Numerical Boundary Control of Multi-Dimensional Hyperbolic Equations

TL;DR

This paper extends stabilization results for multi-dimensional hyperbolic equations to their discretized forms, analyzing the impact of numerical dissipation and confirming findings through simulations with various DG schemes.

Contribution

It introduces a comprehensive analysis of numerical dissipation effects in discretized multi-dimensional hyperbolic problems, extending previous one-dimensional results using dimensional splitting.

Findings

Numerical dissipation effects are explicitly quantified.

Stability results are confirmed through simulations.

Analysis applies to both low-order and high-order DG schemes.

Abstract

Existing theoretical stabilization results for linear, hyperbolic multi-dimensional problems are extended to the discretized multi-dimensional problems. In contrast to existing theoretical and numerical analysis in the spatially one-dimensional case the effect of the numerical dissipation is analyzed and explicitly quantified. Further, using dimensional splitting, the numerical analysis is extended to the multi-dimensional case. The findings are confirmed by numerical simulations for low-order and high-order DG schemes both in the one-dimensional and two-dimensional case.