Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws

Tan, Sirui; Shu, Chi‐Wang

doi:10.1016/j.jcp.2010.07.014

Cited by 130 publications

(213 citation statements)

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“…If there have been no computations before we fill the ghost cells with data obtained by a one sided reconstruction presented in [30]. If the computations from time step t l are available, we can simply use the polynomial (13).…”

Section: Reconstruction For the Generalized Riemann Problem At The Jumentioning

confidence: 99%

“…These can be obtained from the given spatial derivatives ∂ k x u r using the Cauchy-Kowalewski or Lax-Wendroff procedure [18,30]. Note that it is important that the Cauchy-Kowalewski procedure is carried out on the basis of the zeroth order Godunov state, which can be obtained from the classical Riemann Problem at the junction.…”

Section: By Introducing the Notationsmentioning

confidence: 99%

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ADER schemes and high order coupling on networks of hyperbolic conservation laws

Borsche

Kall

2014

Journal of Computational Physics

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In this article we present a method to extend high order finite volume schemes to networks of hyperbolic conservation laws with algebraic coupling conditions. This method is based on an ADER approach in time to solve the generalized Riemann problem at the junction. Additionally to the high order accuracy, this approach maintains an exact conservation of quantities if stated by the coupling conditions. Several numerical examples confirm the benefits of a high order coupling procedure for high order accuracy and stable shock capturing.

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Section: Reconstruction For the Generalized Riemann Problem At The Jumentioning

confidence: 99%

Section: By Introducing the Notationsmentioning

confidence: 99%

ADER schemes and high order coupling on networks of hyperbolic conservation laws

Borsche

Kall

2014

Journal of Computational Physics

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“…We use a characteristic decomposition of the temporal derivative of the solution at the nodal point and estimate the outgoing information using spatial derivatives. This procedure has been applied to (pure) boundary value problems in the context of finite differences (and finite volumes) for example in [39][40][41] and mimics the Cauchy-Kowalewski theorem for sufficiently smooth solutions. It should be mentioned that also in [39] been developed for an arbitrary order of accuracy while we only discuss second-order schemes.…”

Section: Introductionmentioning

confidence: 99%

“…This procedure has been applied to (pure) boundary value problems in the context of finite differences (and finite volumes) for example in [39][40][41] and mimics the Cauchy-Kowalewski theorem for sufficiently smooth solutions. It should be mentioned that also in [39] been developed for an arbitrary order of accuracy while we only discuss second-order schemes. We show that the derived scheme coincides with a second-order discretization in the case of two connected arcs, we validate the discretization by reformulating a classical boundary value problem using coupling conditions and we present numerical results for gas flow in pipe networks.…”

Section: Introductionmentioning

confidence: 99%

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Numerical Discretization of Coupling Conditions by High-Order Schemes

2016

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Abstract. We consider numerical schemes for 2 × 2 hyperbolic conservation laws on graphs. The hyperbolic equations are given on the spatially one-dimensional arcs and are coupled at a single point, the node, by a nonlinear coupling condition. We develop high-order finite volume discretizations for the coupled problem. The reconstruction of the fluxes at the node is obtained using derivatives of the parameterized algebraic conditions imposed at the nodal points in the network. Numerical results illustrate the expected theoretical behavior. 35R02, 35Q35, 35F30

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Nonlinear diffusion‐limited two‐dimensional colliding‐plume simulations with very high‐order numerical approximations

Straka,

Williams,

Kanak

2024

Quart J Royal Meteoro Soc

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Atmospheric numerical models play a crucial role in operational weather forecasting, as well as improving our understanding of atmospheric dynamics via research studies. Maximising their accuracy is of paramount importance. Use of >O7 flux schemes in atmospheric models is largely undocumented, with no studies considering O3–17 fluxes with formal accuracy‐preserving high order interpolation, pressure gradient / divergence, and subgrid‐scale (SGS) turbulent fluxes. Higher order numerical approximations can reduce truncation, amplitude and phase errors, and potentially improve model accuracy and effective resolution.Here, simulations are presented using very high order O3–17 fluxes, with / without high order O2–18 Lagrangian interpolations, pressure gradient / divergence approximations, and SGS turbulent fluxes for a 2D, highly‐viscous (Re ~ 100) diffusion‐limited, nonlinear colliding plumes problem using 25–200 m spatial resolutions. The highest order flux schemes coupled with higher order interpolations, pressure gradient / divergence and SGS flux approximations produced the best solutions, with higher‐order fluxes and interpolations being most important. Overall solution convergence of ~O1–2 with mode‐split (fast sound / slow advective waves) O3 Runge–Kutta temporal schemes was negatively impacted by ≤O1 temporal convergence with SGS fluxes, divergence damping, and especially spatial filters, compared to ~O3 convergence with these inactivated.While very high order schemes were shown to improve solution accuracy, few cost‐effective higher order highly‐viscous test problem solutions (higher order versus finer resolution) were found using theoretical floating‐point operations (FPO) with CFL‐limited or constant stable Courant number‐based time steps. However, employing CPU time, rather than FPOs, demonstrated there was reduced computational burden using higher order approximations. We conclude that O9–17 flux schemes with or without high‐order ≥O4 interpolations, pressure gradient / divergence approximations, and SGS fluxes can improve atmospheric model solution accuracy, without prohibitive computation costs, compared to O3–7 flux with O2 interpolations, pressure gradient / divergence approximations, and SGS fluxes.This article is protected by copyright. All rights reserved.

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Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws

Cited by 130 publications

References 20 publications

ADER schemes and high order coupling on networks of hyperbolic conservation laws

ADER schemes and high order coupling on networks of hyperbolic conservation laws

Numerical Discretization of Coupling Conditions by High-Order Schemes

Nonlinear diffusion‐limited two‐dimensional colliding‐plume simulations with very high‐order numerical approximations

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