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NASA NTRS1992

Design of optimally smoothing multistage schemes for the Euler equations

A recently derived local preconditioning of the Euler equations is shown to be useful in developing multistage schemes suited for multigrid use. The effect of the preconditioning matrix on the spatial Euler operator is to equalize the characteristic speeds. When applied to the discretized Euler equations, the preconditioning has the effect of strongly clustering the operator's eigenvalues in the complex plane. This makes possible the development of explicit marching schemes that effectively damp most high-frequency Fourier modes, as desired in multigrid applications. The technique is the same as developed earlier for scalar convection schemes: placement of the zeros of the amplification factor of the multistage scheme in locations where eigenvalues corresponding to high-frequency modes abound.

Van Leer, Bram↗

NASA NTRS1993

A multigrid nonoscillatory method for computing high speed flows

A multigrid method using different smoothers has been developed to solve the Euler equations discretized by a nonoscillatory scheme up to fourth order accuracy. The best smoothing property is provided by a five-stage Runge-Kutta technique with optimized coefficients, yet the most efficient smoother is a backward Euler technique in factored and diagonalized form. The singlegrid solution for a hypersonic, viscous conic flow is in excellent agreement with the solution obtained by the third order MUSCL and Roe's method. Mach 8 inviscid flow computations for a complete entry probe have shown that the accuracy is at least as good as the symmetric TVD scheme of Yee and Harten. The implicit multigrid method is four times more efficient than the explicit multigrid technique and 3.5 times faster than the single-grid implicit technique. For a Mach 8.7 inviscid flow over a blunt delta wing at 30 deg incidence, the CPU reduction factor from the three-level multigrid computation is 2.2 on a grid of 37 x 41 x 73 nodes.

Li, C. P.↗

NASA NTRS1984

Multigrid applications to three-dimensional elliptic coordinate generation

The multigrid algorithm was applied to solve the coupled set of elliptic quasilinear partial differential equations associated with three-dimensional coordinate generation. The results indicate that the multigrid scheme is more than twice as fast as conventional relaxation schemes on moderate-size grids. Convergence factors of order 0.90 per work unit were achieved on 36,000-point grids. The paper covers the form of transformation, develops the set of generation equations, and gives details on the multigrid approach used. Included are a development of the full-approximation storage scheme, details of the smoothing-rate analysis, and a section devoted to rational programming techniques applicable to the multigrid algorithm.

Thames, F. C.↗

DOE OSTI2017

Numerical Multilevel Upscaling for Incompressible Flow in Reservoir Simulation: An Element-Based Algebraic Multigrid (AMGe) Approach

Here, we study the application of a finite element numerical upscaling technique to the incompressible two-phase porous media total velocity formulation. Specifically, an element-agglomeration-based algebraic multigrid (AMGe) technique with improved approximation properties is used, for the first time, to generate upscaled and accurate coarse systems for the reservoir simulation equations. The upscaling technique is applied to both the mixed system for velocity and pressure and to the hyperbolic transport equations, providing fully upscaled systems. By introducing additional degrees of freedom associated with nonplanar interfaces between agglomerates, the coarse velocity space has guaranteed approximation properties. The employed AMGe technique provides coarse spaces with desirable local mass conservation and stability properties analogous to the original pair of Raviart--Thomas and piecewise discontinuous polynomial spaces, resulting in strong mass conservation for the upscaled systems. Due to the guaranteed approximation properties and the generic nature of the AMGe method, recursive multilevel upscaling is automatically obtained. Furthermore, this technique works for both structured and unstructured meshes. Multiscale mixed finite elements exhibit accuracy for general unstructured meshes but do not in general lead to nested hierarchy of spaces. Multiscale multilevel mimetic finite differences generate nested spaces but lack the adaptivity of the flux representation on coarser levels that the proposed AMGe approach offers. Thus, the proposed approach can be seen as a rigorous bridge that merges the best properties of these two existing methods. The accuracy and stability of the studied multilevel AMGe upscaling technique is demonstrated on two challenging test cases.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2017

On the Preconditioning of a Newton-Krylov Solver for a High-Order reconstructed Discontinuous Galerkin Discretization of All-Speed Compressible Flow with Phase Change for Application in Laser-Based Additive Manufacturing

This dissertation focuses on the development of a fully-implicit, high-order compressible ow solver with phase change. The work is motivated by laser-induced phase change applications, particularly by the need to develop large-scale multi-physics simulations of the selective laser melting (SLM) process in metal additive manufacturing (3D printing). Simulations of the SLM process require precise tracking of multi-material solid-liquid-gas interfaces, due to laser-induced melting/ solidi cation and evaporation/condensation of metal powder in an ambient gas. These rapid density variations and phase change processes tightly couple the governing equations, requiring a fully compressible framework to robustly capture the rapid density variations of the ambient gas and the melting/evaporation of the metal powder. For non-isothermal phase change, the velocity is gradually suppressed through the mushy region by a variable viscosity and Darcy source term model. The governing equations are discretized up to 4th-order accuracy with our reconstructed Discontinuous Galerkin spatial discretization scheme and up to 5th-order accuracy with L-stable fully implicit time discretization schemes (BDF2 and ESDIRK3-5). The resulting set of non-linear equations is solved using a robust Newton-Krylov method, with the Jacobian-free version of the GMRES solver for linear iterations. Due to the sti nes associated with the acoustic waves and thermal and viscous/material strength e ects, preconditioning the GMRES solver is essential. A robust and scalable approximate block factorization preconditioner was developed, which utilizes the velocity-pressure (vP) and velocity-temperature (vT) Schur complement systems. This multigrid block reduction preconditioning technique converges for high CFL/Fourier numbers and exhibits excellent parallel and algorithmic scalability on classic benchmark problems in uid dynamics (lid-driven cavity ow and natural convection heat transfer) as well as for laser-induced phase change problems in 2D and 3D.

42 ENGINEERING↗

NASA NTRS1993

Multigrid direct numerical simulation of the whole process of flow transition in 3-D boundary layers

A new technology was developed in this study which provides a successful numerical simulation of the whole process of flow transition in 3-D boundary layers, including linear growth, secondary instability, breakdown, and transition at relatively low CPU cost. Most other spatial numerical simulations require high CPU cost and blow up at the stage of flow breakdown. A fourth-order finite difference scheme on stretched and staggered grids, a fully implicit time marching technique, a semi-coarsening multigrid based on the so-called approximate line-box relaxation, and a buffer domain for the outflow boundary conditions were all used for high-order accuracy, good stability, and fast convergence. A new fine-coarse-fine grid mapping technique was developed to keep the code running after the laminar flow breaks down. The computational results are in good agreement with linear stability theory, secondary instability theory, and some experiments. The cost for a typical case with 162 x 34 x 34 grid is around 2 CRAY-YMP CPU hours for 10 T-S periods.

Liu, Chaoqun↗

NASA NTRS1993

A mixed method Poisson solver for three-dimensional self-gravitating astrophysical fluid dynamical systems

A key ingredient in the simulation of self-gravitating astrophysical fluid dynamical systems is the gravitational potential and its gradient. This paper focuses on the development of a mixed method multigrid solver of the Poisson equation formulated so that both the potential and the Cartesian components of its gradient are self-consistently and accurately generated. The method achieves this goal by formulating the problem as a system of four equations for the gravitational potential and the three Cartesian components of the gradient and solves them using a distributed relaxation technique combined with conventional full multigrid V-cycles. The method is described, some tests are presented, and the accuracy of the method is assessed. We also describe how the method has been incorporated into our three-dimensional hydrodynamics code and give an example of an application to the collision of two stars. We end with some remarks about the future developments of the method and some of the applications in which it will be used in astrophysics.

Duncan, Comer↗

NASA NTRS1981

Multigrid solvers on parallel computers

Massively parallel computers, as considered in this investigation, are not yet available. However, a large-scale parallel computer cannot usefully be designed before the hypothetical algorithms which will employ it are studied. Most of the studies of parallel partial differential equations (PDE) solvers are based on solution techniques much slower (on sequential machines) than multigrid methods. Multigrid methods are highly parallelizable. Each of their processes can simultaneously be performed at all grid points. The present investigation is concerned with a preliminary exploration of the potential of multigrid, or, more generally, Multi-Level Adaptive Techniques (MLAT) on computers with many processors. Basic processes are considered, taking into account coarse-grid approximation, relaxation, coarse-grid corrections, full multigrid algorithms, nonlinear problems and eigenvalue problems, fine-to-coarse correction, and chains of problems. Details of parallel multigrid processing are also examined.

Brandt, A.↗

NASA NTRS1991

Transonic Navier-Stokes flow computations over wing-fuselage geometries

The steady-state transonic viscous flows over two wing-fuselage configurations are solved numerically. The 3D compressible thin-layer Navier-Stokes equations are solved by means of multistage timestepping similar to the Runge-Kutta method. The timestepping technique is applied with a multigrid scheme to make the transonic Navier-Stokes computations using the Baldwin-Lomax turbulence for closure and converging the solution on a grid of 1.52 million grid points with a C-O topology. Both generic and fighter wing-fuselage geometries are solved, and the results are compared to experimental studies. Solution convergence is found to be fast, and the fine grid generates accurate results for both configurations by means of the Baldwin-Lomax turbulence model. Results from the Johnson-King model calculations produce results that are more consistent with experimental results for the fighter configuration.

Agrawal, S.↗

NASA NTRS1995

Multigrid solution strategies for adaptive meshing problems

This paper discusses the issues which arise when combining multigrid strategies with adaptive meshing techniques for solving steady-state problems on unstructured meshes. A basic strategy is described, and demonstrated by solving several inviscid and viscous flow cases. Potential inefficiencies in this basic strategy are exposed, and various alternate approaches are discussed, some of which are demonstrated with an example. Although each particular approach exhibits certain advantages, all methods have particular drawbacks, and the formulation of a completely optimal strategy is considered to be an open problem.

Mavriplis, Dimitri J.↗

NASA NTRS1995

Pressure updating methods for the steady-state fluid equations

We consider the steady state equations for a compressible fluid. Since we wish to solve for a range of speeds we must consider the equations in conservation form. For transonic speeds these equations are of mixed type. Hence, the usual approach is to add time derivatives to the steady state equations and then march these equations in time. One then adds a time derivative of the density to the continuity equation, a derivative of the momentum to the momentum equation and a derivative of the total energy to the energy equation. This choice is dictated by the time consistent equations. However, since we are only interested in the steady state this is not necessary. Thus we shall consider the possibility of adding a time derivative of the pressure to the continuity equation and similar modifications for the energy equation. This can then be generalized to adding combinations of time derivatives to each equation since these vanish in the steady state. When using acceleration techniques such as residual smoothing, multigrid, etc. these are applied to the pressure rather than the density. Hence, the code duplicates the behavior of the incompressible equations for low speeds.

Fiterman, A.↗

NASA NTRS1995

Multigrid solution strategies for adaptive meshing problems

Mavriplis, Dimitri J.↗

NASA NTRS1993

Computational techniques for high-speed flows with viscous and chemical effects

Algorithms for solving the Euler and the Navier-Stokes equations in conjunction with chemical kinetic equations are presented. The convective flux is estimated from a quasi one dimensional interpolation procedure. Shock, contact, and expansion waves and thermochemical nonequilibrium phenomena are captured by the Lax-Friedrichs technique. Relaxation techniques were developed to enhance their effectiveness in dealing with spatial and temporal stiffness associated with the physical problems. Both explicit and implicit smoothers were implemented into the standard multigrid time stepping method. Unsteady and steady scalar problems are discussed. A perfect gas and equilibrium air shock tube problem is investigated. Numerical schemes and techniques are compared for the problems of shock and boundary layer interaction and three dimensional viscous, nonequilibrium flow encompassing an aerobrake. The results are comparable in accuracy against other high order non-oscillatory techniques. The multigrid methods are assessed using a Mach 8 flow over a complete planar body, a sphere, and a blunt delta wing at 20 deg incidence. Applying an implicit multigrid method on a nested grid of 128 by 64 nodes the reduction factor is 0.25. The central processing unit reduction factor is 2.2 after both the single and multigrid Runge-Kutta solutions converged to machine zero on a grid of 37 by 41 by 73 nodes.

Li, C. P.↗

NASA NTRS1981

Multigrid simulation of asymptotic curved-duct flows using a semi-implicit numerical technique

Asymptotic flows inside curved ducts of rectangular as well as polar cross section are analyzed using the Navier-Stokes equations in terms of the axial velocity and vorticity and the cross-flow stream function. Numerical solutions of the three second-order coupled elliptic partial differential equations governing this flow are obtained efficiently using the coupled alternating-direction implicit (ADI) method as well as the multigrid strongly-implicit (SI) scheme. For the flow configuration studied, the ADI method is found to be more sensitive to the time steps used than is the SI scheme. Use of the multigrid-coupled-strongly-implicit (MG-SI) scheme makes it possible to efficiently obtain fine-grid solutions for configurations having strong secondary flow. It is shown that, for this asymptotic curved-duct flow, the similarity parameter of significance is the Dean's number K rather than the Reynolds number Re. Results are obtained for curved ducts with square cross sections for K up to 900, which here corresponds to Re = 9,000 for this internal flow configuration.

Ghia, K. N.↗

DOE OSTI2014

Parallel time integration software

This package implements an optimal-scaling multigrid solver for the (non) linear systems that arise from the discretization of problems with evolutionary behavior. Typically, solution algorithms for evolution equations are based on a time-marching approach, solving sequentially for one time step after the other. Parallelism in these traditional time-integrarion techniques is limited to spatial parallelism. However, current trends in computer architectures are leading twards system with more, but not faster. processors. Therefore, faster compute speeds must come from greater parallelism. One approach to achieve parallelism in time is with multigrid, but extending classical multigrid methods for elliptic poerators to this setting is a significant achievement. In this software, we implement a non-intrusive, optimal-scaling time-parallel method based on multigrid reduction techniques. The examples in the package demonstrate optimality of our multigrid-reduction-in-time algorithm (MGRIT) for solving a variety of parabolic equations in two and three sparial dimensions. These examples can also be used to show that MGRIT can achieve significant speedup in comparison to sequential time marching on modern architectures.

Source record↗

DOE OSTI2014

Parallel time integration software

Source record↗

NASA NTRS1981

General relaxation schemes in multigrid algorithms for higher order singularity methods

Relaxation schemes based on approximate and incomplete factorization technique (AF) are described. The AF schemes allow construction of a fast multigrid method for solving integral equations of the second and first kind. The smoothing factors for integral equations of the first kind, and comparison with similar results from the second kind of equations are a novel item. Application of the MD algorithm shows convergence to the level of truncation error of a second order accurate panel method.

Oskam, B.↗

NASA NTRS1996

Multigrid Methods for Fully Implicit Oil Reservoir Simulation

In this paper we consider the simultaneous flow of oil and water in reservoir rock. This displacement process is modeled by two basic equations: the material balance or continuity equations and the equation of motion (Darcy's law). For the numerical solution of this system of nonlinear partial differential equations there are two approaches: the fully implicit or simultaneous solution method and the sequential solution method. In the sequential solution method the system of partial differential equations is manipulated to give an elliptic pressure equation and a hyperbolic (or parabolic) saturation equation. In the IMPES approach the pressure equation is first solved, using values for the saturation from the previous time level. Next the saturations are updated by some explicit time stepping method; this implies that the method is only conditionally stable. For the numerical solution of the linear, elliptic pressure equation multigrid methods have become an accepted technique. On the other hand, the fully implicit method is unconditionally stable, but it has the disadvantage that in every time step a large system of nonlinear algebraic equations has to be solved. The most time-consuming part of any fully implicit reservoir simulator is the solution of this large system of equations. Usually this is done by Newton's method. The resulting systems of linear equations are then either solved by a direct method or by some conjugate gradient type method. In this paper we consider the possibility of applying multigrid methods for the iterative solution of the systems of nonlinear equations. There are two ways of using multigrid for this job: either we use a nonlinear multigrid method or we use a linear multigrid method to deal with the linear systems that arise in Newton's method. So far only a few authors have reported on the use of multigrid methods for fully implicit simulations. Two-level FAS algorithm is presented for the black-oil equations, and linear multigrid for two-phase flow problems with strong heterogeneities and anisotropies is studied. Here we consider both possibilities. Moreover we present a novel way for constructing the coarse grid correction operator in linear multigrid algorithms. This approach has the advantage in that it preserves the sparsity pattern of the fine grid matrix and it can be extended to systems of equations in a straightforward manner. We compare the linear and nonlinear multigrid algorithms by means of a numerical experiment.

Molenaar, J.↗