Numerical Schemes for Partial Difference Equation in Physics

This paper examines various numerical schemes for 1-D and 2-D advection and diffusion equations using MATLAB, focusing on stability, accuracy, and performance under different boundary conditions [1]. For 1-D advection, methods such as the upwind, implicit up wind, Beam-Warming(B-W), Lax-Friedrichs(L-F), and Lax-Wendroff(L-W) schemes are evaluated. The implicit upwind scheme delivers consistent results, the Beam-Warming scheme works well under specific conditions, while the upwind scheme shows dissipation and dispersion. In 2-D advection, the upwind and Lax-Friedrichs schemes are tested, with the upwind scheme being more stable with discontinuities but less stable for smooth solutions. For diffusion, the Classical, Dufort-Frankel(D-F), and Crank-Nicolson(C-N) schemes are analyzed. The Crank-Nicolson scheme proves to be the most accurate, while the Classical scheme is fast but mesh-dependent, and the Dufort-Frankel scheme is stable but introduces minor fluctuations. The paper suggests using operator splitting to improve 2-D advection stability.