The application of the spectral method to nonlinear wave propagation

We integrate the Korteweg-de Vries/Burgers equation numerically by using the spectral and pseudospectral method, respectively. Comparing the results with analytic solutions, we show that the aliasing interactions within the pseudospectral method lead to errors increasing in time, while the spectral method gives the correct time evolution. It is shown both analytically and by the numerical solutions that three invariants of the Korteweg-de Vries equation are conserved by both; therefore the number of invariants of any scheme is not decisive for a good approximation of the continuous solutions. Finally, we apply the spectral method to calculate the time evolution of turbulent sound waves in one and two space dimensions.