We propose an Exponential DG approach for numerically solving partial differential equations (PDEs). The idea is to decompose the governing PDE operators into linear (fast dynamics extracted by linearization) and nonlinear (the remaining after removing the former) parts, on which we apply the discontinuous Galerkin (DG) spatial discretization. The resulting semi-discrete system is then integrated using exponential time-integrators: exact for the former and approximate for the latter. By construction, our approach i) is stable with a large Courant number ($Cr>1$); ii) supports high-order solutions both in time and space; iii) is computationally favorable compared to IMEX DG methods with no preconditioner; iv) requires comparable computational time compared to explicit RKDG methods, while having time stepsizes orders magnitude larger than maximal stable time stepsizes for explicit RKDG methods; v) is scalable in a modern massively parallel computing architecture by exploiting Krylov-subspace matrix-free exponential time integrators and compact communication stencil of DG methods. Various numerical results for both Burgers and Euler equations are presented to showcase these expected properties. For Burgers equation, we present a detailed stability and convergence analyses for the exponential Euler DG scheme.

我们提出了一种用于数值求解偏微分方程(PDE)的指数 DG 方法。这个想法是将控制 PDE 算子分解为线性（通过线性化提取的快速动态）和非线性（去除前者后的剩余部分）部分，我们对其应用不连续伽辽金（DG）空间离散化。然后使用指数时间积分器对得到的半离散系统进行积分：前者精确，后者近似。通过构造，我们的方法 i) 在大柯朗数 ($Cr>1$); ii) 支持时间和空间上的高阶解；iii) 与没有预处理器的 IMEX DG 方法相比，在计算上是有利的；iv) 与显式 RKDG 方法相比，需要相当的计算时间，同时时间步长数量级大于显式 RKDG 方法的最大稳定时间步长；v)通过利用 Krylov 子空间无矩阵指数时间积分器和 DG 方法的紧凑通信模板，在现代大规模并行计算架构中可扩展。提供了 Burgers 和 Euler 方程的各种数值结果来展示这些预期属性。对于 Burgers 方程，我们提供了指数欧拉 DG 方案的详细稳定性和收敛性分析。