Development of Eight- and Tenth-Order Higher-Order Compact Finite Difference Scheme for solving system of Nonlinear two-point Boundary Value Problems

Abstract: Boundary value problems (BVPs) frequently arise in applied mathematics, physics, and various fields of engineering, where they serve as models for diverse physical and chemical processes. Obtaining analytical solutions for nonlinear BVPs is often challenging, thus necessitating robust and accurate numerical techniques. Higher-order compact finite difference schemes (HOCFDS) provide an efficient framework for approximating such problems with high accuracy. This paper presents the derivation, analysis, and application of new eighth- and tenth-order compact finite difference schemes for solving systems of nonlinear two-point boundary value problems. The proposed schemes are designed to achieve both superior accuracy and computational efficiency in handling single and coupled nonlinear BVPs. The derivation is based on Taylor series expansion, while stability and convergence are established using modified von Neumann analysis, the energy method, and consistency analysis. To demonstrate the effectiveness of the methods, several nonlinear test problems—including two coupled systems—are considered. The numerical performance of the new schemes is compared with that of the standard central difference method, fourth- and sixth-order compact schemes. Results show that the proposed higher-order compact schemes consistently deliver enhanced accuracy, improved convergence properties, and robustness in handling complex nonlinearities and boundary conditions.