Temporal Oscillations

The previous chapter shows that the wave continuity equation suppresses node-to-node oscillations in space. For a centered three time level approximation of the momentum equations there remain, however, node-to-node oscillations in time for the velocity solution (Kinnmark and Gray, 1982). It is shown in Section 8.2 that three time level momentum equations introduce an additional non-physical root, a numerical artifact. By using different three time level approximations of the momentum equations, the magnitude of this numerical artifact can however be made smaller, as shown in Section 8.3. Except for the nonlinear convective term we do however preserve second order accuracy in time. Finally it is shown, in Section 8.4, that a two time level, Crank-Nicolson type approximation of the momentum equations completely eliminates the numerical artifact. We still maintain second order accuracy in time, except for the non-linear convective terms. If lumping, through integration, is applied to the momentum equations, velocities are computed from a simple block diagonal matrix equation.