On modeling subsurface flow using a novel hybrid Trefftz-MFS method

Abstract

In this paper, a novel hybrid Trefftz–MFS method for modeling subsurface flow problems is presented. The proposed method constructs its nonsingular basis function as a series using T functions from the Trefftz method instead of using the singular solution in the method of fundamental solutions (MFS). Numerical solutions are approximated by superpositioning basis functions that are expressed in terms of many source points. Because of the use of the nonsingular basis function, the position of the source point in the proposed method is not sensitive to the results; thus, it resolves a major issue in the MFS for finding a satisfactory location for the source point. Additionally, the order of the nonsingular basis function can be reduced, because the addition theorem is used for approximating the solution for many source points. Consequently, the ill-posedness resulting from the adoption of higher order terms for the solution with only one source point in the collocation Trefftz method (CTM) is mitigated. The proposed method is validated for several test problems. Application examples are also performed. The results reveal that this novel method provides a promising solution that combines the benefits of the CTM and the MFS, such as the boundary collocation only and high accuracy. Moreover, the limitations to the practical application of the CTM and the MFS can be overcome using the proposed method.