Influence of Numerical Methods on the Computational Performance of High-Dimensional Partial Differential Equations in Mathematics

Abstract

High-dimensional partial differential equations (PDEs) play a critical role in modeling complex phenomena in science, engineering, finance, and environmental systems. However, solving such equations analytically is often difficult due to their complexity and the large number of variables involved. Consequently, numerical methods are widely employed to obtain approximate solutions. Despite their usefulness, the computational performance of different numerical methods varies significantly when applied to high-dimensional PDEs. This study therefore examined the influence of numerical methods on the computational performance of high-dimensional partial differential equations. The study adopted a computational and comparative research design. Three widely used numerical methods—the Finite Difference Method (FDM), Finite Element Method (FEM), and Spectral Method (SM)—were implemented to solve a benchmark high-dimensional PDE model under identical computational conditions. The performance of these numerical methods was evaluated using key computational indicators including computational time, solution accuracy measured by the L2 error norm, and convergence rate. Data for the study were generated through computational experiments and analyzed using comparative performance analysis and regression techniques. The results revealed among others that, the Spectral Method demonstrated the best overall computational performance among the methods considered. Specifically, the Spectral Method recorded the lowest computational time, the highest solution accuracy, and the fastest convergence rate. The study therefore recommended that Researchers and computational scientists should consider using spectral methods when solving high-dimensional PDEs due to their superior accuracy and computational efficiency.