论文标题
非线性保护法的可扩展指数-DG方法：应用于汉堡和欧拉方程
A scalable exponential-DG approach for nonlinear conservation laws: with application to Burger and Euler equations
论文作者
论文摘要
我们提出了一种用于求解部分微分方程（PDE）的指数DG方法。这个想法是将管理PDE操作员分解为线性（通过线性化提取的快速动力学）和非线性（删除前者之后的剩余）部分，我们将不连续的Galerkin（DG）空间离散化应用。然后，使用指数时间集成仪集成了所得的半差异系统：对于前者而言，确切的时间和近似值。通过构造，我们的方法i）稳定，数量很大（Cr> 1）； ii）支持时空和空间上的高阶解决方案；与没有预处理的IMEX DG方法相比，III）在计算上是有利的； iv）与显式RKDG方法相比，需要可比较的计算时间，而时间得出的时间大小要比最大稳定时间得出的幅度大于显式RKDG方法； v）通过利用Krylov-Subspace矩阵无指数时间积分器和DG方法的紧凑型通信模板来利用Krylov-Subspace矩阵，可以扩展到现代平行的计算体系结构中。提出了汉堡和Euler方程的各种数值结果，以展示这些预期的特性。对于汉堡方程，我们为指数Euler DG方案提供了详细的稳定性和收敛分析。
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 detailed stability and convergence analyses for the exponential Euler DG scheme.
