论文标题
基于振幅的广义平面波：2D中标量方程的新的Quasi-Trefftz函数
Amplitude-based Generalized Plane Waves: new Quasi-Trefftz functions for scalar equations in 2D
论文作者
论文摘要
引入了广义平面波（GPW），以利用由可变系数方程建模的问题来利用Trefftz方法。尽管GPW不满足TREFFTZ属性，即它们不是管理方程的确切解决方案，但它们不满足Quasi-Trefftz属性：它们只是近似解决方案。它们导致高阶数值方法，并且该准特性属性对于其数值分析至关重要。 目前的工作介绍了一个新的GPW家族，基于振幅。动机在于基于阶段的GPW近似值的不良行为，在临时状态下，这将通过避免指数级内的高度多项式来驯服。新的Ansatz在振幅中引入了高阶项，而不是最初提出的平面波的相位。新功能的结构及其插值特性的研究由[16]中提出的路线图指导。为了清楚起见，第一个焦点是带有空间变化的波数的二维Helmholtz方程。以下是一系列运算符的扩展，允许在第一阶和二阶项中进行各向异性。数值模拟说明了新的准特性函数的理论研究。
Generalized Plane Waves (GPWs) were introduced to take advantage of Trefftz methods for problems modeled by variable coefficient equations. Despite the fact that GPWs do not satisfy the Trefftz property, i.e. they are not exact solutions to the governing equation, they instead satisfy a quasi-Trefftz property: they are only approximate solutions. They lead to high order numerical methods, and this quasi-Trefftz property is critical for their numerical analysis. The present work introduces a new family of GPWs, amplitude-based. The motivation lies in the poor behavior of the phase-based GPW approximations in the pre-asymptotic regime, which will be tamed by avoiding high degree polynomials within an exponential. The new ansatz is introduces higher order terms in the amplitude rather than the phase of a plane wave as was initially proposed. The new functions' construction and the study of their interpolation properties are guided by the roadmap proposed in [16]. For the sake of clarity, the first focus is on the two-dimensional Helmholtz equation with spatially-varying wavenumber. The extension to a range of operators allowing for anisotropy in the first and second order terms follows. Numerical simulations illustrate the theoretical study of the new quasi-Trefftz functions.
