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.

The wavelet-based eddy-capturing approach with variable thresholding is extended to bluff-body flows, where the obstacle geometry is enforced through Brinkman volume penalization. The use of a spatio-temporally varying threshold allows one to perform adaptive large-eddy simulations with the prescribed fidelity on a near optimal computational mesh. The space-time evolution of the threshold variable is achieved by solving a transport equation based on the Lagrangian path-line diffusive averaging methodology. The coupled wavelet-collocation/volume-penalization approach with variable thresholding is illustrated for a turbulent incompressible flow around an isolated stationary prism with square cross-section. Wavelet-based adaptive large-eddy simulations supplied with the one-equation localized dynamic kinetic energy-based model are successfully performed at moderately high Reynolds number. The present study demonstrates that the proposed variable thresholding methodology for wavelet-based modelling of turbulent flows around solid obstacles is feasible, accurate and efficient.

A new framework for spatio-temporally adaptive turbulence simulations is proposed. The method is based on a variable-fidelity representation that tightly integrates numerics and modelling of subgrid-scale turbulence and aims to capture the flow physics on a near-optimal adaptive mesh. The integration is achieved by combining hierarchical wavelet-based computational modelling with spatially and temporally varying wavelet threshold filtering. The proposed approach provides automatic smooth transition from directly resolving all flow physics to capturing only the energetic/coherent structures, which leads to a dynamically adaptive variable-fidelity approach. The self-regulating continuous switch between different fidelity regimes is accomplished through a two-way feedback mechanism between the modelled dissipation and the local grid resolution, which is based on spatio-temporal variation of the wavelet filtering threshold. The proposed methodology systematically accounts for and exploits the spatial and temporal intermittency of turbulence. Thus, it overcomes the major limitation of all existing wavelet-based multi-resolution techniques, namely, the use of a global thresholding criterion. The procedure consists of tracking the wavelet thresholding factor within a Lagrangian frame by exploiting a Lagrangian path-line diffusive averaging approach, based on either interpolation along characteristics or direct solution of the corresponding evolution equation. This new methodology is tested for linearly forced homogeneous turbulence at different Reynolds numbers and provides very promising results on a benchmark with time-varying prescribed level of turbulence resolution.

In view of the ongoing longtime pursuit of numerical approaches that can capture important flow physics of high Reynolds number flows with fewest degrees of freedom, two important wavelet-based multi-resolution schemes are thoroughly examined, namely, the Coherent Vortex Simulation (CVS) and the Stochastic Coherent Adaptive Large Eddy Simulation (SCALES) with constant and spatially/temporarily variable thresholding. Reynolds number scaling of active spatial modes for CVS and SCALES of linearly forced homogeneous turbulence at high Reynolds numbers is investigated in dynamic study for the first time. This dynamic computational complexity study demonstrates that wavelet-based methods can capture flow-physics while using substantially fewer degrees of freedom than both direct numerical simulation and marginally resolved LES with the same level of fidelity or turbulence resolution, defined as ratio of subgrid scale and the total dissipations. The study provides four important observations: (1) the linear Reynolds number scaling of energy containing structures at a fixed level of kinetic energy, (2) small, close to unity, fractal dimension for constant-threshold CVS and SCALES simulations, (3) constant, close to two, fractal dimension for constant-dissipation SCALES that is insensitive to the level of fidelity, and (4) faster than quadratic decay of the compression ratio as a function of turbulence resolution. The very promising slope for Reynolds number scaling of CVS and SCALES demonstrates the potential of the wavelet-based methodologies for hierarchical multiscale space/time adaptive variable fidelity simulations of high Reynolds number turbulent flows.

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