Comparison of Numerical Methods for the Calculation of Two-Dimensional Turbulence

The estimate derived by Henshaw, Kreiss, and Reyna for the smallest scale present in solutions of the two-dimensional incompressible Navier-Stokes equations is employed to obtain convergent pseudospectral approximations. These solutions are then compared with those obtained by a number of commonly used numerical methods. If the viscosity term is deleted and the energy in high wave numbers removed by setting the amplitudes of all wave numbers above a certain point in the spectrum to zero, the "chopped" solution differs considerably from the convergent solution, even at early times. In the case that the regular viscosity is replaced by a hyperviscosity term, i.e., the square of the Laplacian, we also derive an estimate for the smallest scale present. If the coefficient of hyperviscosity is chosen so that the spectrum of the hyperviscosity solution disappears at the same point as for the regular viscosity solution, the hyperviscosity solution is also completely different from the convergent solution. If we "tune" the hyperviscosity coefficient, then the solutions are similar in amplitude or phase, but not both. The solution obtained by a second-order difference method with twice the number of points as the pseudospectral model, or a fourth-order difference method with the same number of points as the pseudospectral model, is essentially identical to the convergent solution. This is reasonable since most of the energy of the solution is contained in the lower part of the spectrum.