Generalized framework for admissibility preserving Lax-Wendroff Flux Reconstruction for hyperbolic conservation laws with source terms

TL;DR

The work develops a generalized admissibility-preserving framework for Lax-Wendroff Flux Reconstruction (LWFR) applied to hyperbolic conservation laws with source terms, enabling high-order accuracy in a single-stage method. It extends Zhang-Shu's admissibility ideas to LWFR by decomposing the update into time-averaged flux and source components and enforcing admissibility through flux and source term limiting, respectively. The time-averaged flux is corrected via a flux-limiter that blends against a low-order, admissible flux, while the time-averaged sources are similarly limited to maintain mean admissibility; together, these yield an admissibility-preserving LWFR scheme with sources. Numerical tests on the Ten Moment equations demonstrate high-order convergence, robust shock-turbulence behavior, near-vacuum handling, and realistic applicability, confirming the method’s effectiveness and practical impact for complex hyperbolic systems with source terms.

Abstract

Lax-Wendroff Flux Reconstruction (LWFR) is a single-stage, high order, quadrature free method for solving hyperbolic conservation laws. We perform a cell average decomposition of the LWFR scheme that is similar to the one used in the admissibility preserving framework of Zhang and Shu (2010). By performing a flux limiting of the time averaged numerical flux, the decomposition is used to obtain an admissibility preserving LWFR scheme. The admissibility preservation framework is further extended to a newly proposed extension of LWFR scheme for conservation laws with source terms. This is the first extension of the high order LW scheme that can handle source terms. The admissibility and accuracy are verified by numerical experiments on the Ten Moment equations of Livermore et al.

Figures (6)

• Figure 1: Error convergence analysis of a smooth test with source terms for (a) $\rho$, (b) $p_{11}$ variable
• Figure 2: Density plots of numerical solutions with polynomial degree $N=2$ for (a) Sod's problem, (b) Two rarefaction (near vacuum) problem
• Figure 3: Numerical solution for Shu-Osher problem with polynomial degree $N=4$ using TVB and blending limiter and we show (a) Density, (b) $v_1$ profiles. The density plot has an inset plot near the shock which compares number of cells smeared across the shock by blending and TVB limiter.
• Figure 4: Density profile of the numerical solution for Shu-Osher problem with using 1000 degrees of freedom (solution points) for polynomial degrees $N=2,3,4$ using (a) TVB and (b) blending limiter.
• Figure 5: 2-D near vacuum test. Density plot of numerical solution with degree $N=2$ on a $100^2$ element mesh (a) Pseudocolor plot (b) Solution cut along the line $y=0$