Finite volume WENO methods for hyperbolic conservation laws on Cartesian grids with adaptive mesh refinement

Buchmüller, Pawel; Dreher, J.; Helzel, Christiane

doi:10.1016/j.amc.2015.03.078

Cited by 31 publications

(33 citation statements)

References 25 publications

(57 reference statements)

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“…The obtained fields at t = 0.2 are shown on Figures 4.11. They comfort the previous observations: the approximate solution seems correct (see [13,22,12,46,16,15,7] for other recents results), with rather sharp interfaces, while the slip lines are not subject to instabilities. This is confirmed by computations performed with refined meshes, dividing the cells by 2 in each direction (to obtain a 3200 × 800 structured mesh) and by 4 (6400 × 1600 mesh); we give on Figure 4.13 the obtained internal energy field in the reflection structure.…”

Section: One Dimensional Riemann Problemssupporting

confidence: 77%

A MUSCL-type segregated – explicit staggered scheme for the Euler equations

et al. 2018

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Abstract. We present a numerical scheme for the solution of Euler equations based on staggered discretizations and working either on structured schemes or on general simplicial or tetrahedral/hexahedral meshes. The time discretization is performed by a fractional-step or segregated algorithm involving only explicit steps. The scheme solves the internal energy balance, with corrective terms to ensure the correct capture of shocks, and, more generally, the consistency in the Lax-Wendroff sense. To keep the density, the internal energy and the pressure positive, conditionally positivity-preserving convection operators for the mass and internal energy balance equations are designed by a MUSCL-like procedure: first, second-order in space fluxes are computed, then a limiting procedure is applied. This latter is purely algebraic: it does not require any geometric argument and thus works on quite general meshes; moreover, it keeps the pressure constant at contact discontinuities. The construction of the fluxes does not need any Riemann or approximate Riemann solver, and yields thus a particularly simple algorithm. Artificial viscosity is added in order to reduce the oscillations of the scheme. Numerical tests confirm the accuracy of the scheme.

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Section: One Dimensional Riemann Problemssupporting

confidence: 77%

A MUSCL-type segregated – explicit staggered scheme for the Euler equations

et al. 2018

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“…Recently, Buchmüller and Helzel [4] proposed a modification to the dimension-bydimension FV WENO method on Cartesian grids and applied this modified method on adaptive Cartesian meshes [14,15]. Later on a fourth-order quadrature modification flux (QMF) method was introduced and applied on adaptive Cartesian meshes by Tamaki and Imamura [16].…”

Section: Introductionmentioning

confidence: 99%

“…A key technique used in Refs. [4,14,15] is the conversion between faceaveraged values and face-centered values, which helps improve the spatial order of accuracy of the dimension-by-dimension FV WENO method. However, Refs.…”

Section: Introductionmentioning

confidence: 99%

“…However, Refs. [4,14] mainly concentrated on two-dimensional problems and Ref. [15] only gave the fourth-order conversion formulas on three-dimensional (3D) Cartesian grids.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we further develop the modified FV WENO method by deriving sixth-order conversion formulas on 3D Cartesian grids which are not available in Refs. [4,14,15]. The derivation is based on the observation that the differentiation and cell-averaging are exchangeable [16], and it can be extended to even higher order accuracy of conversion.…”

Section: Introductionmentioning

confidence: 99%

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A High-Order Modified Finite Volume WENO Method on 3D Cartesian Grids

Du¹

2019

CiCP

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The modified dimension-by-dimension finite volume (FV) WENO method on Cartesian grids proposed by Buchmüller and Helzel can retain the full order of accuracy of the one-dimensional WENO reconstruction and requires only one flux computation per interface. The high-order accurate conversion between face-averaged values and face-center point values is the main ingredient of this method. In this paper, we derive sixth-order accurate conversion formulas on three-dimensional Cartesian grids. It is shown that the resulting modified FV WENO method is efficient and highorder accurate when applied to smooth nonlinear multidimensional problems, and is robust for calculating non-smooth nonlinear problems with strong shocks.

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A truncation error analysis of third‐order MUSCL scheme for nonlinear conservation laws

Nishikawa

2020

Numerical Methods in Fluids

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This article is a rebuttal to the claim found in the literature that the monotonic upstream‐centered scheme for conservation laws (MUSCL) cannot be third‐order accurate for nonlinear conservation laws. We provide a rigorous proof for third‐order accuracy of the MUSCL scheme based on a careful and detailed truncation error analysis. Throughout the analysis, the distinction between the cell average and the point value will be strictly made for the numerical solution as well as for the target operator. It is shown that the average of the solutions reconstructed at a face by Van Leer's κ‐scheme recovers a cubic solution exactly with κ=1/3, the same is true for the average of the nonlinear fluxes evaluated by the reconstructed solutions, and a dissipation term is already sufficiently small with a third‐order truncation error. Finally, noting that the target spatial operator is a cell‐averaged flux derivative, we prove that the leading truncation error of the MUSCL finite‐volume scheme is third‐order with κ=1/3. The importance of the diffusion scheme is also discussed: third‐order accuracy will be lost when the third‐order MUSLC scheme is used with an incompatible fourth‐order diffusion scheme for convection‐diffusion problems. Third‐order accuracy is verified by thorough numerical experiments for both steady and unsteady problems. This article is intended to serve as a reference to clarify confusions about third‐order accuracy of the MUSCL scheme, as a guide to correctly analyze and verify the MUSCL scheme for nonlinear equations, and eventually as the basis for clarifying high‐order unstructured‐grid schemes in a subsequent article.

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Finite volume WENO methods for hyperbolic conservation laws on Cartesian grids with adaptive mesh refinement

Cited by 31 publications

References 25 publications

A MUSCL-type segregated – explicit staggered scheme for the Euler equations

A MUSCL-type segregated – explicit staggered scheme for the Euler equations

A High-Order Modified Finite Volume WENO Method on 3D Cartesian Grids

A truncation error analysis of third‐order MUSCL scheme for nonlinear conservation laws

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