Implicit finite difference techniques for the advection-diffusion equation using spreadsheets

摘要

This study proposes one-dimensional advection-diffusion equation (ADE) with finite differences method (FDM) using implicit spreadsheet simulation (ADEISS). By changing only the values of temporal and spatial weighted parameters with ADEISS implementation, solutions are implicitly obtained for the BTCS, Upwind and Crank-Nicolson schemes. The ADEISS uses iterative spreadsheet solution technique. Thus, it is not required a solution of simultaneous equations for each time step using matrix algebra. Two examples which, have the numerical and analytical solutions in literature, are solved in order to test the ADEISS performance. Both examples are solved for three schemes. It has been determined that the Crank-Nicolson scheme is in good agreement with the analytical solution; however the results of the BTCS and the Upwind schemes are lower than the analytical solution. The Upwind scheme suffers from considerably numerical diffusion whereas the BTCS scheme does not produce numerical diffusion. Thus, it provided better results than Upwind scheme which are closer to analytical results depending on the selected parameters. The ADEISS implementation is a computationally convenient procedure for the three well-known methods in the literature: The BTCS, Upwind and Crank-Nicolson.