Efficient Alternative Finite Difference WENO Schemes for Hyperbolic Conservation Laws

Abstract

Higher order finite difference Weighted Essentially Non-Oscillatory (WENO) schemes for conservation laws are extremely popular because, for multidimensional problems, they offer high order accuracy at a fraction of the cost of finite volume WENO or DG schemes. Such schemes come in two formulations. The very popular classical finite difference WENO (FD-WENO) method (Shu and Osher, J. Comput. Phys., 83 (1989) 32-78) relies two reconstruction steps applied to two split fluxes. However, the method cannot accommodate different types of Riemann solvers and cannot preserve free stream boundary conditions on curvilinear meshes. This limits its utility. The alternative finite difference WENO (AFD-WENO) method can overcome these deficiencies, however, much less work has been done on this method. The reasons are three-fold. First, it is difficult for the casual reader to understand the intricate logic that requires higher order derivatives of the fluxes to be evaluated at zone boundaries. The analytical methods for deriving the update equation for AFD-WENO schemes are somewhat recondite. To overcome that difficulty, we provide an easily accessible script that is based on a computer algebra system in Appendix A of this paper. Second, the method relies on interpolation rather than reconstruction, and WENO interpolation formulae have not been documented in the literature as thoroughly as WENO reconstruction formulae. In this paper, we explicitly provide all necessary WENO interpolation formulae that are needed for implementing AFD-WENO up to ninth order. The third reason is that AFD-WENO requires higher order derivatives of the fluxes to be available at zone boundaries. Since those derivatives are usually obtained by finite differencing the zone-centered fluxes, they become susceptible to a Gibbs phenomenon when the solution ...

arXiv commentAccepted in Communications on Applied Mathematics and Computation

References in corpus (12)

• Second Order Accurate Schemes for Magnetohydrodynamics With Divergence-Free Reconstruction
• Efficient, High Accuracy ADER-WENO Schemes for Hydrodynamics and Divergence-Free Magnetohydrodynamics
• An HLLC Riemann Solver for Relativistic Flows: I. Hydrodynamics
• 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
• A Subluminal Relativistic Magnetohydrodynamics Scheme with ADER-WENO Predictor and Multidimensional Riemann Solver-Based Corrector
• Total-Variation-Diminishing Implicit-Explicit Runge-Kutta Methods for the Simulation of Double-Diffusive Convection in Astrophysics
• Simple smoothness indicator and multi-level adaptive order WENO scheme for hyperbolic conservation laws
• High-order accurate entropy stable finite difference schemes for one- and two-dimensional special relativistic hydrodynamics
• Physical-constraints-preserving Lagrangian finite volume schemes for one- and two-dimensional special relativistic hydrodynamics
• Entropy stable discontinuous Galerkin methods for ten-moment Gaussian closure equations
• Efficient Finite Difference WENO Scheme for Hyperbolic Systems with Non-Conservative Products

Cited by in corpus (6)

• 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
• Finite difference alternative WENO schemes with Riemann invariant-based local characteristic decompositions for compressible Euler equations
• Physical Constraint Preserving Higher Order Finite Volume Schemes for Divergence-Free Astrophysical MHD and RMHD
• Compact Runge-Kutta flux reconstruction methods for non-conservative hyperbolic equations
• An Alternative Finite Difference WENO-like Scheme with Physical Constraint Preservation for Divergence-Preserving Hyperbolic Systems