Compact Central WENO Schemes for Multidimensional Conservation Laws

TL;DR

This paper introduces a new third-order central WENO scheme for multidimensional conservation laws that combines high accuracy with robustness near discontinuities, using a compact stencil for efficiency.

Contribution

The paper presents a novel third-order central WENO scheme with a compact stencil and an adaptive mechanism to prevent oscillations near discontinuities, applicable in multiple dimensions.

Findings

Achieves third-order accuracy in smooth regions

Automatically switches to second-order near discontinuities

Demonstrates robustness and high resolution in 1D and 2D 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 as a convex combination of interpolants based on different stencils. The heart of the matter is that one of these interpolants is taken as an arbitrary 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 scheme is based on an extremely compact 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.