Compact Central WENO Schemes for Multidimensional Conservation Laws
D. Levy, G. Puppo, G. Russo
TL;DR
The paper develops a compact, third-order central-weighted essentially non-oscillatory (CWENO) reconstruction for multidimensional hyperbolic conservation laws, enabling high-order accuracy on very small stencils without relying on Riemann solvers. In 1D, a centered quadratic polynomial is combined with left/right linear interpolants so that $P_{\text{OPT}}$-consistency is achieved in smooth regions, while symmetric nonlinear weights with smoothness indicators switch to a stable one-sided stencil near discontinuities. The 2D extension uses four corner one-sided interpolants plus a centered quadratic to maintain third-order accuracy on a nine-point stencil, with carefully chosen corner weights and 2D smoothness indicators. The method is validated on scalar and system tests, including the Euler equations, Burgers, and linear advection, demonstrating robust, non-oscillatory, high-resolution performance and practical compactness for multidimensional 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.