Solutions of One Dimensional Parabolic Partial Differential Equations: An Improved Finite Difference Approach

This paper introduces a finite difference scheme derived from the classical Crank-Nicolson method. The proposed scheme offer an improved spatial accuracy while maintaining the second-order temporal accuracy of the original Crank-Nicolson scheme. The higher order of spatial accuracy leads to improved convergence properties. The consistency and stability of the new scheme are analyzed using Taylor series expansion and von Neumann stability analysis, respectively. To validate the efficiency of the proposed scheme, it is implemented in MATLAB to solve the one-dimensional heat equation. To explore the versatility of the scheme, it is further extended to solve the advection-diffusion equation. Numerical experiments demonstrated on diffusion equation show that the new scheme compares favorably with existing methods in terms of convergence and accuracy. The results of the numerical solutions are presented in tabular form to highlight the accuracy and rates of convergence of the method. In addition, graphical plots of the numerical solutions are provided at different time levels to visualize the behavior of the solution over time and to illustrate the consistency between the numerical and analytical results. These visual and numerical comparisons further emphasize the reliability and precision of the proposed scheme. The combination of improved spatial resolution, solid theoretical foundation, and practical implementation demonstrates the schemeˆ as potential for solving time-dependent partial differential equations efficiently and accurately. This makes the scheme a valuable contribution to the field of numerical methods for parabolic-type equations.