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

Abstract

We prove the $L^{\infty }(L^{\infty })$-boundedness of a higher-order shock-capturing streamline-diffusion DG-method based on polynomials of degree $p\ge 0$ 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.