A Third-Order Semi-Discrete Central Scheme for Conservation Laws and Convection-Diffusion Equations

TL;DR

The paper develops a third-order semi-discrete central scheme for multidimensional hyperbolic conservation laws and convection–diffusion equations. It builds on the second-order semi-discrete approach by adopting a CWENO-based third-order reconstruction and incorporating local propagation speeds $a_{j+1/2}$ to reduce dissipation, without requiring Riemann solvers or characteristic decompositions. The method is presented in fully discrete one-dimensional form and in a semi-discrete form, then extended to multiple spatial dimensions with a dimension-by-dimension CWENO approach. Numerical experiments on linear advection, Burgers, Euler Sod, Buckley–Leverett, and incompressible flows demonstrate third-order accuracy, sharp resolution of shocks, and robustness, with flexible time integration using RK3 or DUMKA3.

Abstract

We present a new third-order, semi-discrete, central method for approximating solutions to multi-dimensional systems of hyperbolic conservation laws, convection-diffusion equations, and related problems. Our method is a high-order extension of the recently proposed second-order, semi-discrete method in [16]. The method is derived independently of the specific piecewise polynomial reconstruction which is based on the previously computed cell-averages. We demonstrate our results, by focusing on the new third-order CWENO reconstruction presented in [21]. The numerical results we present, show the desired accuracy, high resolution and robustness of our method.

Figures (25)

• Figure 3.1: Modified Central Differencing
• Figure 5.1: Burgers equation, (\ref{['Burgers']}); $T=2$, N=40.
• Figure 5.2: Burgers equation, (\ref{['Burgers']}); $T=2$, N=80.
• Figure 5.3: Sod problem -- density. N=200, T=0.1644.
• Figure 5.4: Sod problem -- density. N=400, T=0.1644.