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.

[1] Three-dimensional models of the uplift of Beta Regio caused by a mantle plume satisfy constraints on gravity, topography, rheology, and the uplift rate substantially better than two-dimensional models. In particular, the uplift time of Beta Regio is reduced to an acceptable 800 million years. Three-dimensional models give the plume formation depth around 3000 km, which approximately corresponds to the core-mantle boundary. The results depend only weakly on the initial lithospheric thickness (at the time of global resurfacing), including a very thin initial lithosphere. This implies that the thick present-day lithosphere ($400 km) suggested by our models can be reconciled with a thin lithosphere (100-200 km) inferred from the melt generation rates and the flexural rigidity models.

[1] Tharsis province is a major center of Martian volcanic activity characterized by large gravity and topography anomalies. The origin of Tharsis is debated. One hypothesis is that the province was produced by melting associated with a mantle plume from the coremantle boundary. An alternative hypothesis is that Tharsis formed by a plume associated with an impact. Recent studies have shown that this hypothesis is plausible from a geodynamical point of view and that long-lived impact plumes might play a role in areoid evolution. In this study, the magmatic evolution of impact-induced thermochemical mantle plumes is investigated with fully three-dimensional spherical shell simulations of mantle convection. Melt volumes and emplacement rates predicted by the model can satisfy observational constraints on Tharsis development.

[1] Recent evidence suggests that the Beta Regio equatorial highland, generally considered to be a plume-related structure, was emplaced over a relatively short period of time in the recent Venus history (100-400 m.y.). We perform numerical simulations which use this evidence as a new constraint on geodynamic models and find that it is difficult to satisfy simultaneously the constraint on the uplift rate and the constraints on gravity, topography, and rheology. One possible solution is that the effective viscosity contrast between the mantle and the lithosphere does not exceed about 10 4 . If our simple models do capture the basic dynamics of formation of Beta Regio, this value implies that the viscosity of the lithosphere is softer than olivine at subsolidus temperatures and that the Venusian lithosphere may not be much stronger than the terrestrial one.

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.