Fourier spectral methods for Degasperis-Procesi equation with discontinuous solutions
The Degasperis-Procesi (DP) equation is an integrable partial differential equation which models propagation of nonlinear dispersive waves in shallow water in the small amplitude and long wavelength regime. It belongs to the same family as the more familiar Korteweg-de Vries and Camassa-Holm equations, although it has some distinctive features by itself. In particular, it has not only peakon solutions, but also shock waves. This lack of smoothness of the solution constitutes an additional difficulty when designing stable and high-order numerical schemes. This is particularly true in the case of spectral methods, since the presence of discontinuities in nonlinear partial differential equations leads usually to instabilities of the scheme due to the nonlinear mixing of Gibbs oscillations with the approximate solution. In this paper, the Fourier-Galerkin and collocation spectral methods for the DP equation are developed and analyzed. It is shown that both schemes are \(L^2\) stable: if \(u_h(\cdot,t)\) denotes the semidiscretized solution obtained by the spectral method, then \(\|u_h(\cdot,t)\|_{L^2} \leq 2 \|u_h(\cdot,0)\|_{L^2}\). To deal with discontinuities and shocks, an exponential spectral filter is introduced at each time step to stabilize the schemes when the shock is formed in the equation. This shock is accurately resolved by a Gegenbauer reconstruction of an exponentially convergent approximation to a piecewise smooth function using information at the final time step only. The spectral method in space is coupled with an explicit total variation Runge-Kutta discretization in time. The overall scheme is shown to perform successfully in a variety of examples involving shocks.
scientific article; zbMATH DE number 6389877
scientific article