Trefftz Methods for Time Dependent Partial Differential Equations

Abstract

In this paper we present a mesh-free ap- proach to numerically solving a class of second order time dependent partial differential equations which in- clude equations of parabolic, hyperbolic and parabolic- hyperbolic types. For numerical purposes, a variety of transformations is used to convert these equations to stan- dard reaction-diffusion and wave equation forms. To solve initial boundary value problems for these equa- tions, the time dependence is removed by either the Laplace or the Laguerre transform or time differencing, which converts the problem into one of solving a se- quence of boundary value problems for inhomogeneous modified Helmholtz equations. These boundary value problems are then solved by a combination of the method of particular solutions and Trefftz methods. To do this, a variety of techniques is proposed for numerically com- puting a particular solution for the inhomogeneous mod- ified Helmholtz equation. Here, we focus on the Dual Reciprocity Method where the source term is approxi- mated by radial basis functions, polynomial or trigono- metric functions. Analytic particular solutions are pre- sented for each of these approximations. The Trefftz method is then used to solve the resulting homogenous equation obtained after the approximate particular solu- tion is subtracted off. Two types of Trefftz bases are con- sidered, F-Trefftz bases based on the fundamental solu- tion of the modified Helmholtz equation, and T-Trefftz bases based on separation of variables solutions. Var- ious techniques for satisfying the boundary conditions are considered, and a discussion is given of techniques for mitigating the ill-conditioning of the resulting linear systems. Finally, some numerical results are presented il- lustrating the accuracy and efficacy of this methodology.