22,140 research outputs found
Discretizations and Solvers for Coupling Stokes-Darcy Flows With Transport
This thesis studies a mathematical model, in which Stokes-Darcy flow system is coupled with a transport equation. The objective is to develop stable and convergent numerical schemes that could be used in environmental applications. Special attention is given to discretization methods that conserve mass locally. First, we present a global saddle point problem approach, which employs the discontinuous Galerkin method to discretize the Stokes equations and the mimetic finite difference method to discretize the Darcy equation. We show how the numerical scheme can be formulated on general polygonal (polyhedral in three dimensions) meshes if suitable operators mapping from degrees of freedom to functional spaces are constructed. The scheme is analyzed and error estimates are derived. A hybridization technique is used to solve the system effectively. We ran several numerical experiments to verify the theoretical convergence rates and depending on the mesh type we observed superconvergence of the computed solution in the Darcy region.Another approach that we use to deal with the flow equations is based on non-overlapping domain decomposition. Domain decomposition enables us to solve the coupled Stokes-Darcy flow problem in parallel by partitioning the computational domain into subdomains, upon which families of coupled local problems of lower complexity are formulated. The coupling of the subdomain problems is removed through an iterative procedure. We investigate the properties of this method and derive estimates for the condition number of the associated algebraic system. Results from computer tests supporting the convergence analysis of the method are provided. To discretize the transport equation we use the local discontinuous Galerkin (LDG) method, which can be thought as a discontinuous mixed finite element method, since it approximates both the concentration and the diffusive flux. We develop stability and convergence analysis for the concentration and the diffusive flux in the transport equation. The numerical error is a combination of the LDG discretization error and the error from the discretization of the Stokes-Darcy velocity. Several examples verifying the theory and illustrating the capabilities of the method are presented
Segregated Runge–Kutta time integration of convection-stabilized mixed finite element schemes for wall-unresolved LES of incompressible flows
In this work, we develop a high-performance numerical framework for the large eddy simulation (LES) of incompressible flows. The spatial discretization of the nonlinear system is carried out using mixed finite element (FE) schemes supplemented with symmetric projection stabilization of the convective term and a penalty term for the divergence constraint. These additional terms introduced at the discrete level have been proved to act as implicit LES models. In order to perform meaningful wall-unresolved simulations, we consider a weak imposition of the boundary conditions using a Nitsche’s-type scheme, where the tangential component penalty term is designed to act as a wall law. Next, segregated Runge–Kutta (SRK) schemes (recently proposed by the authors for laminar flow problems) are applied to the LES simulation of turbulent flows. By the introduction of a penalty term on the trace of the acceleration, these methods exhibit excellent stability properties for both implicit and explicit treatment of the convective terms. SRK schemes are excellent for large-scale simulations, since they reduce the computational cost of the linear system solves by splitting velocity and pressure computations at the time integration level, leading to two uncoupled systems. The pressure system is a Darcy-type problem that can easily be preconditioned using a traditional block-preconditioning scheme that only requires a Poisson solver. At the end, only coercive systems have to be solved, which can be effectively preconditioned by multilevel domain decomposition schemes, which are both optimal and scalable. The framework is applied to the Taylor–Green and turbulent channel flow benchmarks in order to prove the accuracy of the convection-stabilized mixed FEs as LES models and SRK time integrators. The scalability of the preconditioning techniques (in space only) has also been proven for one step of the SRK scheme for the Taylor–Green flow using uniform meshes. Moreover, a turbulent flow around a NACA profile is solved to show the applicability of the proposed algorithms for a realistic problem.Peer ReviewedPostprint (author's final draft
A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations
A new projection method based on radial basis functions (RBFs) is presented
for discretizing the incompressible unsteady Stokes equations in irregular
geometries. The novelty of the method comes from the application of a new
technique for computing the Leray-Helmholtz projection of a vector field using
generalized interpolation with divergence-free and curl-free RBFs. Unlike
traditional projection methods, this new method enables matching both
tangential and normal components of divergence-free vector fields on the domain
boundary. This allows incompressibility of the velocity field to be enforced
without any time-splitting or pressure boundary conditions. Spatial derivatives
are approximated using collocation with global RBFs so that the method only
requires samples of the field at (possibly scattered) nodes over the domain.
Numerical results are presented demonstrating high-order convergence in both
space (between 5th and 6th order) and time (up to 4th order) for some model
problems in two dimensional irregular geometries.Comment: 34 pages, 8 figure
A pencil distributed finite difference code for strongly turbulent wall-bounded flows
We present a numerical scheme geared for high performance computation of
wall-bounded turbulent flows. The number of all-to-all communications is
decreased to only six instances by using a two-dimensional (pencil) domain
decomposition and utilizing the favourable scaling of the CFL time-step
constraint as compared to the diffusive time-step constraint. As the CFL
condition is more restrictive at high driving, implicit time integration of the
viscous terms in the wall-parallel directions is no longer required. This
avoids the communication of non-local information to a process for the
computation of implicit derivatives in these directions. We explain in detail
the numerical scheme used for the integration of the equations, and the
underlying parallelization. The code is shown to have very good strong and weak
scaling to at least 64K cores
Fluid Solver Independent Hybrid Methods for Multiscale Kinetic equations
In some recent works [G. Dimarco, L. Pareschi, Hybrid multiscale methods I.
Hyperbolic Relaxation Problems, Comm. Math. Sci., 1, (2006), pp. 155-177], [G.
Dimarco, L. Pareschi, Hybrid multiscale methods II. Kinetic equations, SIAM
Multiscale Modeling and Simulation Vol 6., No 4,pp. 1169-1197, (2008)] we
developed a general framework for the construction of hybrid algorithms which
are able to face efficiently the multiscale nature of some hyperbolic and
kinetic problems. Here, at variance with respect to the previous methods, we
construct a method form-fitting to any type of finite volume or finite
difference scheme for the reduced equilibrium system. Thanks to the coupling of
Monte Carlo techniques for the solution of the kinetic equations with
macroscopic methods for the limiting fluid equations, we show how it is
possible to solve multiscale fluid dynamic phenomena faster with respect to
traditional deterministic/stochastic methods for the full kinetic equations. In
addition, due to the hybrid nature of the schemes, the numerical solution is
affected by less fluctuations when compared to standard Monte Carlo schemes.
Applications to the Boltzmann-BGK equation are presented to show the
performance of the new methods in comparison with classical approaches used in
the simulation of kinetic equations.Comment: 31 page
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