Strict Stability of High-Order Compact Implicit Finite-Difference Schemes: The Role of Boundary Conditions for Hyperbolic PDEs, I

Abarbanel, Saul; Chertock, Alina; Yefet, Amir

doi:10.1006/jcph.2000.6420

Cited by 64 publications

(59 citation statements)

References 9 publications

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“…With the recently derived interface treatment, it can be shown that the scheme is strongly stable and mimics the continuous estimate in (15). This was previously done in [13] and we do not repeat those derivations here.…”

Section: Stabilitymentioning

confidence: 75%

“…In Figure 3, a schematic of the mesh is shown to clarify the indexing used in (16). The first derivative with respect to the ξ direction is approximated by D ξ u, where D ξ is a so-called SBP operator decomposed as…”

Section: The Discrete Problemmentioning

confidence: 99%

“…Together with a diagonal norm P, the boundary closure is s-order accurate, making a stable first order approximation of a hyperbolic problem s + 1 order accurate globally [4,9]. For more details on non-standard SBP operators see [15,16,17,18].…”

Section: The Discrete Problemmentioning

confidence: 99%

“…where x, y are the x and y coordinates of the left mesh in the Cartesian coordinate system, arranged as a vector consistent to (16). We also use…”

Section: The Discrete Problemmentioning

confidence: 99%

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A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces

Nikkar

Nordstrm

2017

Journal of Computational Physics

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We introduce an interface/coupling procedure for hyperbolic problems posed on time-dependent curved multi-domains. First, we transform the problem from Cartesian to boundary-conforming curvilinear coordinates and apply the energy method to derive well-posed and conservative interface conditions. Next, we discretize the problem in space and time by employing finite difference operators that satisfy a summation-by-parts rule. The interface condition is imposed weakly using a penalty formulation. We show how to formulate the penalty operators such that the coupling procedure is automatically adjusted to the movements and deformations of the interface, while both stability and conservation conditions are respected.The developed techniques are illustrated by performing numerical experiments on the linearized Euler equations and the Maxwell equations. The results corroborate the stability and accuracy of the fully discrete approximations.

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Section: Stabilitymentioning

confidence: 75%

Section: The Discrete Problemmentioning

confidence: 99%

Section: The Discrete Problemmentioning

confidence: 99%

“…where x, y are the x and y coordinates of the left mesh in the Cartesian coordinate system, arranged as a vector consistent to (16). We also use…”

Section: The Discrete Problemmentioning

confidence: 99%

See 2 more Smart Citations

A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces

Nikkar

Nordstrm

2017

Journal of Computational Physics

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“…The deviation should not be interpreted as a drawback, on the contrary it can serve as an error estimate for the solution in the interior. However, the most important advantage of weak boundary treatment is that combined with the Summation-By-Parts (SBP) operators, it yields stability [1,2,3,4,5,6].…”

mentioning

confidence: 99%

A stable and conservative method for locally adapting the design order of finite difference schemes

Eriksson

Abbas

Nordström

2011

Journal of Computational Physics

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A procedure to locally change the order of accuracy of finite difference schemes is developed. The development is based on existing SummationBy-Parts operators and a weak interface treatment. The resulting scheme is proven to be accurate and stable.Numerical experiments verify the theoretical accuracy for smooth solutions. In addition, shock calculations are performed, using a scheme where the developed switching procedure is combined with the MUSCL technique.

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New compact difference schemes on non‐uniform collocated and staggered grids for convection‐dominated flows

Yadav

Maurya²,

Rajpoot

2022

Numerical Methods in Fluids

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Summary This paper deals with the formulation of the tridiagonal compact difference schemes for derivatives up to second‐order with boundary stencils on non‐uniform grids. A compact scheme for the first derivative with interpolation is also devised on staggered non‐uniform grids. The developed schemes with non‐uniform spacing transform to respective classical compact schemes for the case of uniform mesh spacing. The resolution, numerical diffusion, and anti‐diffusion features of the devised schemes are evaluated using global spectral analysis. Applications to the direct numerical simulation (DNS) of two‐dimensional lid‐driven cavity (LDC) flow governed by Navier‐Stokes equations and wave‐propagation following linear rotating shallow water (LRSWE) equations with variable grid‐spacing are discussed at different choices of parameters. Computed results are also compared with solutions available in the literature.

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Strict Stability of High-Order Compact Implicit Finite-Difference Schemes: The Role of Boundary Conditions for Hyperbolic PDEs, I

Cited by 64 publications

References 9 publications

A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces

A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces

A stable and conservative method for locally adapting the design order of finite difference schemes

New compact difference schemes on non‐uniform collocated and staggered grids for convection‐dominated flows

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