An Extrapolated Second-Order BDF Combined with Local Meshfree Radial Point Interpolation Method for Numerical Simulation of Multi-Dimensional Regularized Long Wave Equation Emerging in Fluids

Abstract This study proposes, an accurate method based on a unification of second-order BDF and a local meshfree method to obtain numerical solutions of one-, two-, three- and four-dimensional regularized long wave equations. The regularized long wave equation is a frequently used important nonlinear partial differential equation in modelling of fluids. The proposed method discretizes the temporal variable via a second-order extrapolated BDF approach in first stage. In second stage, space variables are discretized via local meshfree radial point interpolation method based on thin plate splines. In this way, a full discrete scheme is obtained for getting numerical solution of the considered problem. The use of local radial point interpolation method is promising approach for large scale problems due to sparse matrices obtained in discretization of space variables. The stability and convergence analysis of time discrete scheme are investigated through the medium of energy method. To assess performance of the proposed method, acquired numerical outcomes are subjected to comparisons with analytical solutions and with over ten different existing methods in literature such as finite difference, finite element, meshfree and wavelet methods. The numerical outcomes and their comparisons confirm accuracy and reliability of the proposed method.