Adaptive mesh methods for hyperbolic conservation laws with bound-preserving flux limiters

TL;DR

This work addresses robust, high-order simulation of hyperbolic conservation laws on adaptive moving meshes by introducing bound-preserving flux limiters that decompose high-order updates into BP sub-cell components. The method yields BP CFL conditions (e.g., $\lambda^{n}\alpha^{n} \le \tfrac{1}{2}$ and $\lambda^{(\ell-1)}\alpha^{(\ell-1)} \le \tfrac{1}{6}$) and an accuracy-preserving regime, while enforcing the discrete solution to stay within the invariant domain $\mathcal{G}$. The framework is extended to the Euler equations and the five-equation two-medium flow model using a path-conservative formulation, ensuring positivity and physical bounds even in non-conservative or multiphase contexts. Numerical tests demonstrate that BP moving-mesh schemes achieve high resolution and strong robustness, validating the theoretical BP/AP CFL analyses and highlighting the practical value for complex compressible flows. The approach promises scalable benefits for high-dimensional problems and multiphase systems where invariant-domain preservation on adaptive meshes is critical.

Abstract

In this paper, we develop bound-preserving (BP) finite-volume schemes for hyperbolic conservation laws on adaptive moving meshes. For scalar conservative laws, we rewrite the conventional high-order discretization as a convex combination of first-order counterparts on each sub-cell, which is mathematically equivalent to introducing a bound-preserving flux limiter. Such a limiter is inexpensive to evaluate, with a feature that the corresponding BP CFL conditions depend solely on the first-order sub-cell schemes. A mild CFL restriction is derived under which high-order spatial accuracy is retained. The proposed BP schemes are extend to two nonlinear systems, namely, the Euler equations and the five-equation transport model of two-medium flows. Numerical results demonstrate that the present schemes possess high resolution and strong robustness properties.

Figures (6)

• Figure 1: Example \ref{['exam-GCL']}. D-GCL test. Left: $a=-2$. Right: $a=5$. The output time is $t=2$.
• Figure 2: Example \ref{['exam1']}. Left: trajectory of grid points. Middle: comparison of solutions on uniform and adaptive meshes without BP limiters. Right: comparison of solutions on uniform and adaptive meshes with BP limiters. The output time is $t=1.3$.
• Figure 3: Example \ref{['exam2']}: Trajectory of mesh grids (top left) and distributions of density $\rho$ (top right), velocity $u$ (bottom left), and pressure $p$ (bottom right) of the Riemann problem of Euler system at $t=8\times 10^{-4}$.
• Figure 4: Example \ref{['exam3']}: Trajectory of moving mesh grids (left) and distributions of density at $t=1$(right), where red color indicates the BP moving mesh solutions, and blue color presents the uniform mesh solutions. v1 represents results with the monitor function \ref{['mx1']}.
• Figure 5: Example \ref{['exam3']}: Trajectory of mesh grids (left) and distributions of density at $t=1$(right), v1 and v2 represent results with the monitor function \ref{['mx1']} and \ref{['mx2']}, respectively.

Theorems & Definitions (7)

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