Improved Accuracy of High-Order WENO Finite Volume Methods on Cartesian Grids

Abstract

We propose a simple modification of standard WENO finite volume methods for Cartesian grids, which retains the full spatial order of accuracy of the one-dimensional discretization when applied to nonlinear multidimensional systems of conservation laws.We derive formulas, which allow us to compute high-order accurate point values of the conserved quantities at grid cell interfaces. Using those point values, we can compute a highorder flux at the center of a grid cell interface. Finally, we use those point values to compute high-order accurate averaged fluxes at cell interfaces as needed by a finite volume method.The method is described in detail for the two-dimensional Euler equations of gas dynamics. An extension to the three-dimensional case as well as to other nonlinear systems of conservation laws in divergence form is straightforward. Furthermore, similar ideas can be used to improve the accuracy of WENO type methods for hyperbolic systems which are not in divergence form.Several test computations confirm the high-order accuracy for smooth nonlinear problems.

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

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a b s t r a c tWe present a WENO finite volume method for the approximation of hyperbolic conservation laws on adaptively refined Cartesian grids.On each single patch of the AMR grid, we use a modified dimension-by-dimension WENO method, which was recently developed by Buchmüller and Helzel (2014) [1]. This method retains the full spatial order of accuracy of the underlying one-dimensional WENO reconstruction for nonlinear multidimensional problems, and requires only one flux computation per interface.

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Higher Order Finite Volume Central Schemes for Multi-dimensional Hyperbolic Problems

“…To achieve an accuracy higher than second order one must use higher order accurate averaged fluxes because point value fluxes would only be second-order approximations and, thus, degrade the accuracy to the second order [4,35].…”

“…(16) we just need to obtain point values at the face centers. This averaging procedure has previously been used to obtain higher order averaged flux in the multidimensional PPM (piecewise parabolic method) [35] and WENO schemes [4]. Whereas, we are applying it in the framework of a central scheme which naturally allows the computation of non-oscillatory point values (U + i+1/2,j,k , U − i−1/2,j,k ) at the face-center.…”

“…This method of computing higher-order point value at the face-center has previously been used in the framework of the third order CWENO scheme of Kurganov and Levy [21], fourth order PPM scheme [35] and fourth order WENO scheme [4]. We are applying this method to develop a fourth-order accurate CWENO scheme.…”

“…These initial conditions result in the advection of the initial density profile due to constant velocity components and constant pressure (see for example [4] Example 2. Initial conditions for the 2D Euler vortex evolution problem (see for example [13]) are,…”