Filtering aspects

Abstract: In this report, a Fourier spectral approximation of the solution to the linear convection--diffusion equation for initial conditions of different smoothness, and for Burger's equation for the initial condition f(x) = sin(x), was constructed, and implemented. Three different filters (Cesàro, Lanczos, and 4--th order exponential cutoff) were either applied to the initial condition or to the numerical approximation after the last integration step has been performed. The local error was then calculated in order to compare the performance of the three filters. Filtering was found to improve the local accuracy of the numerical approximation for the linear convection--diffusion equation for a diffusivity constant of 0 and for initial conditions of low smoothness, i.e. discontinuous functions or functions in C^0 or C^1. For initial conditions of infinite smoothness and a larger diffusivity constant, filtering did not improve the local accuracy but rather made it worse. No significant difference was found between filtering the initial condition and filtering after the final integration step. The 4--th order exponential cutoff performed best overall for most initial conditions. For Burger's equation, filtering only improved the local accuracy when applied after the final integration step and only if a discontinuity had started to form. For a discontinuity to form, the diffusivity constant furthermore needed to be sufficiently small. In conclusion, filtering is applicable when solving the linear convection--diffusion equation for a low diffusivity constant and initial conditions of low smoothness. For Burger's equation, filtering is applicable after a discontinuity starts to form. These results were in line with the theory presented in the report.