Multiscale Analysis and Computation for the Three-Dimensional Incompressible Navier–Stokes Equations

In this paper, we perform a systematic multiscale analysis for the three-dimensional incompressible Navier–Stokes equations with multiscale initial data. There are two main ingredients in our multiscale method. The first one is that we reparameterize the initial data in the Fourier space into a formal two-scale structure. The second one is the use of a nested multiscale expansion together with a multiscale phase function to characterize the propagation of the small-scale solution dynamically. By using these two techniques and performing a systematic multiscale analysis, we derive a multiscale model which couples the dynamics of the small-scale subgrid problem to the large-scale solution without a closure assumption or unknown parameters. Furthermore, we propose an adaptive multiscale computational method which has a complexity comparable to a dynamic Smagorinsky model. We demonstrate the accuracy of the multiscale model by comparing with direct numerical simulations for both two- and three-dimensional problems. In the two-dimensional case we consider decaying turbulence, while in the three-dimensional case we consider forced turbulence. Our numerical results show that our multiscale model not only captures the energy spectrum very accurately, it can also reproduce some of the important statistical properties that have been observed in experimental studies for fully developed turbulent flows

Kinetic Solvers with Adaptive Mesh in Phase Space

An Adaptive Mesh in Phase Space (AMPS) methodology has been developed for solving multi-dimensional kinetic equations by the discrete velocity method. A Cartesian mesh for both configuration (r) and velocity (v) spaces is produced using a tree of trees data structure. The mesh in r-space is automatically generated around embedded boundaries and dynamically adapted to local solution properties. The mesh in v-space is created on-the-fly for each cell in r-space. Mappings between neighboring v-space trees implemented for the advection operator in configuration space. We have developed new algorithms for solving the full Boltzmann and linear Boltzmann equations with AMPS. Several recent innovations were used to calculate the discrete Boltzmann collision integral with dynamically adaptive mesh in velocity space: importance sampling, multi-point projection method, and the variance reduction method. We have developed an efficient algorithm for calculating the linear Boltzmann collision integral for elastic and inelastic collisions in a Lorentz gas. New AMPS technique has been demonstrated for simulations of hypersonic rarefied gas flows, ion and electron kinetics in weakly ionized plasma, radiation and light particle transport through thin films, and electron streaming in semiconductors. We have shown that AMPS allows minimizing the number of cells in phase space to reduce computational cost and memory usage for solving challenging kinetic problems

Finite element solution techniques for large-scale problems in computational fluid dynamics

Element-by-element approximate factorization, implicit-explicit and adaptive implicit-explicit approximation procedures are presented for the finite-element formulations of large-scale fluid dynamics problems. The element-by-element approximation scheme totally eliminates the need for formation, storage and inversion of large global matrices. Implicit-explicit schemes, which are approximations to implicit schemes, substantially reduce the computational burden associated with large global matrices. In the adaptive implicit-explicit scheme, the implicit elements are selected dynamically based on element level stability and accuracy considerations. This scheme provides implicit refinement where it is needed. The methods are applied to various problems governed by the convection-diffusion and incompressible Navier-Stokes equations. In all cases studied, the results obtained are indistinguishable from those obtained by the implicit formulations

From Physics Model to Results: An Optimizing Framework for Cross-Architecture Code Generation

Starting from a high-level problem description in terms of partial differential equations using abstract tensor notation, the Chemora framework discretizes, optimizes, and generates complete high performance codes for a wide range of compute architectures. Chemora extends the capabilities of Cactus, facilitating the usage of large-scale CPU/GPU systems in an efficient manner for complex applications, without low-level code tuning. Chemora achieves parallelism through MPI and multi-threading, combining OpenMP and CUDA. Optimizations include high-level code transformations, efficient loop traversal strategies, dynamically selected data and instruction cache usage strategies, and JIT compilation of GPU code tailored to the problem characteristics. The discretization is based on higher-order finite differences on multi-block domains. Chemora's capabilities are demonstrated by simulations of black hole collisions. This problem provides an acid test of the framework, as the Einstein equations contain hundreds of variables and thousands of terms.Comment: 18 pages, 4 figures, accepted for publication in Scientific Programmin

Numerical simulations of fuel droplet flows using a Lagrangian triangular mesh

The incompressible, Lagrangian, triangular grid code, SPLISH, was converted for the study of flows in and around fuel droplets. This involved developing, testing and incorporating algorithms for surface tension and viscosity. The major features of the Lagrangian method and the algorithms are described. Benchmarks of the algorithms are given. Several calculations are presented for kerosene droplets in air. Finally, extensions which make the code compressible and three dimensional are discussed

POD model order reduction with space-adapted snapshots for incompressible flows

We consider model order reduction based on proper orthogonal decomposition (POD) for unsteady incompressible Navier-Stokes problems, assuming that the snapshots are given by spatially adapted finite element solutions. We propose two approaches of deriving stable POD-Galerkin reduced-order models for this context. In the first approach, the pressure term and the continuity equation are eliminated by imposing a weak incompressibility constraint with respect to a pressure reference space. In the second approach, we derive an inf-sup stable velocity-pressure reduced-order model by enriching the velocity reduced space with supremizers computed on a velocity reference space. For problems with inhomogeneous Dirichlet conditions, we show how suitable lifting functions can be obtained from standard adaptive finite element computations. We provide a numerical comparison of the considered methods for a regularized lid-driven cavity problem