守恒律方程的ALE-LDG方法

In this paper, we develop and analyze an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method with a time-dependent approximation space for one dimensional conservation laws, which satisfies the geometric conservation law.For the semi-discrete ALE-DG method, when applied to nonlinear scalar conservation laws, a cell entropy inequality, $mathrm{L}^{2}$ stability and error estimates are proven.More precisely, we prove the sub-optimal ($k+ rac{1}{2}$) convergence for monotone fluxes, and optimal ($k+1$) convergence for an upwind flux, when a piecewise $P^k$ polynomial approximation space is used.For the fully-discrete ALE-DG method, the geometric conservation law and the local maximum principle are proven.Moreover, we state conditions for slope limiters, which ensure total variation stability of the method.