Innovative Hybrid Numerical Schemas for Partial Differential Equations with Variable Coefficients

Abstract

    Abstract This research aims to study and develop innovative hybrid numerical schemes for solving partial differential equations with variable coefficients. These equations are among the most widely used in the mathematical modeling of numerous physical and engineering phenomena, such as heat transfer, fluid dynamics, wave propagation, and heterogeneous materials. The difficulty with these equations lies in their inclusion of variable or discontinuous coefficients, which complicates analytical solutions and presents numerical challenges that affect the accuracy and stability of traditional methods. The research addresses the theoretical foundations of partial differential equations, including their classification, weak formulation, and Sobolev spaces. It also examines the stability and convergence characteristics of classical numerical methods, such as the finite difference method, the finite element method, and the finite volume method. Furthermore, it analyzes the shortcomings of these methods when dealing with multiscale problems or those involving highly variable coefficients. To address these challenges, a hybrid numerical scheme is proposed, combining the advantages of multiple numerical methods within a unified computational framework. This scheme aims to improve numerical accuracy, reduce computational costs, and enhance stability. The proposed method relies on using finite elements in complex and irregular regions, while employing finite differences in regular regions to reduce computation time. Through theoretical analysis and numerical experiments, the study demonstrated that the proposed hybrid method possesses superior ability to handle sharp variations in coefficients and heterogeneous media compared to traditional methods. It also achieved more stable and accurate results with higher computational efficiency. These findings underscore the importance of developing hybrid numerical methods in applied mathematics and scientific computing to address complex engineering and physical problems. .