A radial basis function finite difference (RBF-FD) method for numerical simulation of interaction of high and low frequency waves: Zakharov-Rubenchik equations

Abstract In this study, we examine numerical solutions of Zakharov–Rubenchik system which is a coupled nonlinear partial differential equation. The numerical method in the current study is based on radial basis function finite difference (RBF-FD) meshless method and an explicit Runge–Kutta method. As a radial basis function we choose polyharmonic spline augmented with polynomials. The essential motivation for choosing polyharmonic spline is that it is free of shape parameter which has a crucial role in accuracy and stability of meshless methods. The main benefit of the proposed method is the approximation of the differential operators is performed on local-support domain which produces sparse differentiation matrices. This reduces computational cost remarkably. To see performance of the proposed method, some test problems are solved. L ∞ error norms and conserved quantities such as mass and energy are calculated. Numerical outcomes are presented and compared with other methods available in the literature. From the comparison it can be deduced that the proposed method gives reliable and precise results with low computational cost. Stability of the proposed method is also discussed.

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