Efficient Finite Difference WENO Scheme for Hyperbolic Systems with Non-Conservative Products

Abstract

Higher order finite difference Weighted Essentially Non-Oscillatory (WENO) schemes have been constructed for conservation laws. For multidimensional problems, they offer high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of comparable accuracy. This makes them quite attractive for several science and engineering applications. But, to the best of our knowledge, such schemes have not been extended to non-linear hyperbolic systems with non-conservative products. In this paper, we perform such an extension which improves the domain of applicability of such schemes. The extension is carried out by writing the scheme in fluctuation form. We use the HLLI Riemann solver of Dumbser and Balsara (2016) as a building block for carrying out this extension. Because of the use of an HLL building block, the resulting scheme has a proper supersonic limit. The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme, thus expanding its domain of applicability. Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions, making it very easy for users to transition over to the present formulation. For conservation laws, the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO, with two major advantages:- 1) It can capture jumps in stationary linearly degenerate wave families exactly. 2) It only requires the reconstruction to be applied once. Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of accuracy for smooth multidimensional flows. Stringent Riemann ... *Abstract truncated, see PDF*

arXiv commentAccepted in Communications on Applied Mathematics and Computation

References in corpus (7)

• Efficient, High Accuracy ADER-WENO Schemes for Hydrodynamics and Divergence-Free Magnetohydrodynamics
• On the accuracy of WENO and CWENO reconstructions of third order on nonuniform meshes
• Adaptive Mesh Refinement for Hyperbolic Systems based on Third-Order Compact WENO Reconstruction
• High Order Space-Time Adaptive WENO Finite Volume Schemes for Non-Conservative Hyperbolic Systems
• Continuum Mechanics and Thermodynamics in the Hamilton and the Godunov-type Formulations
• Total-Variation-Diminishing Implicit-Explicit Runge-Kutta Methods for the Simulation of Double-Diffusive Convection in Astrophysics
• An entropy stable high-order discontinuous Galerkin spectral element method for the Baer-Nunziato two-phase flow model

Cited by in corpus (5)

• Efficient Alternative Finite Difference WENO Schemes for Hyperbolic Conservation Laws
• Efficient Alternative Finite Difference WENO Schemes for Hyperbolic Systems with Non-Conservative Products
• Well-balanced high order finite difference WENO schemes for a first-order Z4 formulation of the Einstein field equations
• Physical Constraint Preserving Higher Order Finite Volume Schemes for Divergence-Free Astrophysical MHD and RMHD
• An Alternative Finite Difference WENO-like Scheme with Physical Constraint Preservation for Divergence-Preserving Hyperbolic Systems