Coarse Grid Solution Research Articles

Error Control and Adaptivity for Low-Mach-Number Compressible Flows

An error estimation and grid adaptation strategy is presented for low-Mach-number, compressible, isentropic flows in two dimensions. There is very little literature investigating error control and grid refinement strategies combined with low-Mach-number compressible flows simultaneously, although these two concepts have been treated separately. The error control and the refinement procedure is based on the adjoint formulation in which the adjoint function is connected to local residual error, as well as linear variation of the functional with respect to a coarse grid solution. The benefit of the presented local grid refinement strategy based on some prechosen relevant engineering quantity is that it quantifies the specific locations in the domain that most affect the approximation of this quantity while maintaining the computational efficiency. Moreover, this approach is broader in the sense of not requiring a priori knowledge of the flow compared to standard gradient-based or global refinement approaches

A Finite Difference Domain Decomposition Method Using Local Corrections for the Solution of Poisson's Equation

We present a domain decomposition method for computing finite difference solutions to the Poisson equation with infinite domain boundary conditions. Our method is a finite difference analogue of Anderson's Method of Local Corrections. The solution is computed in three steps. First, fine-grid solutions are computed in parallel using infinite domain boundary conditions on each subdomain. Second, information is transferred globally through a coarse-grid representation of the charge, and a global coarse-grid solution is found. Third, a fine-grid solution is computed on each subdomain using boundary conditions set with the global coarse solution, corrected locally with fine-grid information from nearby subdomains. There are three important features of our algorithm. First, our method requires only a single iteration between the local fine-grid solutions and the global coarse representation. Second, the error introduced by the domain decomposition is small relative to the solution error obtained in a single-grid calculation. Third, the computed solution is second-order accurate and only weakly dependent on the coarse-grid spacing and the number of subdomains. As a result of these features, we are able to compute accurate solutions in parallel with a much smaller ratio of communication to computation than more traditional domain decomposition methods. We present results to verify the overall accuracy, confirm the small communication costs, and demonstrate the parallel scalability of the method.

Risk Management for Petroleum Reservoir Production: A Simulation-Based Study of Prediction

We consider numerical solutions of the Darcy and Buckley–Leverett equations for flow in porous media. These solutions depend on a realization of a random field that describes the reservoir permeability. The main content of this paper is to formulate and analyze a probability model for the numerical coarse grid solution error. We explore the extent to which the coarse grid oil production rate is sufficient to predict future oil production rates. We find that very early oil production data is sufficient to reduce the prediction error in oil production by about 30%, relative to the prior probability prediction.

A Two-Grid Finite Difference Scheme for Nonlinear Parabolic Equations

We present a two-level finite difference scheme for the approximation of nonlinear parabolic equations. Discrete inner products and the lowest-order Raviart--Thomas approximating space are used in the expanded mixed method in order to develop the finite difference scheme. Analysis of the scheme is given assuming an implicit time discretization. In this two-level scheme, the full nonlinear problem is solved on a "coarse" grid of size H. The nonlinearities areexpanded about the coarse grid solution and an appropriate interpolation operator is used to provide values of the coarse grid solution on the fine grid in terms of superconvergent node points. The resulting linear but nonsymmetric system is solved on a "fine" grid of size h. Some a priori error estimates are derived which show that the discrete L\infty (L2 ) and L2 (H1 ) errors are $O(h^2 + H^{4-d/2} + \Delta t)$, where $d \geq 1$ is the spatial dimension.

Hierarchical composite grid method for global-local analysis of laminated composite shells

A hierarchical version of the composite grid method (denoted as HFAC), which exploits the solution of the shell model in studying local effects via a 3D solid model, is developed. Convergence studies on a beam/2D model problem indicate that the spectral radius of the point iteration matrix for the HFAC method is O(1) and O(( L H ) 2) with exact and approximate auxiliary coarse grid solutions, respectively, where L and H are the span and the thickness of the beam, respectively. Numerical experiments in multidimensions confirm these findings.

Perspective: A Method for Uniform Reporting of Grid Refinement Studies

This paper proposes the use of a Grid Convergence Index (GCI) for the uniform reporting of grid refinement studies in Computational Fluid Dynamics. The method provides an objective asymptotic approach to quantification of uncertainty of grid convergence. The basic idea is to approximately relate the results from any grid refinement test to the expected results from a grid doubling using a second-order method. The GCI is based upon a grid refinement error estimator derived from the theory of generalized Richardson Extrapolation. It is recommended for use whether or not Richardson Extrapolation is actually used to improve the accuracy, and in some cases even if the conditions for the theory do not strictly hold. A different form of the GCI applies to reporting coarse grid solutions when the GCI is evaluated from a “nearby” problem. The simple formulas may be applied a posteriori by editors and reviewers, even if authors are reluctant to do so.

Computation of three-dimensional hypersonic flows in chemical nonequilibrium

An algorithm for the simulation of three-dimensi onal hypersonic flows in chemical nonequilibrium is presented. The basic flow solver is based on a quasiconservative formulation of the Euler or Navier-Stokes equations. The Jacobi matrices are split according to the sign of their eigenvalues. The derivatives of the conservative variables are split accordingly. A third-order upwind space discretization is used in conjunction with an optimized three-stage Runge-Kutta explicit time stepping scheme. The chemistry source terms are treated point-implicitly. For inviscid flow, the code is applied to the complete HERMES 1.0 configuration. The influence of mesh resolution is studied by comparing a fine grid with a coarse grid solution. The coarse grid solution is usually sufficient to describe global flow phenomena. The analysis of local flow details requires refined meshes. For viscous flow, the flow about generic configurations (double-ellipse, hemisphere-cylinder-flare, hyperbola-flare) is investigated by performing grid sensitivity studies as well as by comparing different transport models. UE to the initiation of the HERMES program, large research efforts have been directed to the development of space vehicles in Europe. The re-entry trajectories for space vehicles include large regions where the airflow is in chemical and thermal nonequilibrium. Nonequilibrium effects are important when the time scale for chemical reactions or energy exchange tchcm is comparable to a characteristic time scale for global flow changes fnow. If r chem « f flow, the process is in equilibrium, and if fchem » £now, the process is frozen. Since the density of air is low and the vehicle velocity on the reentry trajectory is high in the upper atmosphere, nonequilibrium effects have to be considered for altitudes larger than about 50 km. It is important to take nonequilibrium influences into account in order to be able to correctly simulate the aerodynamic behavior (pitching moment, c.p.), as well as heat-loads on the surface of the space vehicle (recombination, wall catalycity, radiation) at high altitudes. Although thermal nonequilibrium may be important near the vehicle nose and may influence the subsequent flowfield by convection, it is neglected here. Especially for the heat loads, a viscous flow simulation is necessary because the flow in the immediate vicinity of the vehicle surface is viscosity dominated. The numerical simulation of nonequilibriu m real gas flows requires very specialized and detailed physical models and numerical methods. Due to the complicated numerical formulation, code implementation, and due to the excessive computational requirements for numerical calculations, only recently nonequilibrium reacting flow computations have been undertaken. Numerical methods implemented so far use a full-conservative flux-splitting approach for the convective fluxes (either flux-vector or flux-differen ce).1-6 The intention of the work described in this article was to develop a solution method and computer code for the numerical simulation of nonequilibrium real gas flows about the complete HERMES configuration based on earlier developments for perfect gas flows7-8 and equilibrium real gas flows.9-10 In contrast to other solution methods, we employ the quasiconservative formulation with matrix splitting originally developed by Weiland7 and the shockfitting technique implemented by Pfitzner, 11 which is a postcorrection technique based on the ideas of Pandolfi. 12 The quasiconservative matrix splitting technique allows an accurate flow description in both inviscid and viscosity dominated regions. Using the shock-fitting technique, the location of the bow shock is given precisely. This is especially important for local bow shock—structure interactions (e.g., bow shock— winglet interaction). Additionally, the shock-fitting procedure enhances the efficiency of the method by limiting the computational domain automatically to the region where flow changes occur. In the near future, the solution method will be extended to include thermal nonequilibrium as well as advanced wall catalycity models and models for the transport terms. This work will be performed in the frame of the HERMES technology program in cooperation with international partners. In Sec. II we describe the physical model for viscous flows in chemical nonequilibrium (equation of state, chemistry terms, transport terms). In Sec. Ill we present the solution method (mixed quasi-/full-conservative formulation, matrix splitting, and discretization). Results are shown in Sec. IV for inviscid flows about the complete HERMES configuration and for viscous flows over generic configurations. A general conclusion is given in Sec. V.

A local adaptive grid procedure for incompressible flows with multigridding and equidistribution concepts

AbstractAn adaptive grid solution procedure is developed for incompressible flow problems in which grid refinement based on an equidistribution law is performed in high‐error‐estimate regions that are flagged from a preliminary coarse grid solution. Solutions on the locally refined and equidistributed meshes are obtained using boundary conditions interpolated from the preliminary coarse grid solution, and solutions on both the refined and coarse grid regions are successively improved using a multigrid approach. For this purpose, suitable correction terms for the coarse grid equations are derived for all variables in the flagged regions. This procedure with Local Adaptation, Multigridding and Equidistribution (LAME) concepts is applied to various flow problems to demonstrate the accuracy improvements obtained using this method.

Global convergence of inexact Newton methods for transonic flow

AbstractIn computational fluid dynamics, non‐linear differential equations are essential to represent important effects such as shock waves in transonic flow. Discretized versions of these non‐linear equations are solved using iterative methods. In this paper an inexact Newton method using the GMRES algorithm of Saad and Schultz is examined in the context of the full potential equation of aerodynamics. In this setting, reliable and efficient convergence of Newton methods is difficult to achieve. A poor initial solution guess often leads to divergence or very slow convergence. This paper examines several possible solutions to these problems, including a standard local damping strategy for Newton's method and two continuation methods, one of which utilizes interpolation from a coarse grid solution to obtain the initial guess on a finer grid. It is shown that the continuation methods can be used to augment the local damping strategy to achieve convergence for difficult transonic flow problems. These include simple wings with shock waves as well as problems involving engine power effects. These latter cases are modelled using the assumption that each exhaust plume is isentropic but has a different total pressure and/or temperature than the freestream.

An adaptive grid solution procedure for convection-diffusion problems

Abstract A computationally efficient and stable adaptive grid solution procedure is developed for convection diffusion problems. In this method, grid refinement and adaptation is based on an equidistribution law but is only performed in regions with high error estimates that are flagged from a preliminary coarse grid solution. The equidistribution law is implicit in the grid generation procedure which requires the solution of two Poisson equations with control functions that are obtained directly from the error estimates or weighting functions at the grid points. Solution on the refined, equidistributed mesh in the flagged region is obtained with boundary conditions interpolated from the coarse grid results. Accurate solutions in both the flagged region and the coarse grid regions of the domain are obtained with a multigrid approach that requires successive solutions on the refined, equidistributed mesh in the flagged region and on the coarse mesh in the entire domain. The adaptive grid method including the multigrid calculations can be extended to several levels of refinement. The acronym LAME is coined for this procedure in view of its Local Adaptation, Multigridding, and Equidistribution features. The method is shown to be stable, computationally efficient, and accurate by applying it to three test problems and comparing with conventional calculations on a fixed curvilinear grid.

Adaptive grid refinement for numerical weather prediction

An adaptive atmospheric flow model is described and results of integrations with this model are presented. The adaptive technique employed is that of Berger and Oliger. The technique uses a finite difference method to integrate the dynamical equations first on a coarse grid and then on finer grids which have been placed based on a Richardson-type estimate of the truncation error in the coarse grid solution. By correctly coupling the integrations on the various grids, periodically re-estimating the error, and recreating the finer grids, uniformily accurate solutions are economically produced. The “primitive” hydrostatic equations of meteorology are solved for the advection of a barotropic cyclone and for the development of a baroclinic disturbance which results from the perturbation of an unstable jet. These integrations demonstrate the feasibility of using multiple, rotated, overlapping fine grids. Direct computations of the truncation error are used to confirm the accuracy of the Richardson-type truncation error estimates.

Multi-cell vortices computed in large-scale difference solution to the incompressible Euler equations

Large-scale computational results are presented for a three-dimensional flow containing a vortex sheet shed from a delta wing. A series of results computed under mesh refinement are compared with a vortex-element solution in which the vortex sheet is tracked as a non-diffusing discontinuity. The results are analyzed for the position of the vortex features captured in the Euler flowfield, the accuracy of the pressure field, and for the diffusion of the vortex sheet. The conclusions are that, with a sufficiently fine mesh, the dynamics of shed vortex flow are reasonably resolved and can be studied by grid-based solutions. The numerical solution indicates that the non-axisymmetric disturbance caused by the trailing edge of the wing sets up a torsional wave on the vortex core and produces a structure with multiple cells of vorticity. Although observed in coarse grid solutions too, this effect becomes better resolved with mesh refinement to 614,000 grid volumes. It is argued that this phenomenon is real, not numerical, and that it is not a form of vortex bursting. Instead, the conjecture is made that the phenomenon is the result of a vortex instability.

The Fast Adaptive Composite Grid (FAC) Method for Elliptic Equations

The fast adaptive composite grid (FAG) method is a systematic process for solving differential boundary value problems. FAC uses global and local uniform grids both to define the composite grid problem and to interact for its fast solution. It can with little added cost substantially improve accuracy of the coarse grid solution and is very suitable for vector and parallel computation. This paper develops both the theoretical and practical aspects of FAC as it applies to elliptic problems, 1. Introduction. The need for local resolution in physical models occurs frequently in practice. Special local features of the forcing function, operator coefficients, boundary, and boundary conditions can demand resolution in restricted regions of the domain that is much finer than the required global resolution. It is important that the discretization and solution processes account for this locally, that is, that the local phenomena do not precipitate a dramatic increase in the overall computation. Unfortunately, this objective of efficiently adapting to local features is often in conflict with the solution process: equation solvers can degrade or even fail to apply in the presence of varying discretization scales; data structures that account for irregular grids can be cumbersome; the computer architecture may not be able to effectively account for grid irregularity (e.g., vectorizability may be inhibited); etc. In fact, even the discretization process itself may find difficulty with this objective: for finite differences, it is problematic to develop accurate difference formulae for irregular grids; for finite elements, this objective is reflected in the substantial overhead costs needed to automate the discretization. The fast adaptive composite grid method (FAC (11)) is a discretization and solution method designed to achieve efficient local resolution by constructing the

The fast adaptive composite grid (FAC) method for elliptic equations

The fast adaptive composite grid (FAC) method is a systematic process for solving differential boundary value problems. FAC uses global and local uniform grids both to define the composite grid problem and to interact for its fast solution. It can with little added cost substantially improve accuracy of the coarse grid solution and is very suitable for vector and parallel computation. This paper develops both the theoretical and practical aspects of FAC as it applies to elliptic problems.

On the solution of nonlinear two-point boundary value problems on successively refined grids

The solution of nonlinear two-point boundary value problems by adaptive finite difference methods ordinarily proceeds from a coarse to a fine grid. Grid points are inserted in regions of high spatial activity and the coarse grid solution is then interpolated onto the finer mesh. The resulting nonlinear difference equations are often solved by Newton's method. As the size of the mesh spacing becomes small enough. Newton's method converges with only a few iterations. In this paper we derive an estimate that enables us to determine the size of the critical mesh spacing that assures us that the interpolated solution for a class of two-point boundary value problems will lie in the domain of convergence of Newton's method on the next finer grid. We apply the estimate in the solution of several model problems.

MULTIGRID TECHNIQUES FOR THE NUMERICAL SOLUTION OF THE DIFFUSION EQUATION

An accurate numerical solution of diffusion problems containing large local gradients can be obtained with a significant reduction in computational time by using a multi-grid computational scheme. The spatial domain is covered with sets of uniform square grids of different sizes. The finer grid patterns overlap the coarse grid patterns. The finite-difference expressions for each grid pattern are solved Independently by iterative techniques. Two interpolation methods were used to establish the values of the potential function on the fine grid boundaries with information obtained from the coarse grid solution. The accuracy and computational requirements for solving a test problem by a simple multigrid and a multilevel-multigrid method were compared. The multilevel-multigrid method combined with a Taylor series interpolation scheme was found to be best.