Publications

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This paper describes the design of Teko, an object-oriented C++ library for implementing advanced block preconditioners. Mathematical design criteria that elucidate the needs of block preconditioning libraries and techniques are explained and shown to motivate the structure of Teko. For instance, a principal design choice was for Teko to strongly reflect the mathematical statement of the preconditioners to reduce development burden and permit focus on the numerics. Additional mechanisms are explained that provide a pathway to developing an optimized production capable block preconditioning capability with Teko. Finally, Teko is demonstrated on fluid flow and magnetohydrodynamics applications. In addition to highlighting the features of the Teko library, these new results illustrate the effectiveness of recent preconditioning developments applied to advanced discretization approaches.

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This work introduces a set of design principles and new algorithmic tools for enforcing maximum principles and/or positivity preservation in continuous finite element approximations to convection-dominated transport problems. Enabling a linear first-order advection equation as a model problem, we address the design of first-order artificial diffusion operators and their higherorder counterparts at the element matrix level. The proposed methodology leads to a nonlinear high-resolution scheme capable of resolving moving fronts and internal/boundary layers as sharp localized nonoscillatory features. The amount of numerical dissipation depends on the difference between the solution value at a given node and a local maximum or minimum. The shockcapturing numerical diffusion coefficient is designed to vanish as the nodal values approach a mass-weighted or linearity-preserving average. The universal applicability and simplicity of the element-based limiting procedure makes it an attractive alternative to edge-based algebraic flux correction.

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New large eddy simulation (LES) turbulence models for incompressible magnetohydrodynamics (MHD) derived from the variational multiscale (VMS) formulation for finite element simulations are introduced. The new models include the variational multiscale formulation, a residual-based eddy viscosity model, and a mixed model that combines both of these component models. Each model contains terms that are proportional to the residual of the incompressible MHD equations and is therefore numerically consistent. Moreover, each model is also dynamic, in that its effect vanishes when this residual is small. The new models are tested on the decaying MHD Taylor Green vortex at low and high Reynolds numbers. The evaluation of the models is based on comparisons with available data from direct numerical simulations (DNS) of the time evolution of energies as well as energy spectra at various discrete times. A numerical study, on a sequence of meshes, is presented that demonstrates that the large eddy simulation approaches the DNS solution for these quantities with spatial mesh refinement.

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In the context of a posteriori error estimation for nonlinear time-dependent partial differential equations, the state-of-the-practice is to use adjoint approaches which require the solution of a backward-in-time problem defined by a linearization of the forward problem. One of the major obstacles in the practical application of these approaches is the need to store, or recompute, the forward solution to define the adjoint problem and to evaluate the error representation. This study considers the use of data compression techniques to approximate forward solutions employed in the backward-in-time integration. The development derives an error representation that accounts for the difference between the standard-approach and the compressed approximation of the forward solution. This representation is algorithmically similar to the standard representation and only requires the computation of the quantity of interest for the forward solution and the data-compressed reconstructed solution (i.e.scalar quantities that can be evaluated as the forward problem is integrated). This approach is then compared with existing techniques, such as checkpointing and time-averaged adjoints. Finally, we provide numerical results indicating the potential efficiency of our approach on a transient diffusion-reaction equation and on the Navier-Stokes equations. These results demonstrate memory compression ratios up to 450× while maintaining reasonable accuracy in the error-estimates.