A Multidomain spectral method for solving elliptic equations
A Multidomain spectral method for solving elliptic equations
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DOI：
10.1016/s0010-4655(02)00847-0
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发表时间：
2002-02
影响因子：
6.3
通讯作者：
H. Pfeiffer;Lawrence E. Kidder;M. Scheel;S. Teukolsky
中科院分区：
文献类型：
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作者：
H. Pfeiffer;Lawrence E. Kidder;M. Scheel;S. Teukolsky
提出了一种求解耦合非线性椭圆型偏微分方程的新方法。该求解器基于伪谱配置和区域分解，可以处理一维到三维的问题。它有三个明显的特点。首先，将求解偏微分方程、满足边界条件和不同子域之间匹配的组合问题转化为标准线性和非线性求解器易于访问的一组方程。其次，支持触摸和重叠子域；实现了具有切比雪夫基函数的矩形块和具有球谐展开的球壳。第三，代码非常灵活：可以在运行时选择域分解以及每个域中并置点的分布，求解器很容易适应新的pde。该代码已用于求解广义相对论初值问题的方程，并应在许多其他问题中有用。我们将新方法与有限差分码进行了比较，发现它在运行时间和精度上都有优势，至少对于这里考虑的光滑问题。
We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three distinct features. First, the combined problem of solving the PDE, satisfying the boundary conditions, and matching between different subdomains is cast into one set of equations readily accessible to standard linear and nonlinear solvers. Second, touching as well as overlapping subdomains are supported; both rectangular blocks with Chebyshev basis functions as well as spherical shells with an expansion in spherical harmonics are implemented. Third, the code is very flexible: The domain decomposition as well as the distribution of collocation points in each domain can be chosen at run time, and the solver is easily adaptable to new PDEs. The code has been used to solve the equations of the initial value problem of general relativity and should be useful in many other problems. We compare the new method to finite difference codes and find it superior in both runtime and accuracy, at least for the smooth problems considered here.