A multiple-direction Trefftz method for solving the multi-dimensional wave equation in an arbitrary spatial domain

In this paper we first express the wave equation in terms of the Minkowskian polar coordinates and generate a set of complete hyperbolic type Trefftz bases: rkcosh⁡(kθ) and rksinh⁡(kθ), which are further transformed to wave polynomials as the trial solution bases for the one-dimensional wave equation. In order to stably solve the wave propagation problems long-term we develop a multiple-scale Trefftz method (MSTM), of which the scales are determined a priori by the collocation points. Then we derive a very simple method of multi-dimensional wave polynomials, equipped with different spatial directions which being the normalized wavenumber vectors, as the polynomial Trefftz bases for solving the multi-dimensional wave equations, which is named a multiple-direction Trefftz method (MDTM). Several numerical examples of two- and three-dimensional wave equations demonstrate that the present method is efficient and stable.