A Powerful High-Order B-Spline Galerkin Technique for Numerical Solutions of Advection-Diffusion Equation

The present paper proposes a novel numerical approach for solving the advection-diffusion equation using quintic B-spline functions combined with the Galerkin finite element method. The advection-diffusion equation is a fundamental partial differential equation that is used in a variety of scientific and engineering domains, including fluid dynamics, heat transfer, and environmental modeling. The finite difference scheme is applied for time derivative, and Crank-Nicolson scheme is used for space parameters. The proposed approach, which is found to be accurate and effective, is validated with two numerical problems. The numerical results indicate that the present scheme performs well and correctly with the higher-order B-spline functions. To see the performance of the method, the error norms \(L_\infty \) and \(L_2\) are measured.