$L^\infty(L^\infty)$-boundedness of DG($p$)-solutions for nonlinear conservation laws with boundary conditions

TL;DR

This paper proves that a high-order discontinuous Galerkin method for scalar conservation laws remains bounded in the $L^(L^)$ norm, ensuring stability for multi-dimensional problems with boundary conditions.

Contribution

It establishes the $L^(L^)$-boundedness of a higher-order shock-capturing DG method for scalar conservation laws in multiple dimensions.

Findings

Proves $L^(L^)$-boundedness of DG solutions

Applicable to multi-dimensional scalar conservation laws

Includes boundary condition considerations

Abstract

We prove the $L^\infty(L^\infty)$-boundedness of a higher-order shock-capturing streamline-diffusion DG-method based on polynomials of degree $p\geq 0$ for general scalar conservation laws. The estimate is given for the case of several space dimensions and for conservation laws with initial and boundary conditions.