A third-order upwind scheme for the advection–diffusion equation using spreadsheets

Abstract

In this study, numerical solution of advection–diffusion equation with third-order upwind scheme by using spreadsheet simulation (ADE-TUSS) is carried out. This is a user-friendly and a flexible solution algorithm for the numerical solution of the one dimensional advection–diffusion equation (ADE). The ADE-TUSS algorithm is based on the description of ADE by using the third-order upwind scheme (TU) for advection term and second-order central finite representation. For the solution of the governing equations, spreadsheet simulation (SS) technique is used instead of conventional solution techniques. The solution of ADE can be obtained for explicit, implicit and Crank–Nicolson schemes by only changing the values of temporal weighted parameter in the ADE-TUSS algorithm. It is clear in the literature that numerical diffusion causes great deviations in the model results when the first- or second-order upwind schemes are used. In order to decrease the numerical diffusion and to obtain oscillation free results, an artificial diffusion term is usually defined or sizes of the time step and/or grid sizes are set small values. Reduce of the grid sizes and/or time step increases the computational time and generally requires writing a fairly complex code when high order schemes are used. However, numerical solution of ADE by taking into account TU scheme has been carried out by using the iterative spreadsheet solution technique in the proposed solution algorithm. In order to simulate transient solution, a simple macro that carries out time cycle is defined with the help of VBA feature of spreadsheets. One of the most important advantages of ADE-TUSS algorithm is that it does not require the matrix algebra at each time step of the transient solutions. In order to test the ADE-TUSS model, two examples having numerical and analytical solutions are solved. Results showed that use of the high-order schemes in the spreadsheet simulation is very applicable for the numerical solution of ADE. Moreover, numerical diffusion problem is drastically prevented by using the ADE-TUSS model.