An explicit mapped tent pitching scheme for Maxwell equations
An explicit mapped tent pitching scheme for Maxwell equations
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麦克斯韦方程的显式映射帐篷倾斜方案
DOI：
10.1007/978-3-030-39647-3_28
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Wintersteiger, Christoph
中科院分区：
文献类型：
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作者：
Gopalakrishnan, Jay;Hochsteger, Matthias;Schöberl, Joachim;Wintersteiger, Christoph
电磁波以光速传播。因此，在空间和时间中的某个点处的字段仅取决于依赖锥内的字段值。帐篷俯仰法引入了一个特殊的“因果”时空网格，尊重这种有限的传播速度。它不仅限于麦克斯韦方程组，而且可以应用于一般的双曲型方程组。帐篷俯仰法需要一个数值方案来离散该网格上的方程。不连续Galerkin（DG）方法是特别感兴趣的，因为它们提供了一个系统的途径来建立高阶方法。对于帐篷底部的给定初始条件，离散方程可以在每个单独的帐篷内求解，直到帐篷顶部。帐篷顶部的计算解为后面的帐篷提供了初始条件。这种方法是高度并行的，因为许多帐篷可以独立求解。使用这种帐篷式网格的方法可以追溯到[5，7]。最近的作品[1，6，8]通过制定局部变分问题来开发帐篷内的时空DG（SDG）方法，并为此建立和求解线性系统。虽然这些系统是局部的，但矩阵的大小可以以多项式的阶数快速增长，特别是在四维时空帐篷中。在这方面
Electromagnetic waves propagate at the speed of light. Thus, the field at a certain point in space and time depends only on field values within a dependency cone. A tent pitching method introduces a special “causal” spacetime mesh that respects this finite speed of propagation. It is not limited to Maxwell equations, but can be applied to general hyperbolic equations. A tent pitching method requires a numerical scheme to discretize the equation on that mesh. Discontinuous Galerkin (DG) methods are of particular interest since they offer a systematic avenue to build high order methods. For a given initial condition at the bottom of a tent, the discrete equations may be solved within each individual tent, up to the tent top. The computed solution at the tent top provides initial conditions for the tents that follow later in time. This method is highly parallel, since many tents can be solved independently. Methods using such tent-pitched meshes may be traced back to [5, 7]. More recent works [1, 6, 8] develop Spacetime DG (SDG) methods within tents by formulating local variational problems, for which linear systems are set up and solved. Although these systems are local, the matrix size can grow rapidly with the polynomial order, especially in four-dimensional spacetime tents. In this context