The Fourier method for nonsmooth initial data

TLDR: In this paper, the Fourier method is applied to very general linear hyperbolic Cauchy problems with nonsmooth initial data, and it is shown that applying appropriate smoothing techniques applied to the equation gives stability and that this smoothing combined with a certain smoothing of the initial data leads to infinite order accuracy away from the set of discontinuities of the exact solution modulo a small easily characterized exceptional set.

Abstract: Application of the Fourier method to very general linear hyperbolic Cauchy problems having nonsmooth initial data is considered, both theoretically and computationally. In the absence of smoothing, the Fourier method will, in general, be globally inaccurate, and perhaps unstable. Two main results are proven: the first shows that appropriate smoothing techniques applied to the equation gives stability; and the second states that this smoothing combined with a certain smoothing of the initial data leads to infinite order accuracy away from the set of discontinuities of the exact solution modulo a very small easily characterized exceptional set. A particular implementation of the smoothing method is discussed; and the results of its application to several test problems are presented, and compared with solutions obtained without smoothing. Introduction. In recent years the Fourier method for the numerical approximation of solutions to hyperbolic initial value problems has been used quite successfully. In fact, if the initial function is C°° and the coefficients of the equation are constant the method converges arbitrarily fast, i.e. is limited in practice only by the method of time discretization which is chosen. This is the reason that the Fourier method is caled "infinite order" accurate. However, the situation is drastically different when the initial function is not smooth. We take as a model the one space dimension scalar problem ut = ux to be solved for 2ir periodic u on the interval n < x < n with initial values <fix), having a simple jump discontinuity at x = 0, but otherwise smooth and 2tt periodic. In this simple example the rate of convergence is globally only second order. (In fact, if any value for i^(0) except the average of the right and left limits is chosen the method degenerates further to be globally only first order.) This means that even in regions where the exact solution is smooth, i.e. away from the line x = t, the error is 0(h2), where h is the mesh width. The analysis of this and related examples is carried out in Section 1 of this paper. There we show that such a large global error occurs in general situations. However, we note here that in certain constant coefficient problems special time discretizations may give better accuracy on mesh points than predicted for the semidiscrete problem. We analyze this phenomenon, which we believe is limited to constant coefficient problems,in the next section. Thus, even in the simplest cases, discontinuous initial data causes a large error unless we modify the Fourier method. Received July 25, 1977; revised February 15, 1978. AMS (MOS) subject classifications (1970). Primary 65M10; Secondary 35L45.