A HIGH ACCURACY NUMERICAL METHOD FOR SOLVING THE NONLINEAR KORTEWEG-DE VRIES EQUATION

The aliasing which was noted and discussed by N. A. Philips in the numericalsolution of a nonlinear equation is discussed, using the Walsh-Hadamard transform scheme,and it is pointed out that errors will appear, if we leave out the aliasing terms. For a long time the Korteweg-de Vries equation has been integrated numerically, usingthe second-order difference scheme, the second-order Shuman’s difference scheme, the spectralderivative scheme (the pseudospectral scheme), the Walsh-Hadamard-Fourier transform scheme,the spectral transform scheme as well as the spectral appended-zero transform scheme,respectively. Comparing the numerical solutions with the analytic solution at the 9600th timestep, we find that: (i) in these numerical methods for solving the Korteweg-de Vries equa-tion we have not discovered the nonlinear computational instability induced by the aliasingterms (the folding terms); (ii) the accuracy of the spectral derivative scheme, the Walsh-Hadamard-Fourier transform scheme and the spectral transform scheme, in which the deriva-tives are given, using the truncated Fourier ttansform scheme and the aliasing terms areretained, are higher than that of the other schemes; (iii) the application of the spectralappended-zero transform scheme, in which the aliasing terms are removed, leads to the loss ofaccuracy and increase in computational amount; and (iv) the second-order difference schemesare less accurate and show both large amplitude and lagging phase errors.