Fast wavelet-Taylor Galerkin method for linear and non-linear wave problems

In this paper, we extend a wavelet-Taylor Galerkin method for linear and non-linear wave problems introduced in [B.V. Rathish Kumar, Mani Mehra, Wavelet-Taylor Galerkin method for Burger’s equations, BIT Numer. Math. 45 (3) (2005) 543–560, B.V. Rathish Kumar, Mani Mehra, Time accurate solution of Korteweg–de Vries equation using wavelet Galerkin method, Appl. Math. Comput. 162 (1) (2005) 447–460], where the authors have taken the advantage of the good approximation properties of the wavelets but not the useful feature of wavelets to compress the solution with a localized structure, which might occur intermittently anywhere in the computational domain. Here this method takes the advantage of the wavelet bases capabilities to compress operators and sparse representation of solutions with a localized structure, which occur in wave problems. We illustrate the performance of the method through linear advection–diffusion problem and non-linear waves problems taken from shallow water theory and non-linear optics.