A localized RBF meshfree method for the numerical solution of the KdV-Burger's equation

This paper formulates a local Radial Basis Functions (LRBFs) collocation method for the numerical solution of the non-linear dispersive and dissipative KdV-Burger's (KdVB) equation. This equation models physical problems, such as irrotational incompressible flow, considering a shallow layer of an inviscid fluid moving under the influence of gravity and the motion of solitary waves. The local type of approximations used, leads to sparse algebraic systems that can be solved efficiently. The Inverse Multiquadrics (IMQ), Gaussian (GA) and Multiquadrics (MQ) Radial Basis Functions (RBF) interpolation are employed for the construction of the shape functions. Accuracy of the method is assessed in terms of the L₂ and L_∞perror norms and three conservative properties related to mass, momentum and energy. Additionally we investigate how both the accuracy and the stability of the proposed method are affected from the number of nodes in the support domain, the parameter dependent RBFs, the condition number of the resulting algebraic systems and finally the time step length. Numerical experiments demonstrate the accuracy and the robustness of the method for solving nonlinear dispersive and dissipative problems, while stability analysis demonstrates that the numerical scheme is conditionally stable.