Strong stability preserving hybrid methods

Huang, Chengming

doi:10.1016/j.apnum.2008.03.030

Cited by 24 publications

(33 citation statements)

References 30 publications

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“…They can be The second-order methods are of particular interest because they are also optimal second-order SSP methods in their respective classes. For instance, the optimal second-order two-stage methods have been proposed as SSP methods in [12], while the optimal second-order two-step methods have been proposed in [15]. The present results imply that these methods, which in some cases were obtained by nonlinear optimization, are indeed optimal, even among much larger classes of methods than those considered in [12,15].…”

Section: The Feasibility Problemmentioning

confidence: 59%

“…Take any (ψ 1 , ..., ψ k ) ∈ Π s,k,p , and let R be the threshold factor of this method. Writing out explicitly the first two order conditions (i.e., (12) for p = 0, 1) gives…”

Section: Lemma 1 a Polynomial ψ(Z) Is Absolutely Monotonic At Z = −ξmentioning

confidence: 99%

“…For instance, the optimal second-order two-stage methods have been proposed as SSP methods in [12], while the optimal second-order two-step methods have been proposed in [15]. The present results imply that these methods, which in some cases were obtained by nonlinear optimization, are indeed optimal, even among much larger classes of methods than those considered in [12,15]. These results also imply the optimality of the thirdorder SSP general linear methods with up to three stages or three steps in [12,15].…”

Section: The Feasibility Problemmentioning

confidence: 99%

“…The monotonicity property is guaranteed under the maximum timestep ∆t max = c∆t FE , where the SSP coefficient c depends only on the numerical integration method (not on F ). Considerable research effort has been devoted to finding methods with the largest value of c among various classes (see, e.g., [7,14,12]). …”

Section: Introductionmentioning

confidence: 99%

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Computation of optimal monotonicity preserving general linear methods

Ketcheson

2009

Math. Comp.

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Abstract. Monotonicity preserving numerical methods for ordinary differential equations prevent the growth of propagated errors and preserve convex boundedness properties of the solution. We formulate the problem of finding optimal monotonicity preserving general linear methods for linear autonomous equations, and propose an efficient algorithm for its solution. This algorithm reliably finds optimal methods even among classes involving very high order accuracy and that use many steps and/or stages. The optimality of some recently proposed methods is verified, and many more efficient methods are found. We use similar algorithms to find optimal strong stability preserving linear multistep methods of both explicit and implicit type, including methods for hyperbolic PDEs that use downwind-biased operators.

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Section: The Feasibility Problemmentioning

confidence: 59%

“…Take any (ψ 1 , ..., ψ k ) ∈ Π s,k,p , and let R be the threshold factor of this method. Writing out explicitly the first two order conditions (i.e., (12) for p = 0, 1) gives…”

Section: Lemma 1 a Polynomial ψ(Z) Is Absolutely Monotonic At Z = −ξmentioning

confidence: 99%

Section: The Feasibility Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Computation of optimal monotonicity preserving general linear methods

Ketcheson

2009

Math. Comp.

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“…In our construction of HBO(d, k, p), we replace the forward Euler (FE) method, y n+1 = y n + ∆t f (t n , y n ), (1.2) used by Gottlieb et al and Huang [7,11] in establishing strong stability preserving (SSP) Runge-Kutta (RK) methods as convex combinations of FE methods, by rewriting HBO(d, k, p) as a convex combination of the special d-derivative extension of FE which we denote by S(d) series and which has the form:…”

Section: Introductionmentioning

confidence: 99%

On contractivity preserving 4- to 7-step predictor-corrector HBO series for ODEs

Nguyen‐Ba¹,

Giordano²,

Nguyen-Thu³

et al. 2017

J. of Mod. Meth. in Numer. Math.

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The contractivity-preserving 2-and 3-step predictor-corrector series methods for ODEs (T. Nguyen-Ba, A. Alzahrani, T. Giordano and R. Vaillancourt, On contractivity-preserving 2-and 3-step predictor-corrector series for ODEs, J. Mod. Methods Numer. Math. 8:1-2 (2017), pp. 17-39. doi:10.20454/jmmnm.2017.1130) are expanded into new optimal, contractivity-preserving (CP), d-derivative, k-step, predictor-corrector, Hermite-Birkhoff-Obrechkoff series methods, denoted by HBO(d, k, p), k = 4, 5, 6, 7, with nonnegative coefficients for solving nonstiff first-order initial value problems y = f (t, y), y(t 0 ) = y 0 . The main reason for considering this class of formulae is to obtain a set of methods which have larger regions of stability and generally higher upper bound p u of order p of HBO(d, k, p) for a given d. Their stability regions have generally a good shape and grow generally with decreasing p − d. A selected CP HBO method: 6-derivative 4-step HBO of order 14, denoted by HBO(6,4,14) which has maximum order 14 based on the CP conditions compares satisfactorily with Adams-Cowell of order 13 in PECE mode, denoted by AC(13), in solving standard N-body problems over an interval of 1000 periods on the basis of the relative error of energy as a function of the CPU time. HBO(6,4,14) also compares well with AC(13) in solving standard N-body problems on the basis of the growth of relative positional error, relative energy error and 10000 periods of integration. The coefficients of HBO(6,4,14) are listed in the appendix.

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Strong Stability Preserving Time Discretizations: A Review

Gottlieb

2015

Lecture Notes in Computational Science and Engineering

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Strong stability preserving hybrid methods

Cited by 24 publications

References 30 publications

Computation of optimal monotonicity preserving general linear methods

Computation of optimal monotonicity preserving general linear methods

On contractivity preserving 4- to 7-step predictor-corrector HBO series for ODEs

Strong Stability Preserving Time Discretizations: A Review

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