Rational approximation for initial boundary problems with the fronts
A spectral method with adaptive rational approximation is proposed. In traditional spectral polynomial interpolation, the interpolation points are fixed, usually at the roots or extrema of orthogonal polynomials. Free selection of interpolation points is impossible due to the effect described in the Runge example. The key feature of rational interpolation is the free distribution of interpolation nodes without the occurrence of the Runge phenomenon. Nevertheless, in practice it is very important to implement rational approximation effectively. Here rational approximation is implemented using the barycentric Lagrange form. This leads to fast computations and numerical stability comparable with the polynomial interpolation. It is shown that rational interpolation has significant advantages over polynomial on functions that have singularities in the form of fronts. The key idea is that rational interpolation allows adapting interpolation points according to function singularities. An effective method of grid adaptation that accounts for singularity location was used. Method was generalized to the case of several singularities, for example, for solutions with several fronts. For the solutions of the Burgers equation with singularities in the form of fronts, it is shown that rational interpolation has significant advantages over polynomial. The implementation of spectral method is described, and calculations results on model problems, including problems with two fronts, are presented.