One-dimensional heat and advection-diffusion equation based on improvised cubic B-spline collocation, finite element method and Crank-Nicolson technique

Abstract

In this study, we numerically investigate the solution of one-dimensional heat and advection-diffusion equation by employing improvised quartic order cubic B-spline using Crank-Nicolson method and finite element method (FEM). The numerical technique discretizes the spatial derivatives variables using Crank-Nicolson method and time derivatives using finite element method. It is demonstrated that the method is unconditionally stable with Von-Neumann process and accurate to convergence order Oh4+∆t2. The numerical results are simulated and error bounds are calculated using finite difference scheme on a piecewise uniform mesh. Finally, to support our claim, five test problems are considered and the experimental results also compared to existing methods using MATLAB as well as MATHEMATICA software tools. The error norms L2,L∞,root mean square (RMS) error with computational time and order of convergence are investigated for each example. The method is computationally efficient with less memory storage. The accuracy and efficiency of present scheme is measured in terms of absolute errors and also compared with analytical and published results in literature.