Arbitrary high-order schemes for the linear advection and wave equations: application to hydrodynamics and aeroacoustics

Pino, Stéphane Del; Jourdren, Hervé

doi:10.1016/j.crma.2006.01.013

Cited by 21 publications

(4 citation statements)

References 6 publications

(16 reference statements)

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“…A proof that all odd order schemes constructed in [2,11,12] satisfy the technical p = q = 2k + 1 of Theorem 5 is given in [3]. In the rest of this work, we study general conditions such that A-stability in L 1 is true and we use them to analyze the numerical results obtained with the O3 scheme.…”

Section: A-stability In Lmentioning

confidence: 99%

“…This estimate is sharper than (2). Using very high order schemes as in advocated in [2,11,12] means p is large. In this case p p+1 is very close to 1.…”

Section: Theorem 2 Consider the Scheme Defined By (1) And The Mean Inmentioning

confidence: 99%

“…Since we are interested in the numerical analysis of high order and very high order schemes for transport schemes that have been introduced in the recent works [2,11,12], we shall need general notations. We begin with some standard Fourier considerations.…”

Section: General Schemesmentioning

confidence: 99%

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Finite volume transport schemes

2007

Numer. Math.

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We analyze arbitrary order linear finite volume transport schemes and show asymptotic stability in L 1 and L ∞ for odd order schemes in dimension one. It gives sharp fractional order estimates of convergence for BV solutions. It shows odd order finite volume advection schemes are better than even order finite volume schemes. Therefore the Gibbs phenomena is controlled for odd order finite volume schemes. Numerical experiments sustain the theoretical analysis. In particular the oscillations of the Lax-Wendroff scheme for small Courant numbers are correlated with its non stability in L 1 . A scheme of order three is proved to be stable in L 1 and L ∞ .

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Section: A-stability In Lmentioning

confidence: 99%

“…This estimate is sharper than (2). Using very high order schemes as in advocated in [2,11,12] means p is large. In this case p p+1 is very close to 1.…”

Section: Theorem 2 Consider the Scheme Defined By (1) And The Mean Inmentioning

confidence: 99%

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Finite volume transport schemes

2007

Numer. Math.

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“…It is a strong improvement with respect to the literature [6,1,10]. This new error estimate is based on new estimates for the advection equation on a regular periodic grid for which we use sharp properties of semi-lagrangian schemes also equal to shifted Strang's stencil [14,8,9], see also [2,3,4,5]. We will make use of the connection of these numerical schemes with B-Splines techniques [11,13].…”

Section: Introductionmentioning

confidence: 94%

Enhanced Convergence Estimates for Semi-Lagrangian Schemes Application to the Vlasov--Poisson Equation

Charles¹,

Després²,

Mehrenberger³

2013

SIAM J. Numer. Anal.

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We prove enhanced error estimates for high order semi-lagrangian discretizations of the Vlasov-Poisson equation. It provides new insights into optimal numerical strategies for the numerical solution of this problem. The new error estimate O min ∆x ∆t , 1 ∆x p + ∆t 2 is based on advanced error estimates for semi-lagrangian schemes, also equal to shifted Strang's schemes, for the discretization of the advection equation.

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Higher Order Time Stepping for Second Order Hyperbolic Problems and Optimal CFL Conditions

Gilbert

Joly

2008

Partial Differential Equations

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Arbitrary high-order schemes for the linear advection and wave equations: application to hydrodynamics and aeroacoustics

Cited by 21 publications

References 6 publications

Finite volume transport schemes

Finite volume transport schemes

Enhanced Convergence Estimates for Semi-Lagrangian Schemes Application to the Vlasov--Poisson Equation

Higher Order Time Stepping for Second Order Hyperbolic Problems and Optimal CFL Conditions

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