A fully discrete energy stability analysis is carried out for linear advection-diffusion problems discretized by generalized upwind summation-by-parts~(upwind gSBP) schemes in space and implicit-explicit Runge-Kutta~(IMEX-RK) schemes in time. Hereby, advection terms are discretized explicitly while diffusion terms are solved implicitly. In this context, specific combinations of space and time discretizations enjoy enhanced stability properties. In fact, if the first and second-derivative upwind gSBP operators fulfill a compatibility condition, the allowable time step size is independent of grid refinement, although the advective terms are discretized explicitly. In one space dimension it is shown that upwind gSBP schemes represent a general framework including standard discontinuous Galerkin~(DG) schemes on a global level. While previous work for DG schemes has demonstrated that the combination of upwind advection fluxes and the central-type first Bassi-Rebay~(BR1) scheme for diffusion does not allow for grid-independent stable time steps, the current work shows that central advection fluxes are compatible with BR1 regarding enhanced stability of IMEX time stepping. Furthermore, unlike previous discrete energy stability investigations for DG schemes, the present analysis is based on the discrete energy provided by the corresponding SBP norm matrix and yields time step restrictions independent of the discretization order in space since no finite-element-type inverse constants are involved. Numerical experiments are provided confirming these theoretical findings.

翻译：针对采用广义迎风求和分部（upwind gSBP）格式进行空间离散、隐式-显式龙格-库塔（IMEX-RK）格式进行时间离散的线性对流扩散问题，开展了全离散能量稳定性分析。其中，对流项采用显式离散，而扩散项采用隐式求解。在此框架下，特定的时空离散组合展现出增强的稳定性特性。事实上，若一阶与二阶导数迎风gSBP算子满足相容性条件，尽管对流项采用显式离散，可允许的时间步长仍与网格细化无关。在一维空间中，研究表明迎风gSBP格式构成了一个通用框架，可在全局层面包含标准间断伽辽金（DG）格式。虽然先前针对DG格式的研究表明，迎风对流通量与扩散项的中央型第一Bassi-Rebay（BR1）格式组合无法实现网格无关的稳定时间步长，但当前工作证明：在IMEX时间推进的增强稳定性方面，中央对流通量与BR1格式具有相容性。此外，不同于先前针对DG格式的离散能量稳定性研究，本分析基于对应SBP范数矩阵所提供的离散能量，所得时间步长限制与空间离散阶数无关（因无需引入有限元型逆常数）。数值实验结果证实了这些理论发现。