Radial basis function partition of unity procedure combined with the reduced-order method for solving Zakharov–Rubenchik equations

Abstract

A meshless radial basis function based on partition of unity (RBF-PU) method is proposed to solve Zakharov–Rubenchik equations. In this local method, the domain is split into overlapping patches forming a covering of it and also, it provides accurate results for PDEs. Time discretization is performed using a second-order implicit explicit backward difference method (IMEX-BDF2). Although the proper orthogonal decomposition (POD) is applied to reduce the dimension of the governing model, the computational complexity of the reduced model for nonlinear terms still depends on the number of variables of the full model. To overcome this subject, we employ the discrete empirical interpolation method (DEIM). Two problems with different situations are solved by the proposed method and the comparison of numerical findings with the conservative compact difference scheme and RBF-FD method shows that the presented method provides accurate results at a low computing cost.