This work concerns the analysis of the discontinuous Galerkin spectral element method (DGSEM) with implicit time stepping for the numerical approximation of nonlinear scalar conservation laws in multiple space dimensions. We consider either the DGSEM with a backward Euler time stepping, or a space-time DGSEM discretization to remove the restriction on the time step. We design graph viscosities in space, and in time for the space-time DGSEM, to make the schemes maximum principle preserving and entropy stable for every admissible convex entropy. We also establish well-posedness of the discrete problems by showing existence and uniqueness of the solutions to the nonlinear implicit algebraic relations that need to be solved at each time step. Numerical experiments in one space dimension are presented to illustrate the properties of these schemes.

翻译：本研究分析了采用隐式时间步进的间断伽辽金谱元法（DGSEM）在多维非线性标量守恒律数值逼近中的应用。我们分别考察了采用后向欧拉时间步进的DGSEM格式以及时空DGSEM离散格式，后者可消除时间步长的限制。我们在空间维度（以及时空DGSEM的时间维度）设计了图粘性项，使得格式对于所有容许凸熵均满足最大原理保持特性与熵稳定性。通过证明每个时间步需求解的非线性隐式代数关系解的存在唯一性，我们建立了离散问题的适定性。文中给出了一维空间的数值实验以验证这些格式的特性。