We investigate the properties of the high-order discontinuous Galerkin spectral element method (DGSEM) with implicit backward-Euler time stepping for the approximation of hyperbolic linear scalar conservation equation in multiple space dimensions. We first prove that the DGSEM scheme in one space dimension preserves a maximum principle for the cell-averaged solution when the time step is large enough. This property however no longer holds in multiple space dimensions and we propose to use the flux-corrected transport limiting [Boris and Book, J. Comput. Phys., 11 (1973)] based on a low-order approximation using graph viscosity to impose a maximum principle on the cell-averaged solution. These results allow to use a linear scaling limiter [Zhang and Shu, J. Comput. Phys., 229 (2010)] in order to impose a maximum principle at nodal values within elements. Then, we investigate the inversion of the linear systems resulting from the time implicit discretization at each time step. We prove that the diagonal blocks are invertible and provide efficient algorithms for their inversion. Numerical experiments in one and two space dimensions are presented to illustrate the conclusions of the present analyses.

翻译：我们研究了具有隐式后向欧拉时间步进的高阶间断伽辽金谱元法（DGSEM）在多个空间维度上逼近双曲线性标量守恒方程的性质。首先证明，在时间步长足够大的情况下，一维空间DGSEM格式能保持单元平均解的极大值原理。然而这一性质在多个空间维度中不再成立，为此我们提出基于图粘性的低阶逼近通量校正输运限制方法[Boris and Book, J. Comput. Phys., 11 (1973)]，对单元平均解施加极大值原理。这些结果允许使用线性缩放限制器[Zhang and Shu, J. Comput. Phys., 229 (2010)]，在单元内的节点值上施加极大值原理。随后，我们研究了每个时间步中时间隐式离散化所产生的线性系统的求逆问题。证明了对角块的可逆性，并给出了高效的求逆算法。通过一维和二维空间的数值实验，验证了本文分析结论。