A SPECTRAL METHOD WITH SUBCELL RESOLUTION FOR SHOCK-WAVE

A spectral method for the numerical solution of hyperbolic partial dif ferential equations is presented. Any discontinuous solutions which ma y occur are assumed to be the sum of a step function and a smooth func tion. A series approximation is then applied to the smooth function in order to eliminate the Gibbs phenomenon. By adding the step function onto the series we then have an accurate approximation to the solution at any given time. When discretising in time, a modification is added to the numerical flux to account for the advection of sharp discontin uities across cells. We evaluate the method by its application to thre e standard test problems involving the scalar wave equation, the invis cid Burgers equation and the Euler equations of gas dynamics. For all experiments we observe an absence of the Gibbs phenomenon with discont inuities captured to within a single mesh interval and high accuracy i s observed in smooth regions.