Compact Central WENO Schemes for Multidimensional Conservation Laws

TLDR

The authors introduce a new third‑order central scheme for solving systems of conservation laws in one and two space dimensions. The scheme reconstructs a piecewise‑polynomial interpolant from cell averages using a compact third‑order CWENO reconstruction that blends interpolants on multiple stencils, with weights chosen for third‑order accuracy in smooth regions and an embedded switch to a one‑sided second‑order reconstruction near discontinuities, and it advances the solution exactly in time on an extremely compact three‑point stencil in one dimension and similarly compact stencils in higher dimensions. The accuracy, robustness, and high‑resolution properties of the scheme are demonstrated on a variety of one‑ and two‑dimensional test problems.

Abstract

We present a new third-order central scheme for approximating solutions of systems of conservation laws in one and two space dimensions. In the spirit of Godunov-type schemes, our method is based on reconstructing a piecewise-polynomial interpolant from cell-averages which is then advanced exactly in time. In the reconstruction step, we introduce a new third-order, compact, central weighted essentially nonoscillatory (CWENO) reconstruction, which is written as a convex combination of interpolants based on different stencils. The heart of the matter is that one of these interpolants is taken as a suitable quadratic polynomial, and the weights of the convex combination are set as to obtain third-order accuracy in smooth regions. The embedded mechanism in the WENO-like schemes guarantees that in regions with discontinuities or large gradients, there is an automatic switch to a one-sided second-order reconstruction, which prevents the creation of spurious oscillations. In the one-dimensional case, our new third-order reconstruction is based on an extremely compact three-point stencil. Analogous compactness is retained in more space dimensions. The accuracy, robustness, and high-resolution properties of our scheme are demonstrated in a variety of one- and two-dimensional problems.