A spectral method for the numerical solution of hyperbolic partial differential equations is presented. Any discontinuous solutions which may occur are assumed to be the sum of a step function and a smooth function. 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 discontinuities across cells. We evaluate the method by its application to three standard test problems involving the scalar wave equation, the inviscid Burgers equation and the Euler equations of gas dynamics. For all experiments we observe an absence of the Gibbs phenomenon with discontinuities captured to within a single mesh interval and high accuracy is observed in smooth regions.
Crossley, P. S., Saunders, R., Causon, D. M., & Mingham, C. G. (1996). A spectral method with subcell resolution for shock wave calculations. Applied Numerical Mathematics, 21(2), 141–153. https://doi.org/10.1016/0168-9274(95)00134-4