NUMERICAL SOLUTION OF 1D CONVECTION-DIFFUSION EQUATION USING IMPLICIT EULER METHOD ON A NON-UNIFORM MESH

Abstract

This thesis focuses on the numerical solution of the one-dimensional convection-diffusion equation using implicit Euler Method implemented on a non-uniform mesh. The convection diffusion equation is a fundamental partial differential equation that arises in various physical and engineering problems involving the transport of mass, heat, or momentum. Accurately solving this equation, particularly in convection dominated regimes, presents significant nu merical challenges such as artificial oscillations and smearing near steep gradients or bound ary layers. To address these issues, a non-uniform mesh is employed to provide higher res olution in regions with rapid variations in the solution, while maintaining coarser discretiza tion where the solution is smoother. An implicit Euler Method is adapted to accommodate variable grid spacing, ensuring enhanced accuracy in both convection and diffusion terms. The resulting system of algebraic equations is solved using appropriate numerical solvers. Comparative analysis with uniform mesh solutions demonstrates that the non-uniform mesh approach significantly improves accuracy and stability, especially in capturing sharp solution features with fewer grid points. The findings of this work contribute to the development of efficient and accurate numerical techniques for solving convection-diffusion problems en countered in scientific computing and engineering applications