Spacetime Wavelet Method for the Solution of Nonlinear Partial Differential Equations

Abstract

We propose a high-order spacetime wavelet method for the solution of nonlinear partial differential equations with a user-prescribed accuracy. The technique utilizes wavelet theory with a priori error estimates to discretize the problem in both the spatial and temporal dimensions simultaneously. We also propose a novel wavelet-based recursive algorithm to reduce the system sensitivity stemming from steep initial and/or boundary conditions. The resulting nonlinear equations are solved using the Newton–Raphson method. We parallelize the construction of the tangent operator along with the solution of the system of algebraic equations. We perform rigorous verification studies using the nonlinear Burgers' equation. The application of the method is demonstrated by solving the Sod shock tube problem using the Navier–Stokes equations. The numerical results of the method reveal high-order convergence rates for the function as well as its spatial and temporal derivatives. We solve multiscale problems with steep gradients in both the spatial and temporal directions with a priori error estimates.