A Multiresolution Adaptive Wavelet Method for Nonlinear Partial Differential Equations

Abstract

Modern computational science and engineering applications are inherently multiphysics and multiscale. Therefore, novel algorithms are required to solve partial differential equations (PDEs) with features evolving on a wide range of spatial and temporal scales. In this work, we propose a wavelet-based method which is ideally suited for problems developing localized structures, which might occur intermittently anywhere in the computational domain, or change their locations and scales in space and time. Wavelet-based numerical methods have been shown to be efficient for modeling multiscale and multiphysics problems because they provide spatial adaptivity through the use of multiresolution basis functions. In our work, we develop the Multiresolution Wavelet Toolkit (MRWT) that requires far fewer unknowns than other algorithms when applied to problems with a great range of spatial and temporal scales. In addition, the computational grid can be refined locally and our a priori error estimates safeguard the accuracy of the numerical solution. MRWT is designed to execute in a hybrid CPU/GPU environment and scale to ~10^3-10^4 cores. Consequently, MRWT provides high fidelity simulations with significant data compression. We explain how this technique uses differentiable wavelet basis functions and second-generation wavelets, to solve nonlinear PDEs on finite domains. Moreover, we provide a priori error estimates for the wavelet representation of fields, their derivatives, and the aliasing errors associated with the nonlinear terms in PDEs. Then, by projecting fields and spatial derivative operators onto the wavelet basis, our estimates are used to construct a sparse multiresolution spatial discretization which guarantees the prescribed accuracy for each field. Additionally, MRWT utilizes a predictor-corrector procedure within the time advancement loop to dynamically adapt the computational grid, maintaining the prescribed accuracy of the solutions of the PDEs as they evolve. We show verification of the MRWT algorithm, demonstrating mathematical correctness and physics simulation capabilities. Furthermore, spatial convergence is achieved at a rate which agrees with a priori estimates. Finally, we apply MRWT to problems with shocks in gasses and to high-strain rate damage nucleation and propagation in nonlinear solids.