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.
翻译：Trefftz 方法是一种高阶的 Galerkin 方法,其中所有离散函数都是PDE 的元素解决方案。 它们只有在PDE 是线性且其系数是质性常数时才可行。 我们引入了一种“ quasi- Trefftz” 的不连续 Galerkin 方法, 用于音波方程式的离散, 配有小于线性波速: 离散函数是元素性近似 PDE 解决方案。 我们显示, 新的离散函数具有与古典 Trefftz 方案相同的极佳近似特性, 并证明 DG 方案具有稳定性和高度的趋同性。 我们为新的离散空间引入了多边基函数, 并描述一种简单的算法来计算它们。 我们提出的技术是由先前为与可变系数的时间- 调问题开发的通用平流波波法所启发的; 事实证明, 在所考虑的时间- 度波方程式的情况下, 准- Trefftz 法允许多基函数 。
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