Assessing the influence of numerical diffusion on the accuracy of numerical solution of the shallow water equations

To control the nonlinear numerical instability, throughout the time evolution of the Eulerian form of the nonlinear rotating shallow water equations, it is necessary to add numerical diffusion to the solution. It is clear that, this extra numerical diffusion degrades the accuracy of the numerical solution and should be kept as small as possible. In a conventional approach a hyper-diffusion is used to maintain the numerical stability. In the present work, the influence of using different orders of hyper-diffusion on the accuracy of the numerical solution of the shallow water equations generated by some high-order numerical methods is examined. Furthermore, application of an eighth-order compact spatial filter as an alternative way to provide the numerical diffusion is considered.
In the present work, the vorticity-divergence-mass representation of the shallow water equations is considered for numerical simulation. To advance the governing equations in time the semi-implicit approach combined with the three level leapfrog method for the time discretization of the temporal derivatives, is used. The second-order central, the fourth-order compact, the sixth-order super compact finite difference and the pseudo-spectral methods are applied to spatial differencing of the shallow water equations.
For a quantitative assessment of accuracy, the global measures of the distribution of mass between potential vorticity iso-levels, called mass error, is used. In addition, based on the results of some recent investigations, decomposing a flow into a balanced part representing vortical flow and an unbalanced part representing freely propagating inertia-gravity waves has found significant usefulness in the accuracy analysis of the numerical solution of the primitive equations. Therefore, in this work the representation of balance and imbalance are also used for quantitative assessment of the numerical accuracy.
It is found that the numerical diffusion plays a crucial role in the accuracy of numerical solution. Results show that using lower order hyper-diffusion terms degrades the numerical accuracy. Furthermore, results show that using higher orders of hyper-diffusion for the sixth-order super compact and pseudo-spectral method is essential. In addition, it is observed that based on the quantitative measures of the mass error, using a sixth-order hyper-diffusion term or using the eighth-order compact spatial filter has nearly a similar effect on the numerical accuracy of the shallow water equations generated by the sixth-order super compact finite difference and the pseudo-spectral methods. The same conclusion is valid based on the quantitative measures of the imbalance in particular for hyper-diffusion terms with orders greater than two.