论文标题
带有分段平滑系数的波方程的时空准Trefftz DG方法
A space-time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients
论文作者
论文摘要
TREFFTZ方法是高阶Galerkin方案，其中所有离散函数都是要近似PDE的元素解决方案。仅当PDE是线性并且其系数是分段常数时，它们才可行。我们引入了一种“准特雷夫兹”不连续的Galerkin方法，用于使用分段平滑波动的声波方程离散：离散函数是元素近似PDE溶液。我们表明，新的离散化具有与经典Trefftz One相同的出色近似属性，并且证明了DG方案的稳定性和高阶收敛性。我们为新离散空间介绍了多项式基础函数，并描述了一种简单的算法来计算它们。我们提出的技术的灵感来自先前为具有可变系数的时间谐波问题而开发的广义平面波。事实证明，在考虑的时间域波方程中，准Trefftz方法允许多项式基础函数。
Trefftz methods are high-order Galerkin schemes in which all discrete functions are elementwise solution of the PDE to be approximated. They are viable only when the PDE is linear and its coefficients are piecewise constant. We introduce a 'quasi-Trefftz' discontinuous Galerkin method for the discretisation of the acoustic wave equation with piecewise-smooth wavespeed: the discrete functions are elementwise approximate PDE solutions. We show that the new discretisation enjoys the same excellent approximation properties as the classical Trefftz one, and prove stability and high-order convergence of the DG scheme. We introduce polynomial basis functions for the new discrete spaces and describe a simple algorithm to compute them. The technique we propose is inspired by the generalised plane waves previously developed for time-harmonic problems with variable coefficients; it turns out that in the case of the time-domain wave equation under consideration the quasi-Trefftz approach allows for polynomial basis functions.
