The polynomial Trefftz method for solving backward and inverse source wave problems

The Trefftz method is a truly meshless boundary-type method, because the trial solutions automatically satisfy the governing equation. In order to stably solve the high-dimensional backward wave problems and the one-dimensional inverse source problems, we develop a multiple-scale polynomial Trefftz method (MSPTM), of which the scales are determined a priori by the collocation points. The MSPTM can retrieve the missing initial data and unknown time varying wave source. The present method can also be extended to solve the higher-dimensional wave equations long-term through the introduction of a director in the two-dimensional polynomial Trefftz bases. Several numerical examples reveal that the MSPTM is efficient and stable for solving severely ill-posed inverse problems of wave equations under large noises.