The Fourier method for nonsmooth initial data

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.