Exploring Sixth-Order Compact Finite Difference Schemes for Hyperbolic PDE: A Case Study on the Nonlinear Goursat Problem

Abstract

This study underscores the significant potential of using sixth-order compact finite difference method to address the nonlinear Goursat problem, a major challenge in mathematical modeling. This approach paves the way for advancement in numerical studies, particularly through the linearization of the nonlinear Goursat problem, which facilitates the implementation of linear scheme. The research employs numerical analysis to assess the accuracy of this new scheme and discusses the theoretical outcomes of multiple series of numerical experiments. Through error analysis, the Von Neumann method, and Taylor series expansion, the results demonstrate that the proposed method achieves superior accuracy compared to traditional approaches and is unconditionally stable and consistent. Additionally, the study confirms the convergence of the new scheme through the Lax equivalence theorem. Moreover, the research highlights the cost effectiveness of this new method, showcasing its ability to achieve high levels of accuracy with fewer computational resources. By thoroughly examining the underlying principles and real world applications, the significant benefits of utilizing advanced high-order compact finite difference methods are emphasized.