Galerkin-compact finite difference residual corrections for nonlinear second order wave equations

Abstract

In this study, numerical solutions for second-order nonlinear wave equations were obtained, and then were subsequently refined the numerical solution using residual corrections. The proposed method uniquely combines the Galerkin method with the compact finite difference (CFD) method, representing a novel approach in this field. We initially develop a rigorous formulation of second-order wave equations using the Galerkin weighted residual method and obtain numerical results. To solve these equations, third degree Bernstein polynomials are utilized as basis functions in the trial solution. Then we apply our proposed residual correction scheme to refine the numerical solution where fourth-order CFD method are used to solve the associated error wave equations in compliance with the error boundary, and initial conditions. The improved approximations are obtained by adding error values that were obtained based on the estimates of the error wave equation to the weighted residual values. Here Galerkin method and residual correction come together and produce highly accurate results. We also discuss the stability and convergence analysis of our proposed residual correction scheme. Numerical outcomes and absolute errors are compared with exact solutions and solutions found in published literature numerically for different values of space and time step sizes to verify our proposed residual correction scheme. High precision is obtained in case of residual corrections.