Summation by parts methods for spherical harmonic decompositions of the wave equation in any dimensions
Summation by parts methods for spherical harmonic decompositions of the wave equation in any dimensions
复制标题
任意维波动方程球谐分解的分部求和法
DOI：
10.1088/0264-9381/30/14/145003
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发表时间：
2013
影响因子：
3.5
通讯作者：
D. Garfinkle
中科院分区：
文献类型：
--
作者：
C. Gundlach;J. Martin;D. Garfinkle
我们研究了n + 2个时空维的波动方程的数值方法，写在球坐标中，在Sn上分解为球谐函数，并在剩余的坐标r和t中进行有限差分。当整个物理问题具有球对称性时，这种方法是有用的，对于关于球形背景的微扰理论，或者存在具有球形拓扑的边界。关键的数值困难来自原点r = 0处的低阶1/r项。作为一个玩具模型，我们考虑平坦空间线性波动方程的形式，其中p = 2l + n和l是领先的球谐指数。我们提出了一类部分求和（SBP）有限差分方法，该方法将离散能量保持到边界项，从而保证了能量范数的稳定性和收敛性。我们明确地构建SBP计划，是二阶和四阶精度的内部点和对称边界r = 0，和一阶和二阶精度的外边界r = R。
We investigate numerical methods for wave equations in n + 2 spacetime dimensions, written in spherical coordinates, decomposed in spherical harmonics on Sn, and finite-differenced in the remaining coordinates r and t. Such an approach is useful when the full physical problem has spherical symmetry, for perturbation theory about a spherical background, or in the presence of boundaries with spherical topology. The key numerical difficulty arises from lower order 1/r terms at the origin r = 0. As a toy model for this, we consider the flat space linear wave equation in the form , , where p = 2l + n and l is the leading spherical harmonic index. We propose a class of summation by parts (SBP) finite-differencing methods that conserve a discrete energy up to boundary terms, thus guaranteeing stability and convergence in the energy norm. We explicitly construct SBP schemes that are second- and fourth-order accurate at interior points and the symmetry boundary r = 0, and first- and second-order accurate at the outer boundary r = R.