

k = 0, 1, . . . , i ? 1, i = 1, 2, . . . , s,

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where the Φj (t) and bj are nonlinear polynomials in αik , βik and the optimal zvalue equals the CFL coe?cient. The notation bj is used in the usual sense of the Butcher array representation of a Runge-Kutta method, and the symbol Tq stands for the set of all rooted trees of order equal to q. The number of constraints corresponding to the order conditions (3.1e) is 1,2,4,8 or 17 for orders p=1,2,3,4 or 5, respectively. We remark that this approach was ?rst used for the optimization of SSPRK schemes with nonnegative coe?cients in [24] and was ?rst proposed for the optimization of SSPRK schemes with unrestricted coe?cients in [20]. This formulation is e?ective for the nonnegative coe?cient problem [24]. To study the unrestricted case, we wish to remove the nonsmooth absolute value function from our formulation, since general-purpose software for nonsmooth optimization problems is not expected to be as e?cient or robust as general-purpose software designed for smooth problems. In particular, the GAMS User’s Guide [1, p. 70] states Smooth functions can be used routinely in nonlinear models, but non-smooth ones may cause numerical problems and should be used only if unavoidable, and only in a special mode type called DNLP. However, the use of DNLP model type is strongly discouraged and the use of binary variables is recommended to model non-smooth functions. Following this recommendation, we introduce a sign variable for each level, σ(k) = +1 if βk+1,k , βk+2,k , . . . , βsk ≥ 0, ?1 otherwise,

indicating the sign of the coe?cients at level k, 1 ≤ k ≤ s, and de?ning a variable, ? βik , to represent the absolute value of βik . To formulate the problem in these new ? ? ? variables, we replace conditions (3.1c) and (3.1f) with βk+1,k , βk+2,k , . . . , βsk ≥ 0 ?ik ≥ 0, respectively. Our updated formulation (given in terms of σ(k), and αik ? z β ? ? βik and αik ) is completed by replacing each βik by σ(k)βik in the order conditions (3.1e). This mixed integer nonlinear programming (MINLP) formulation is comprised of polynomial objective and constraint functions, and thus is suitable for optimization in BARON.3 However, the nonlinear order conditions take a simpler form when written in terms of the Butcher array entries aik and bk rather than αik , βik , so we prefer to solve for the σ(k), αik and the (unsigned) Butcher array entries directly. ? The βik are formed, where needed, as a linear combination of αik and Butcher array entries using (2.4). While this results in more complicated constraints on the bound, z, the overall speed of computation sometimes experiences a noteworthy improvement (e.g., a factor of two or more was observed in some tests), since the complicated nonlinear order conditions are simpli?ed. 3.2. Parallelization. As prescribed above, the optimal SSP scheme is given as the global solution to a MINLP problem. While this is a suitable mathematical formulation for a variety of GAMS optimization software using the GAMS MINLP model type [2], the total computational e?ort can be reduced signi?cantly by instead solving 2s NLP problems, each corresponding to one of the possible sign combinations.

3 Note that a variety of nonlinear functions are allowed in BARON [26, p. 10]: “In addition to multiplication and division, GAMS/BARON can handle nonlinear functions that involve exp(x), ln(x), exp(x), ln(x), xα for real α, β x for real β, xy , and |x|.”

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This leads to a total of 2s cases, which we treat in a parallel fashion using the GAMS NLP model type. The total time savings was not found to have a predictable dependency on the number of stages. For example, for third order and s = 3, 4, 5, we found that treating separate NLP problems required about 40%, 60% and 40% of the total CPU time of a single MINLP problem, respectively. It is interesting that in the optimization of linear multistep SSP schemes, both approaches take about the same overall CPU time if product disaggregation [27] (or distributing products over their sums) is used in the MINLP problem [18]. Without product disaggregation the MINLP problem is much slower [18]. For many models, product disaggregation leads to overall improved e?ciency by making use of tighter linear programming relaxations; see [27] for details. Product disaggregation was not found to improve the e?ciency of the MINLP problem for the more complicated Runge-Kutta case considered here. Nonetheless, it is possible that the tightness of linear programming relaxations plays some role in the relative e?ciencies of the two approaches. We remark that the number of cases can be reduced somewhat by using the fact that the number of nonnegative levels must be equal or exceed the CFL coe?cient [20]: Lemma 1. Suppose a consistent s-stage SSPRK method (2.1) has coe?cients βik ≥ 0 at distinct stages, i.e., βik ≥ 0 for all i and k = k1 , k2 , . . . , k with 0 ≤ k1 < k2 < · · · < k ≤ s ? 1. Then the CFL coe?cient C of the method satis?es C ≤ . We illustrate the usage of this lemma for eight stages and order three in Section 4.1. 3.3. Parameter bounds. For an e?cient search and to guarantee optimality, the variables must be bounded. Fortunately we know that all of the Butcher array entries are bounded by the inverse of the CFL coe?cient [20]: Lemma 2. If a method of the form (2.1) with αik ≥ 0 has a CFL coe?cient c ≥ m > 0, then 1 1 ≤ aik ≤ for all k = 1, 2, . . . , i ? 1, i = 2, 3, . . . , s, m m 1 1 ? ≤ bk ≤ (3.3) for all k = 1, 2, . . . , s. m m According to the lemma, if we can ?nd any feasible solution with a CFL coe?cient of m, we know that each of the Butcher array entries of the optimal solution must be bounded in absolute value by 1/m. Fortunately, ?nding good feasible solutions in BARON is a relatively inexpensive task which typically takes no more than a few seconds in the preprocessing step for orders four or less. If a better bound is not known, we may choose zero as a lower bound on the variable z, since the CFL coe?cient must be positive to be of any practical value. Note that Lemma 1 may be used to derive an upper bound if such a bound is not immediately clear from the problem or from numerical experience. We remark that, in practice, computational speed was much more critically a?ected by the bounds on the Butcher array entries than by bounds on z. (3.2) ? 3.4. Optimization software. Deterministic global branch-and-bound software is particularly appropriate for this formulation since it can guarantee optimality of a

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solution to within the given tolerance, provided bounds are supplied on each of the variables. Brie?y, branch-and-bound methods solve an optimization problem by constructing and solving a related (but much easier to solve) relaxed problem over successively re?ned partitions in the feasible space. In a minimization, the objective of this relaxation bounds the objective of the original problem from below, while local optimization and other bounding heuristics on the original problem give an upper bound on the desired objective. With enough re?nement, the di?erence between the upper and lower bounds becomes su?ciently small, and the procedure terminates with the current upper bounding solution. See [28, 29] for a detailed exposition on deterministic branch-and-bound optimization methods. As explained in [28, p. 4], “convergence of this algorithm (branch-and-bound) is well established as long as the partitioning and bounding schemes obey certain properties (cf. [12]).” In this project we use BARON 5.0 from the GAMS suite of software to solve our reformulated problem. Given su?cient CPU time, BARON will provide the optimal solution within the prescribed tolerances as long as the usersupplied variable bounds (or the ones BARON infers from the problem constraints) are such that all variables and expressions are bounded [28]. If these bounds are missing, BARON reports upon termination: *** User did not provide appropriate variable bounds *** *** Globality is therefore not guaranteed *** Thus the results in this article will be either guaranteed optimal or numerically optimal. The former means that BARON is able to guarantee that the problem has been solved (globally) to within the given tolerances. Numerically optimal results are not guaranteed to be globally optimal and are instead based on extensive numerical searching. Globality will not be guaranteed if some variable bound is missing (see above) or if BARON is terminated before a guarantee of optimality can be found (e.g., due to practical limitations on CPU time). Because our SSP-based formulation involves polynomial functions which are (typically) de?ned on bounded sets, it is very naturally treated by global branchand-bound optimization software such as BARON. On the other hand, BARON cannot be directly applied to the contractivity-based formulation proposed in [3] since BARON cannot treat the nonlinear objective function appearing there (see footnote 3). We recommend the use of BARON over Matlab’s Optimization Toolbox (cf. [24]) for several reasons. Not only does BARON have the potential to guarantee optimality of the solutions found, but it often ?nds solutions with larger e?ective CFL coe?cients. Furthermore, it is straightforward to satisfy constraints to full double precision accuracy within GAMS by using the result from BARON as a starting point for a local optimization with MINOS [17]. Throughout this manuscript, all schemes are provided to the full precision of the optimization software (15 decimal digits). We conjecture that schemes satisfying the order conditions to higher precision could be designed by taking our time-stepping schemes as initial guesses for local optimizations in higher precision arithmetic. Alternatively, near-optimal schemes that satisfy the order conditions exactly may be sought among schemes with fractional coe?cients. One approach to ?nding such methods is to rewrite the corresponding class of schemes in terms of its parameters (using, for example, a symbolic computation package such as Maple). Assuming

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that this step can be achieved, the parameters may be chosen as fractions which give a scheme that closely approximates the optimal scheme, yet satis?es the order conditions exactly. See Gottlieb, Shu and Tadmor [8] for several examples using this technique. To automate the construction of GAMS models, modi?cations of Macdonald’s Maple scripts [16] for nonnegative coe?cient SSPRK schemes were used. All optimizations were carried out on a (shared) cluster of 96 dual 1.2 GHz Athlon processors. 3.5. BARON parameters. In all computations, MINOS was chosen as the NLP solver and CPLEX [2] was chosen as the linear programming (LP) solver. We terminate a run when the upper and lower bounds on the global maximum di?er by 10?10 or less. This is accomplished by setting epsa ( a ) = 10?10 and epsr ( r ) = 0. In di?cult problems, where a guarantee of optimality is not practical, a CPU time limit is also assumed. The number of probing problems (pdo) [28] had a strong e?ect on the speed of computation and on the memory requirements. Larger values led to smaller memory requirements. The e?ect on computational speed was more di?cult to predict. We have found that probing on the unsigned Butcher array variables was usually satisfactory when a guarantee of global optimality was sought. The related parameter pxdo [28] was found to have a weaker in?uence. Typically we took pxdo=pdo which corresponds to a probing strategy whereby optimization problems were solved over all the probing variables [28]. In di?cult problems where CPU time limits the computation, the default value (no probing) worked well. 4. Optimal schemes We now give new existence and optimality results in the context of e?ective CFL coe?cient. We allow for the possibility of downwind-biased operations. This contrasts with most previous work which focuses on optimizing CFL coe?cients for methods with nonnegative coe?cients. Following [22, 21], we report our schemes in optimal α ? β form in this section. Butcher arrays are easily recovered via equation (2.4). 4.1. Third-order schemes. As discussed in the introduction, SSP(3,3) and SSP(4,3) are optimal three- and four-stage third-order schemes. Moving on to methods with ?ve stages and order-three gives a numerically optimized scheme, SSP(5,3), with a CFL coe?cient of approximately 2.65. See Table 1. BARON guarantees optimality of this scheme in about 90 minutes. Considering six stages and order-three gives a numerically optimized scheme, SSP(6,3), with a CFL coe?cient of approximately 3.52. See Table 1 for the optimal α ? β form of this scheme to double precision. BARON guarantees the optimality of this scheme for the general case of unrestricted coe?cients in about eight days of CPU time. We remark that the bulk of this computational work is used to verify that there are no nonnegative coe?cient schemes that exceed the theoretical e?ciency of SSP(6,3). Moving up to seven stages, it is no longer practical to verify the optimality of the nonnegative coe?cient case directly. However, seven-stage schemes which have a time-stepping restriction that agrees with the contractivity bound R(A, b) for linear constant-coe?cient problems [14] can be found in less than two seconds during

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preprocessing (see, e.g., Table 2). Because the SSP and contractivity restrictions are equivalent [3] and the time-stepping restriction for nonlinear problems cannot exceed that for linear problems, we conclude that SSP(7,3) is an optimal sevenstage, third-order nonnegative coe?cient SSPRK scheme. Using BARON, we can verify (in 14 hours) that this scheme exceeds the theoretical e?ciency of any scheme involving downwind-biased operators. Thus, SSP(7,3) is an optimal seven-stage, third-order SSPRK scheme. This approach can also be applied to the eight-stage case to guarantee that the SSP(8,3) scheme shown in Table 2 is an optimal third-order SSPRK scheme. The primarily computational e?ort comes from verifying that schemes with downwindbiased operators are suboptimal. It takes about ?ve days to check the most computationally intensive case. In this example, Lemma 1 is particularly useful for reducing the number of cases corresponding to the signs of the coe?cients. BARON’s preprocessing step immediately (i.e., in less than one second) ?nds a feasible nonnegative coe?cient scheme with a CFL coe?cient exceeding 5.1, which implies that the CFL coe?cient of the optimal scheme must also exceed 5.1. Thus, Lemma 1 indicates that the number of nonnegative levels, , satis?es > 5.1. But the number of nonnegative levels is an integer, so must satisfy 6 ≤ ≤ 8. Over this range we 8 8 have ( 8 ) + ( 7 ) + ( 6 ) = 37 cases to check, which is a signi?cant reduction over the 8 8 full 2 = 256 cases that correspond to all possibilities. 4.2. An optimal fourth-order scheme. Unfortunately optimality is more di?cult to study for fourth-order schemes since the linear contractivity restriction does not provide a sharp bound on the time-stepping restriction for the nonlinear case. This implies that we must resort to applying global optimization software to all cases, including the nonnegative coe?cient case. It was practical to treat the class of ?ve-stage, fourth-order schemes (approximately one day was required to verify the nonnegative coe?cient case). Here, BARON guarantees that the SSP(5,4) scheme given in [15, 24] is optimal. See Table 3 for this scheme in optimal α ? β form (in double precision) or [15] for this scheme in Butcher array form (in quadruple precision). 4.3. Fifth-order schemes. We were unable to guarantee optimality of any ?fthorder schemes using our approach. Instead we carry out extensive numerical searches with a limited amount of CPU time for each job. For a large numbers of stages, it is impractical to check all 2s possibilities. However, extensive numerical searches for seven-, eight-, nine- and ten-stage schemes using one, two and three negative levels were carried out. In all cases we found that for a ?xed number of stages the maximal CFL coe?cient was a strictly increasing function of the number of nonnegative levels. This fact encouraged us to limit our searches to schemes with precisely one, two or three negative levels. Results based on this approach follow. For six stages, the best scheme that was found had a CFL coe?cient of 0.19 which is too small to be of practical use. Using seven or more stages, however, reasonable CFL coe?cients are observed. The best CFL coe?cients using 14 hours per job are reported in Table 4. We also report the fraction of total CPU time required to ?nd a feasible solution with a CFL coe?cient that is within 1% of the best found. Our formulation under BARON ?nds good solutions quickly, and we suspect that these schemes may be optimal or nearly optimal.

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Table 1. The coe?cients of the optimal SSPRK (5,3) and SSPRK (6,3) schemes.

Stages s = 5 1.000000000000000 0.000000000000000 1.000000000000000 0.355909775063327 0.000000000000000 0.644090224936674 0.367933791638137 0.000000000000000 0.000000000000000 0.632066208361863 0.000000000000000 0.000000000000000 0.237593836598569 0.000000000000000 0.762406163401431 0.377268915331368 0.000000000000000 0.377268915331368 0.000000000000000 0.000000000000000 0.242995220537396 0.000000000000000 0.000000000000000 0.000000000000000 0.238458932846290 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.287632146308408 CFL number c = 2.65062919143939 βik αik

S. J. RUUTH

Stages s = 6 1.000000000000000 0.000000000000000 1.000000000000000 0.000000000000000 0.000000000000000 1.000000000000000 0.476769811285196 0.098511733286064 0.000000000000000 0.424718455428740 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 1.000000000000000 0.000000000000000 0.000000000000000 0.155221702560091 0.000000000000000 0.000000000000000 0.844778297439909 0.284220721334261 0.000000000000000 0.284220721334261 0.000000000000000 0.000000000000000 0.284220721334261 0.000000000000000 0.000000000000000 0.000000000000000 0.120713785765930 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.284220721334261 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.240103497065900 CFL number c = 3.51839230899685 βik αik

Table 2. The coe?cients of the optimal SSPRK (7,3) and SSPRK (8,3) schemes.

Stages s = 7 1.000000000000000 0.000000000000000 0.000000000000000 0.184962588071072 0.180718656570380 0.000000000000000 0.000000000000000 0.233213863663009 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 αik 1.000000000000000 0.000000000000000 0.000000000000000 0.314831034403793 0.000000000000000 0.000000000000000 0.233213863663009 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 1.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000

0.815037411928928 0.000000000000000 0.504450309025826 0.000000000000000 0.000000000000000 1.00000000000000 0.120199000000000 0.000000000000000 0.00000000000000 0.879801000000000 βik

0.233213863663009 0.000000000000000 0.190078023865845 0.000000000000000 0.000000000000000 0.117644805593912 0.000000000000000 0.000000000000000 0.000000000000000 0.233213863663009 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.205181790464579 CFL number c = 4.28790975070412

GLOBAL OPTIMIZATION

Stages s = 8 1.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.421366967085359 0.000000000000000 0.000000000000000 0.000000000000000 0.195804015330143 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 αik 1.000000000000000 0.000000000000000 0.000000000000000 0.005949401107575 0.004254010666365 0.000000000000000 0.000000000000000 0.195804015330143 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 1.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.104380143093325 0.000000000000000

1.000000000000000 0.000000000000000 0.000000000000000 0.243265240906726 0.000000000000000

0.572683631807067 0.000000000000000 0.995745989333635 0.000000000000000 0.000000000000000 0.652354615999950 0.000000000000000 0.000000000000000 0.000000000000000 1.000000000000000 βik

0.195804015330143 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000

0.195804015330143 0.000000000000000 0.000000000000000 0.000000000000000 0.000000000000000 CFL number c =

0.112133754621673 0.000000000000000 0.194971062960412 0.000000000000000 0.000000000000000 0.127733653231944 0.000000000000000 0.000000000000000 0.000000000000000 0.195804015330143 5.10714756443533

197

198

Table 3. The coe?cients of the optimal SSPRK (5,4) scheme.

Stages 1 0.444370493651235 0.555629506348765 αik 0.620101851488403 0.178079954393132 0 0.391752226571890 0 βik 0 0 0 CFL coe?cient 0.368410593050371 0 0 0 0.251891774271694 0 0 1.50818004918983 0.544974750228521 0.063692468666290 0.226007483236906 0 0 0 0.379898148511597 0 0.821920045606868 5

0.517231671970585 0.096059710526147 0.386708617503269

S. J. RUUTH

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Table 4. The CFL coe?cients of some numerically optimal ?fthorder SSPRK schemes. We also report the fraction of total CPU time required to ?nd a feasible solution with a CFL coe?cient that is within 1% of the best found. stages 7 8 9 10 CFL coe?cient 1.1785083484719 1.8756849616413 2.6957882892949 3.3953368327742 Fraction of CPU time 0.076 0.054 0.00028 0.00052

Table 5. Coe?cients of SSP(10,5). CFL coe?cient is 3.39533683277420.

(i, k) (1,0) (2,0) (2,1) (3,0) (3,1) (3,2) (4,0) (4,1) (4,2) (4,3) (5,0) (5,1) (5,2) (5,3) (5,4) (6,0) (6,1) (6,2) (6,3) (6,4) (6,5) (7,0) (7,1) (7,2) (7,3) (7,4) (7,5) (7,6) αik 1 0.258168167463650 0.741831832536350 0 0.037493531856076 0.962506468143924 0.595955269449077 0 0.404044730550923 0 0.331848124368345 0 0 0.008466192609453 0.659685683022202 0.086976414344414 0 0 0 0 0.913023585655586 0.075863700003186 0 0.267513039663395 0 0 0 0.656623260333419 βik 0.173586107937995 0 0.218485490268790 0 0.011042654588541 0.283478934653295 0 0 0.118999896166647 0 0.025030881091201 0 0 -0.002493476502164 0.194291675763785 0 0 0 0 0 0.268905157462563 0 0 0.066115378914543 0 0 0 0.193389726166555 (i, k) (8,0) (8,1) (8,2) (8,3) (8,4) (8,5) (8,6) (8,7) (9,0) (9,1) (9,2) (9,3) (9,4) (9,5) (9,6) (9,7) (9,8) (10,0) (10,1) (10,2) (10,3) (10,4) (10,5) (10,6) (10,7) (10,8) (10,9) αik 0.005212058095597 0 0 0.407430107306541 0 0 0 0.587357834597862 0.122832051947995 0 0 0 0 0 0 0 0.877167948052005 0.075346276482673 0.000425904246091 0 0 0 0.064038648145995 0.354077936287492 0 0 0.506111234837749 βik 0 0 0 -0.119996962708895 0 0 0 0.172989562899406 0.000000000000035 0 0 0 0 0 0 0 0.258344898092277 0.016982542367506 0 0 0 0 0.018860764424857 0.098896719553054 0 0 0.149060685217562

The seven-, eight- and nine-stage schemes obtained using this approach are reported in Ruuth and Spiteri [20]. The ten-stage scheme is reported in Table 5. Note that the e?ective CFL coe?cient of this scheme is 0.3395, exceeding that of the popular SSPRK scheme SSP(3,3). We consider this to be a possible reason for further study of downwind-biased spatial discretizations for hyperbolic conservation laws. As we shall see in the next section, another possible reason to use these schemes arises when storage considerations are paramount. 5. Low-storage schemes There are computational problems for which memory management considerations are at least as important as stability considerations when choosing a numerical time discretization method, e.g., direct numerical simulation of Navier-Stokes equations requiring high spatial resolution in three dimensions. In such cases, sstage explicit Runge-Kutta methods that use less than the usual storage are very desirable (see, e.g., [32]).

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It is commonly assumed that two-register low-storage schemes cannot utilize downwind-biased operators without destroying the low-storage property. See, e.g., [8]. However, if we assume that the β’s at a level are the same sign, then the low? storage property is preserved since either F (U (j) ) or F (U (j) ) appears, but not both. As we shall see, downwind-biased operators provide a means to obtain improved theoretical e?ciency in certain low-storage SSPRK schemes. 5.1. Williamson schemes. We begin our discussion with SSPRK schemes of Williamson type [32] that require two units of storage per step,4 although more general methods requiring more storage per step are possible. These schemes take the form (5.1a) (5.1b) dU (i) U (i) = = Ai dU (i?1) + ?tF (U (i?1) ), U (i?1) + Bi dU (i) , i = 1, 2, . . . , s,

where U (0) = U n , U n+1 = U (s) , and A1 ≡ 0. Again, we note that there is a relation between the coe?cients Ai , Bi and the coe?cients αik , βik or equivalently the usual quantities in the Butcher array. We denote the numerically optimal sstage, order-p low-storage SSPRK scheme by Williamson(s,p), and remark that Bi = ai+1,i , 1 ≤ i ≤ s ? 1, and Bs = bs in the usual Butcher notation, so that the Bi are bounded according to Lemma 2. Unfortunately, an analytical bound on the Ai is not known for general schemes of this form. For this reason, we do not expect BARON to be able to guarantee optimality except perhaps in specialized cases where it can determine bounds on the Ai based on the constraints of the problem (see [28, 29] for further details on the construction of bounds based on problem constraints). Nonetheless, improvements to the schemes reported in [24] are possible via our approach. Table 6. The coe?cients of the ?rst few numerically optimal Williamson low-storage schemes of order three: Williamson(3,3), Williamson(4,3) and Williamson(5,3). BARON guarantees optimality in the three-stage case.

Stages 3 Ai 0 -2.915493957701923 0 0 -0.449336503268844 0 -4.661555711601366 0 -4.344339134485095 0 -3.770024161386381 -0.046347284573284 Bi 0.924574112262461 0.287712943868770 0.626538293270800 1.086620745813428 0.854115548251602 -1.576604558206099 -0.278475500113052 0.713497331193829 0.133505249805329 0.713497331193829 0.149579395628565 0.384471116121269 CFL coe?cient 0.322349301195940

4

0.634274456962008

5

1.40154693827206

4 Note that if some form of error control is required, then additional storage for the current solution vector is also needed [13].

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First, we remark that in the ?rst-order case SSP(s,1) is the optimal scheme since this is the provably optimal ?rst-order, s-stage scheme and it may be written in low-storage form with two registers. Moving on to second order, it is clear that a traditional implementation of any two-stage scheme must be low-storage in the sense we are considering, so the optimal low-storage method with s = p = 2 corresponds to SSP(2,2) [24]. The SSP(2,2) scheme has an e?ective CFL coe?cient of 1/2. Extensive searching with more stages (s = 3, 4, 5, 6) did not ?nd any schemes with improved e?ective CFL coe?cients, and we conjecture that an optimal scheme with an even number of stages is just SSP(2,2) repeated (s/2) times: dU (i?1) U (i?1) dU (i) U (i) = = = = ?tF (U (i?2) ), U (i?2) + (2/s) dU (i?1) , ?dU (i?1) + ?tF (U (i?1) ), U (i?1) + (1/s) dU (i) , i = 2, 4, 6, . . . , s,

where U (0) = U n , U n+1 = U (s) . We further remark that all our numerical optimizations for low-storage schemes with an odd number of stages lead to schemes with e?ective CFL coe?cients that were strictly less than 1/2. Thus, we conjecture that low-storage schemes of this type and with an odd number of stages have smaller e?ective CFL coe?cients than the simple SSP(2,2) scheme. The results for the coe?cients Ai , Bi are given in Table 6 for up to ?ve stages and order three. We note that the optimal three-stage, order-three, low-storage method reported in Table 6 agrees with that reported in [7]. In this particular case, BARON is able to use the constraints of the problem to automatically construct bounds on the Ai , and subsequently guarantee the optimality of the scheme. In the more complicated four- and ?ve-stage cases, the software was unable to determine such bounds, and a guarantee of optimality was not obtained. Using four stages, the third-order numerically optimal scheme involved two downwind-biased operators per step and had a CFL coe?cient of about 0.634. This represents a 20% improvement over the nonnegative coe?cient scheme reported in [24]. For completeness, Table 7 supplies the coe?cients of that nonnegative coe?cient scheme to 15 digits. Similar to [24], we ?nd that the optimal ?ve-stage scheme does not require downwind-biased operators. However, here we ?nd a numerically optimized scheme with a 40% larger CFL coe?cient. See Table 6.

Table 7. The coe?cients of the four-stage Williamson low-storage scheme of order three having nonnegative coe?cients. This result comes from extensive searching. Stages 4 Ai 0 -4.946517279341980 0 -0.151274934922161 Bi CFL coe?cient 1.032161930751755 0.528418106518184 0.187941555751458 0.152152605134959 0.656749852605931

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5.2. Van der Houwen and Wray schemes. Van der Houwen and Wray devised another class of low-storage schemes which alternate information between two available storage registers at each successive stage [30, 31, 33]. See Kennedy, Carpenter and Lewis [13] for details. Starting with X (j) and F (U (j) ) stored in Registers 1 and 2 respectively, two intermediate stages would proceed according to (5.3a) (5.3b) (5.3c) (5.3d) (Register 1) U (i+1) (Register 2) X (i+1) (Register 2) U (i+2) (Register 1) X (i+2) = X (i) + ai+1,i ?tF (U (i) ), = U (i+1) + (bi ? ai+1,i )?tF (U (i) ), = X (i+1) + ai+2,i+1 ?tF (U (i+1) ), = U (i+2) + (bi+1 ? ai+2,i+1 )?tF (U (i+1) ),

where aij and bj are the usual Butcher array entries [13]. By overwriting, the three vectors U , F , and X never fully coexist [13]. In particular, during the function evaluation, the previous solution vector is overwritten. While this will not be acceptable in all situations, compressible Navier-Stokes equations provide a situation where this may be pro?tably utilized [13]. Full details on implementing these schemes are given in the comprehensive article of Kennedy, Carpenter and Lewis [13]. To distinguish these methods from Williamson schemes, we shall refer to them as vdH schemes (cf. [13]). Note that these schemes are easily generalized to accommodate more than two registers of storage. We shall consider the cases where two or three registers of storage are available, and for simplicity refer to the numerically optimal r-register, s-stage, pth-order scheme as vdHr(s,p), and the corresponding numerically optimal nonnegative coe?cient scheme as vdHr+(s,p).

Table 8. The coe?cients of the ?rst few numerically optimal vdH low-storage schemes of order three with two registers of storage: vdH2(3,3), vdH2(4,3) and vdH2(5,3). BARON guarantees optimality in all cases.

Stages 3 ai+1,i 0.755726313669390 0.386954492646558 4 0.410502506371045 0.508294264771036 0.309067503393721 5 0.674381436593749 0.116638367147961 0.674381436593749 0.162995387938952 bi 0.245170292105110 0.184896041116058 0.569933666778832 0.222722477423144 0.167687843505189 0.151218171982708 0.458371507088958 0.174481959220521 0.116638367147961 0.162995387938952 0.106256369067643 0.439627916624922 1.482840341885634 1.067414323404809 CFL coe?cient 0.838384821388215

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5.2.1. Two registers of storage. A two-register, s-stage vdH Runge-Kutta scheme is speci?ed by the Butcher array entries ai+1,i , 1 ≤ i ≤ s ? 1, and bi , 1 ≤ i ≤ s. The remaining aij entries are de?ned according to aij = bj , so the Butcher array takes the particularly simple form [13] 0 c2 c3 . . . a21 b1 . . .

a32 b2 . . . a43 b3 . . . b3 a54 .. . ··· ..

cs b1 b2

. bs?2 bs?2

aa,s?1 bs?1

bs

Because each of the entries is bounded according to Lemma 2, we may directly apply the algorithms of Section 3 to derive optimal schemes. Similar to the case of Williamson low-storage schemes, we know that SSP(s,1) is the optimal ?rst-order, s-stage vdH SSPRK scheme and that SSP(2,2) is the optimal second-order, two-stage vdH SSPRK scheme. For order two, BARON guarantees that three- and ?ve-stage schemes both have e?ective CFL coe?cients that are less than 0.5, and we conjecture that any scheme with an odd number of stages will have a smaller e?ective CFL coe?cient than SSP(2,2). For four- and six-stage schemes and order-two BARON guarantees that an optimal scheme is just SSP(2,2) repeated (s/2) times. We conjecture that this also holds for any even number of stages. The results for up to ?ve stages and order three are given in Table 8. The three-stage scheme is guaranteed optimal by BARON and agrees to single precision with the RK3(2)3[2R+]N contractive scheme reported in [13]. We also report new four- and ?ve-stage stage schemes which are also guaranteed to be optimal. It is interesting to note that in each case these schemes have improved CFL coe?cients over the corresponding Williamson schemes. 5.2.2. Three registers of storage. A three-register, s-stage vdH SSPRK scheme is speci?ed by the Butcher array entries ai+1,i , 1 ≤ i ≤ s ? 1, ai+2,i , 1 ≤ i ≤ s ? 2, and bi , 1 ≤ i ≤ s. Similar to the two-register case, the remaining aij entries are simply Table 9. The coe?cients of the optimal ?ve-stage vdH lowstorage scheme of order three with three registers of storage: vdH3(5,3). BARON guarantees optimality.

Stages 5 ai+1,i 0.390109487215461 0.390109487215461 0.226019654979362 0.207634341633736 ai+2,i 0.390109487215461 0.226019654979362 0.120298131155176 bi 0.226019654979362 0.120298131155176 0.109515534018255 0.189023599836438 0.355143080010770 CFL coe?cient 2.56338292907932

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de?ned according to aij = bj , and the Butcher array takes the form [13] 0 c2 c3 . . . a21 a31 b1 . . .

a32 a42 b2 . . . b2 a43 a53 .. . ··· a54 .. . bs?3 bs?3 ..

cs b1

. as,s?2 bs?2

as,s?1 bs?1

bs

which allows us to apply the algorithms of Section 3 to derive optimal schemes. It is clear that the optimal SSP(3,3) and SSP(4,3) schemes ?t within this class of schemes, so they must be optimal three-register vdH SSPRK schemes [13]. Moving on to ?ve stages, we ?nd a new ?ve-stage scheme which BARON guarantees to be optimal (see Table 9). It is noteworthy that the CFL coe?cient for this scheme is only about 3% smaller than the optimal SSP(5,3) scheme. Moving up to fourth order and ?ve stages proved more computationally intensive. Nonetheless, BARON was able to guarantee that the optimal scheme (see Table 10) involved one downwind-biased operator and had a CFL coe?cient of 0.935. It took 12 days of CPU time for BARON to verify that schemes involving only nonnegative coe?cients must have smaller CFL coe?cients. The best nonnegative coe?cient scheme (see Table 11) had a CFL coe?cient of 0.531, an 11% improvement over the scheme found in [13]. The optimality of this scheme was guaranteed by BARON in 21 days of CPU time. Table 10. The coe?cients of the optimal ?ve-stage vdH lowstorage scheme of order four with three registers of storage: vdH3(5,4). BARON guarantees optimality.

Stages 5 ai+1,i 0.537734210467782 0.000000000000000 -0.472823582086688 0.796906902183600 ai+2,i 0.000000000000000 0.596326927126672 -0.545978886296898 bi 0.299925395513371 0.449146588599927 -0.137488645434953 0.225363552896202 0.163053108425452 CFL coe?cient 0.935322006941531

Table 11. The coe?cients of the optimal ?ve-stage vdH lowstorage scheme of order four with three registers of storage and nonnegative coe?cients: vdH3+(5,4). BARON guarantees optimality.

Stages 5 ai+1,i 0.216747619157064 0.513374951629630 0.415710952208246 0.366498283222966 ai+2,i 0.059060301553258 0.113274666138869 0.080866763251240 bi 0.049733200550301 0.370241294854764 0.051983506105255 0.235595750064777 0.292446248424903 CFL coe?cient 0.530770344137093

GLOBAL OPTIMIZATION

205

6. Conclusions We have studied explicit high-order strong-stability-preserving Runge-Kutta methods with downwind-biased spatial discretizations. We have developed a practical approach to guarantee the optimality of a variety of schemes of up to order four and have applied the technique to guarantee the optimality of SSP(5,3), SSP(5,4) as well as several new third-order schemes: SSP(6,3), SSP(7,3) and SSP(8,3). We also ?nd an e?cient ?fth-order scheme that has an e?ective CFL coe?cient that exceeds the popular SSP(3,3) scheme. Several new results for low-storage schemes were found. Our approach guarantees the optimality of the known Williamson(3,3) and vdH2(3,3) schemes. It also guarantees the optimality of the third-order low-storage schemes vdH2(4,3) and vdH2(5,3). Interestingly, we found that in two instances, Williamson(4,3) and vdH3(5,4), signi?cantly improved schemes arise when downwind-biased operators are utilized, and we ?nd and guarantee the optimality of the corresponding schemes. Our approach also derives two nonnegative coe?cient schemes, vdH3+(5,4) and Williamson(5,3), which are more e?cient than the corresponding schemes that were previously proposed in the literature. These results demonstrate that global branchand-bound software may be applied to our mathematical formulation to achieve a practical, constructive way of guaranteeing the optimality of SSPRK schemes. This manuscript has considered the optimization of explicit SSPRK schemes. However, we expect that general implicit schemes, e.g., (6.1a) (6.1b) (6.1c) (6.1d) U n+1 U (0) U

(i)

= U n,

s

=

k=0

αik U (k) + ?t max(βik , 0)F (U (k) ) ? ?t min(βik , 0)F (U (k) ) , i = 1, 2, . . . , s,

+

= U (s) ,

could also be treated using the techniques outlined in this paper. To proceed, assume Euler’s method applied forward in time combined with the spatial discretization F (·) is strongly stable under the CFL restriction ?t ≤ ?tF E . Also assume Euler’s method applied backward in time combined with the spatial dis? cretization F (·) is strongly stable under the same CFL restriction ?t ≤ ?tF E . Then the implicit (or backward) Euler method applied to these problems is unconditionally stable [11], and it easily seen that the method (6.1) will be SSP provided ?t ≤ C?tF E , where C is the CFL coe?cient C ≡ min {cik : 1 ≤ i ≤ s, 0 ≤ k ≤ s} , where cik = ∞

αik |βik |

if βik = 0 and i = k, otherwise.

The development of appropriate optimization techniques then follows in a similar manner to the techniques described in this paper. See also [3] for relevant discussions on the underlying theory for general implicit SSPRK schemes with nonnegative coe?cients. Nonlinearly stable explicit multistep methods with guaranteed optimal CFL coe?cients have been derived using the ideas described in this article [18], and we believe the derivation of optimal general linear methods (cf. [21, 8]) is another

206

S. J. RUUTH

natural application area for these techniques. Ultimately, one might consider developing techniques to guarantee optimality over a combination of properties such as error constant, nonlinear stability and linear stability (cf. [13, 16]). On the other hand, our approach is not suitable to show the nonexistence of a scheme, since as the CFL coe?cient tends to zero, we no longer enjoy the property that our intervals are bounded. Similarly, certain schemes such as the Williamson class of low-storage schemes include parameters which are not obviously bounded. This typically prohibits the use of our approach to guarantee optimality. Nonetheless, as software and hardware advances take place and improved bounds on variables are derived, we expect that the utility of the techniques outlined in this paper for optimal explicit SSPRK schemes will experience a correspondingly rapid growth in applicability and importance. Acknowledgments The author thanks Colin Macdonald, Nick Sahinidis, and Luis Vicente for helpful discussions related to this project. References

1. A. Brooke, D. Kendrick, A. Meeraus, and R. Raman, Gams-a users guide, GAMS Development Corporation, Washington, 1998. 2. Gams-the solver manuals, GAMS Development Corporation, Washington, 2001. 3. L. Ferracina and M. N. Spijker, An extension and analysis of the Shu-Osher representation of Runge-Kutta methods, Math. Comp. 74 (2005), 201–219. MR2085408 , Stepsize restrictions for the total-variation-diminishing property in general Runge4. Kutta methods, SIAM J. Numer. Anal. 42 (2004), no. 3, 1073–1093. MR2113676 5. S. Gottlieb and L. J. Gottlieb, Strong stability preserving properties of Runge-Kutta time discretization methods for linear constant coe?cient operators, J. Scienti?c Computation 18 (2003), no. 1, 83–109. MR1958936 (2003m:65161) 6. P. E. Gill, W. Murray, and M. H. Wright, Practical optimization, Academic Press, San Diego, 1981. MR0634376 (83d:65195) 7. S. Gottlieb and C.-W. Shu, Total variation diminishing Runge-Kutta schemes, Math. Comput. 67 (1998), no. 221, 73–85. MR1443118 (98c:65122) 8. S. Gottlieb, C.-W. Shu, and E. Tadmor, Strong-stability-preserving high-order time discretization methods, SIAM Review 43 (2001), no. 1, 89–112. MR1854647 (2002f:65132) 9. I. Higueras, Representations of Runge-Kutta methods and strong stability preserving methods, Technical Report No. 2, Departamento de Matematica e Informatica, Universidad Publica de Navarra, 2003. , On strong stability preserving time discretization methods, J. Scienti?c Computation 10. 21 (2004), no. 2, 193–223. MR2069949 (2005d:65112) 11. W. Hundsdorfer, S. J. Ruuth, and R. J. Spiteri, Monotonicity-preserving linear multistep methods, SIAM J. Numer. Anal. 41 (2003), no. 2, 605–623. MR2004190 (2004g:65093) 12. R. Horst and H. Tuy, Global optimization: Deterministic approaches, third ed., Springer Verlag, Berlin, 1996. MR1274246 (94m:90004) 13. C. A. Kennedy, M. H. Carpenter, and R. M. Lewis, Low-storage, explicit Runge-Kutta schemes for the compressible Navier-Stokes equations, Appl. Numer. Math. 35 (2000), no. 3, 177–219. MR1793508 (2001k:65111) 14. J. F. B. M. Kraaijevanger, Absolute monotonicity of polynomials occurring in the numerical solution of initial value problems, Numer. Math. 48 (1986), 303–322. MR0826471 (87c:65084) 15. J.F.B.M. Kraaijevanger, Contractivity of Runge-Kutta methods, BIT 31 (1991), 482–528. MR1127488 (92i:65120) 16. C. B. Macdonald, High-order embedded Runge-Kutta pairs for the time evolution of hyperbolic conservation laws, Master’s thesis, Simon Fraser University, Burnaby, BC, Canada, 2003. 17. B. A. Murtagh and M. A. Saunders, MINOS 5.1 User’s Guide, Report SOL 83-20R, Department of Operations Research, Stanford University, 1987.

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18. S. J. Ruuth and W. Hundsdorfer, High-order linear multistep methods with general monotonicity and boundedness properties, J. Comput. Phys. 209 (2005), no. 1, 226–248. MR2145787 19. S. J. Ruuth and R. J. Spiteri, Two barriers on strong-stability-preserving time discretization methods, J. Scienti?c Computation 17 (2002), no. 4, 211–220. MR1910562 20. Steven J. Ruuth and Raymond J. Spiteri, High-order strong-stability-preserving Runge–Kutta methods with downwind-biased spatial discretizations, SIAM J. Numer. Anal. 42 (2004), no. 3, 974–996. MR2112790 21. Chi-Wang Shu, Total-variation-diminishing time discretizations, SIAM J. Sci. Statist. Comput. 9 (1988), no. 6, 1073–1084. MR0963855 (90a:65196) 22. Chi-Wang Shu and Stanley Osher, E?cient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988), no. 2, 439–471. MR0954915 (89g:65113) 23. M. N. Spijker, Contractivity in the numerical solution of initial value problems, Numer. Math. 42 (1983), 271–290. MR0723625 (85b:65067) 24. Raymond J. Spiteri and Steven J. Ruuth, A new class of optimal high-order strongstability-preserving time-stepping schemes, SIAM J. Numer. Anal. 40 (2002), no. 2, 469–491. MR1921666 (2003g:65083) , Nonlinear evolution using optimal fourth-order strong-stability-preserving Runge25. Kutta methods, Mathematics and Computers in Simulation 62 (2003), nos. 1–2, 125–135, Special issue on “Nonlinear Waves: Computation and Theory II”. MR1983581 26. N. V. Sahinidis and M. Tawarmalani, GAMS The Solver Manuals, GAMS Development Corporation, Washington, 2004, pp. 9–20. 27. M. Tawarmalani, S. Ahmed, and N. V. Sahinidis, Product disaggregation in global optimization and relaxations of rational programs, Optimization and Engineering 3 (2002), 281–303. MR1955959 (2003k:90051) 28. M. Tawarmalani and N. V. Sahinidis, Convexi?cation and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications, Nonconvex Optimization and Its Applications, vol. 65, Kluwer Academic Publishers, Dordrecht, 2002. MR1961018 (2004a:90001) 29. M. Tawarmalani and N. V. Sahinidis, Global optimization of mixed-integer nonlinear programs: A theoretical and computational study, Mathematical Programming 99 (2004), 563– 591. MR2051712 (2004m:90096) 30. P.J. van der Houwen, Explicit Runge-Kutta formulas with increased stability boundaries, Numer. Math. 20 (1972), no. 2, 149–164. MR0317547 (47:6094) , Construction of integration formulas for initial value problems, North-Holland, Am31. sterdam, 1977. MR0519726 (58:24960) 32. J. H. Williamson, Low-storage Runge-Kutta schemes, J. Comput. Phys. 35 (1980), no. 1, 48–56. MR81a:65070 MR0566473 (81a:65070) 33. A.A. Wray, Minimal storage time advancement schemes for spectral methods, Tech. report, NASA Ames Research Center, Mo?ett Field, CA, 1986. Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 E-mail address: sruuth@sfu.ca

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