Spacetime Wavelet Method for the Solution of Nonlinear Partial Differential Equations

TL;DR

This work introduces a spacetime high-order wavelet solver for nonlinear PDEs with shocks, combining a priori error estimates and a Newton-Raphson framework to solve space-time systems without sequential time stepping. A novel wavelet-based recursive initialization builds viable initial guesses and improves system conditioning, enabling robust convergence for problems with steep gradients. Demonstrations on Burgers' equation variants and the Sod shock-tube/Navier–Stokes system show high-order convergence for the solution and all spatial/temporal derivatives, and strong parallel performance from simultaneous space-time discretization. The method provides a pathway to accurate, scalable multiscale simulations where traditional time-m marching methods struggle with stability, timestep restrictions, or expensive adaptivity.

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 solving 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 problems with steep gradients in both the spatial and temporal directions with a priori error estimates.

Figures (19)

• Figure 1: Wavelet synthesis of initial guess $\hat{\mathcal{F}}^{i = 0, \ j+1}$ from previous solution $\hat{\mathcal{F}}^{j}$.
• Figure 2: Spacetime solution and profile for shock advection problem with $c(t) = 1$ at $j=7$ (Eq. (\ref{['eq:modBurg']})) and $p_x = 6$, $p_t = 4$.
• Figure 3: Solution and derivative convergence for the shock advection problem with $c(t) = 1$ at $j =5, 6, 7$.
• Figure 4: $6$th-order solution and derivative convergence for $j = 5,6,7$ with $p_x=p_t=8$.
• Figure 5: Quadratic convergence rate of Newton-Raphson method while solving the shock advection problem.