Parallel adaptive wavelet collocation method for PDEs

Abstract

Please cite this article in press as: A. Nejadmalayeri et al., Parallel adaptive wavelet collocation method for PDEs, J. Comput. Phys. (2015), http://dx. AbstractA parallel adaptive wavelet collocation method for solving a large class of Partial Differential Equations is presented. The parallelization is achieved by developing an asynchronous parallel wavelet transform, which allows one to perform parallel wavelet transform and derivative calculations with only one data synchronization at the highest level of resolution. The data are stored using tree-like structure with tree roots starting at a priori defined level of resolution. Both static and dynamic domain partitioning approaches are developed. For the dynamic domain partitioning, trees are considered to be the minimum quanta of data to be migrated between the processes. This allows fully automated and efficient handling of non-simply connected partitioning of a computational domain. Dynamic load balancing is achieved via domain repartitioning during the grid adaptation step and reassigning trees to the appropriate processes to ensure approximately the same number of grid points on each process. The parallel efficiency of the approach is discussed based on parallel adaptive wavelet-based Coherent Vortex Simulations of homogenous turbulence with linear forcing at effective non-adaptive resolutions up to 2048 3 using as many as 2048 CPU cores.

Effects of isothermal stratification strength on vorticity dynamics for single-mode compressible Rayleigh-Taylor instability

“…where φ k are scaling functions on the coarsest level, c k are the corresponding coarse-level wavelet coefficients, ψ l are the scaling interpolating functions on any arbitrary level, d l are the coefficients to which the thresholding is applied, l and k represent physical grid points, and α and j represent the wavelet family and level of resolution, respectively [41,42]. The effect of setting any one of the coefficients d l to zero is the removal of a grid point at that level of resolution.…”

“…The effective resolution is set by a base grid size and the limit put on j (referred to as j max herein). This results in the error being O(ε) and the resolution in a single direction being p · 2 (jmax−1) , where p is the base resolution [41][42][43].…”

Wall-resolved wavelet-based adaptive large-eddy simulation of bluff-body flows with variable thresholding

“…The numerical solutions have been obtained by employing the fourth-order AWC method, the linearized Crank-Nicolson split-step time-integration method with adaptive time stepping and the parallel version of the AWC solver (Nejadmalayeri et al 2015). The spatial discretization of the computational domain Ω is achieved by employing eight nested wavelet-collocation grids for the wavelet decomposition (2.5), where J = 8.…”

Adaptive Wavelet Algorithm for Solving Nonlinear Initial–boundary Value Problems With Error Control

“…These algorithms achieve spatial adaptivity with multiresolution wavelet basis functions (Jawerth and Sweldens, 1994). Notable accomplishments of wavelet solvers include: significant data compression (Bertoluzza, 1996;Beylkin and Keiser, 1997;Liandrat and Tchamitchian, 1990), bounded energy conservation (Qian and Weiss, 1993;Ueno et al, 2003), modeling stochastic systems (Kong et al, 2016), and solving coupled systems of nonlinear PDEs (Dubos and Kevlahan, 2013;Nejadmalayeri et al, 2015;Paolucci et al, 2014a,b;Sakurai et al, 2017). While past solvers have had many successes, they are not without shortcomings.…”