Quantification of numerically induced mixing and dissipation in discretisations of shallow water equations

摘要

The concepts of numerical mixing and numerical dissipation are developed here for one-dimensional advection-diffusion equations for velocity and a scalar tracer where depth changes due to transport divergence are considered. It is shown for the explicit first-order upwind (FOU) scheme that the numerical mixing of the scalar tracer due to the discretisation of the tracer advection (i.e., the decay of the tracer variance) can be calculated by the difference of the advected tracer square and the square of the advected tracer, divided by the time step. This method is generalised for other advection schemes. An equivalent method is introduced for the numerical dissipation which is a kinetic energy loss due to the discretisation of the velocity advection. In a one-dimensional application the performance of both analysis methods is demonstrated by comparing the numerical mixing and dissipation to physical mixing and dissipation for two advection schemes, which are the FOU and a monotone TVD scheme. It is finally discussed how the one-dimensional analysis method can be generalised for three-dimensional primitive equation models of the coastal ocean.