Stability and Convergence Analysis of Fully Discrete Fourier Collocation Spectral Method for 3-D Viscous Burgers' Equation

This paper analyzes the stability and convergence of the Fourier pseudospectral method coupled with a variety of specially designed time-stepping methods of up to fourth order, for the numerical solution of a three dimensional viscous Burgers’ equation. There are three main features to this work. The first is a lemma which provides for an L2 and H 1 bound on a nonlinear term of polynomial type, despite the presence of aliasing error. The second feature of this work is the development of stable time-stepping methods of up to fourth order for use with pseudospectral approximations of the three dimensional viscous Burgers’ equation. Finally, the main result in this work is that the pseudospectral method coupled with the carefully designed time-discretizations is stable provided only that the time-step and spatial grid-size are bounded by two constants over a finite time. It is notable that this stability condition does not impose a restriction on the time-step that is dependent on the spatial grid size, a fact that is especially useful for three dimensional simulations.