Approximate multidimensional Riemann solvers are essential building blocks in designing globally constraint-preserving finite volume time domain (FVTD) and discontinuous Galerkin time domain (DGTD) schemes for computational electrodynamics (CED). In those schemes, we can achieve high-order temporal accuracy with the help of Runge-Kutta or ADER time-stepping. This paper presents the design of a multidimensional approximate Generalized Riemann Problem (GRP) solver for the first time. The multidimensional Riemann solver accepts as its inputs the four states surrounding an edge on a structured mesh, and its output consists of a resolved state and its associated fluxes. In contrast, the multidimensional GRP solver accepts as its inputs the four states and their gradients in all directions; its output consists of the resolved state and its corresponding fluxes and the gradients of the resolved state. The gradients can then be used to extend the solution in time. As a result, we achieve second-order temporal accuracy in a single step. In this work, the formulation is optimized for linear hyperbolic systems with stiff, linear source terms because such a formulation will find maximal use in CED. Our formulation produces an overall constraint-preserving time-stepping strategy based on the GRP that is provably L-stable in the presence of stiff source terms. We present several stringent test problems, showing that the multidimensional GRP solver for CED meets its design accuracy and performs stably with optimal time steps. The test problems include cases with high conductivity, showing that the beneficial L-stability is indeed realized in practical applications.

翻译：近似多维黎曼求解器是计算电磁学（CED）中设计全局约束保持的时域有限体积（FVTD）和时域间断伽辽金（DGTD）格式的关键构建模块。在这些格式中，我们借助龙格-库塔（Runge-Kutta）或ADER时间推进方法可实现高阶时间精度。本文首次提出了一种多维近似广义黎曼问题（GRP）求解器的设计。该多维黎曼求解器的输入为结构化网格上边缘周围的四个状态，其输出由解析状态及其相关通量组成。相比之下，多维广义黎曼问题（GRP）求解器的输入为四个状态及其在各方向上的梯度，其输出包含解析状态、对应通量以及解析状态的梯度。这些梯度可用于在时间上延拓解，从而在单步内实现二阶时间精度。本研究针对含有刚性线性源项的线性双曲系统优化了公式表述，因其在CED中具有广泛应用价值。基于GRP的公式表述构建了全局约束保持的时间推进策略，在刚性源项存在时被证明具有L-稳定性。我们通过多个严格测试问题验证了该CED多维GRP求解器达到设计精度，并以最优时间步长稳定运行。测试问题包含高电导率情形，证明在实际应用中确实实现了有益的L-稳定性。