双曲型方程的轴对称全谱码
Axisymmetric fully spectral code for hyperbolic equations
摘要：
本文提出了一种求解二阶轴对称双曲型方程的全拟谱格式。以切比雪夫多项式为基函数，数值网格基于Lobbato（两个空间方向）和Radau（时间方向）配置点。该方法解决了两个问题，以前的算法被限制到一个空间维度，即（i）的逆稠密矩阵和（ii）收购一个足够好的初始猜测的非线性方程组。对于第一个问题，我们使用迭代双共轭梯度稳定的方法，我们配备了一个预条件的基础上单对角隐式龙格库塔（“SDIRK”）方法。SDIRK方法还为代码提供了一个很好的初始猜测。数值解是正确的机器精度，我们没有观察到任何限制的时间步长相比，空间分辨率。作为应用，我们解决了广义相对论性波动方程的黑洞时空在所谓的双曲面片，并再现了一些最近的结果在文献中。
英文摘要：
We present a fully pseudo-spectral scheme to solve axisymmetric hyperbolic equations of second order. With the Chebyshev polynomials as basis functions, the numerical grid is based on the Lobbato (for two spatial directions) and Radau (for the time direction) collocation points. The method solves two issues of previous algorithms which were restricted to one spatial dimension, namely, (i) the inversion of a dense matrix and (ii) the acquisition of a sufficiently good initial-guess for non-linear systems of equations. For the first issue, we use the iterative bi-conjugate gradient stabilized method, which we equip with a pre-conditioner based on a singly diagonally implicit Runge-Kutta ("SDIRK"-) method. The SDIRK-method also supplies the code with a good initial-guess. The numerical solutions are correct up to machine precision and we do not observe any restriction concerning the time step in comparison with the spatial resolution. As an application, we solve general-relativistic wave equations on a black-hole space-time in so-called hyperboloidal slices and reproduce some recent results available in the literature.
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