On the accuracy of WENO and CWENO reconstructions of third order on nonuniform meshes

Abstract

Third order WENO and CWENO reconstruction are widespread high order reconstruction techniques for numerical schemes for hyperbolic conservation and balance laws. In their definition, there appears a small positive parameter, usually called $\epsilon$, that was originally introduced in order to avoid a division by zero on constant states, but whose value was later shown to affect the convergence properties of the schemes. Recently, two detailed studies of the role of this parameter, in the case of uniform meshes, were published. In this paper we extend their results to the case of finite volume schemes on non-uniform meshes, which is very important for h-adaptive schemes, showing the benefits of choosing $\epsilon$ as a function of the local mesh size $h_j$. In particular we show that choosing $\epsilon =h_j^2$ or $\epsilon =h_j$ is beneficial for the error and convergence order, studying on several non-uniform grids the effect of this choice on the reconstruction error, on fully discrete schemes for the linear transport equation, on the stability of the numerical schemes. Finally we compare the different choices for $\epsilon$ in the case of a well-balanced scheme for the Saint-Venant system for shallow water flows and in the case of an h-adaptive scheme for nonlinear systems of conservation laws and show numerical tests for a two-dimensional generalisation of the CWENO reconstruction on locally adapted meshes.

References in corpus (3)

• Adaptive Mesh Refinement for Hyperbolic Systems based on Third-Order Compact WENO Reconstruction
• Well-balanced high order schemes on non-uniform grids and entropy residuals
• Representation of the Lagrange reconstructing polynomial by combination of substencils

Cited by in corpus (24)

• Central WENO schemes for hyperbolic conservation laws on fixed and moving unstructured meshes
• CWENO: uniformly accurate reconstructions for balance laws
• ADER discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics
• Space-time adaptive ADER-DG schemes for dissipative flows: compressible Navier-Stokes and resistive MHD equations
• Optimal definition of the nonlinear weights in multidimensional Central WENOZ reconstructions
• Third and fourth order well-balanced schemes for the shallow water equations based on the CWENO reconstruction
• Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws
• Simple smoothness indicator and multi-level adaptive order WENO scheme for hyperbolic conservation laws
• High-order well-balanced methods for systems of balance laws: a control-based approach
• Central WENO schemes through a global average weight
• Efficient implementation of adaptive order reconstructions
• A variable high-order shock-capturing finite difference method with GP-WENO
• On a third order CWENO boundary treatment with application to networks of hyperbolic conservation laws
• Efficient Finite Difference WENO Scheme for Hyperbolic Systems with Non-Conservative Products
• Efficient, Divergence-Free, High Order MHD on 3D Spherical Meshes with Optimal Geodesic Meshing
• Efficient Alternative Finite Difference WENO Schemes for Hyperbolic Conservation Laws
• Well-balanced high order finite difference WENO schemes for a first-order Z4 formulation of the Einstein field equations
• Higher Order Central Schemes for Multi-dimensional Hyperbolic Problems
• A class of well-balanced algorithms for relativistic fluids on a Schwarzschild background
• Portability of Fortran's `do concurrent' on GPUs
• An Alternative Finite Difference WENO-like Scheme with Physical Constraint Preservation for Divergence-Preserving Hyperbolic Systems
• Improving accuracy of the fifth-order WENO scheme by using the exponential approximation space
• On direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes for the HPR model of nonlinear hyperelasticity
• One- and multi-dimensional CWENOZ reconstructions for implementing boundary conditions without ghost cells