High-Order Combined Compact Finite Difference Method to Solve Partial Differential Equations

In this study, we introduce an 8th-order combined compact finite difference (CCD) method for solving partial dif-ferential equations (PDEs). By employing the 8th-order CCD approach to discretize spatial derivatives, we transform PDEs into a system of ordinary differential equations (ODEs). We then apply the RK(8)7 method to solve these ODEs efficiently. Our analysis demonstrates the convergence of this technique and highlights its accuracy through various numerical examples. These findings indicate the potential of the proposed high-order CCD method for delivering precise solutions to complex PDEs, making it a valuable tool for numerical analysis and a wide range of applications.