Gauss-Seidel Relaxation Research Articles

A multigrid method for steady incompressible navier‐stokes equations based on flux difference splitting

AbstractThe steady Navier–Stokes equations in primitive variables are discretized in conservative form by a vertex‐centred finite volume method Flux difference splitting is applied to the convective part to obtain an upwind discretization. The diffusive part is discretized in the central way.In its first‐order formulation, flux difference splitting leads to a discretization of so‐called vector positive type. This allows the use of classical relaxation methods in collective form. An alternating line Gauss–Seidel relaxation method is chosen here. This relaxation method is used as a smoother in a multigrid method. The components of this multigrid method are: full approximation scheme with F‐cycles, bilinear prolongation, full weighting for residual restriction and injection of grid functions.Higher‐order accuracy is achieved by the flux extrapolation method. In this approach the first‐order convective fluxes are modified by adding second‐order corrections involving flux limiting. Here the simple MinMod limiter is chosen. In the multigrid formulation the second‐order discrete system is solved by defect correction.Computational results are shown for the well known GAMM backward‐facing step problem and for a channel with a half‐circular obstruction.

A multigrid conjugate residual method for the numerical solution of the Hartree-Fock equation for diatomic molecules

Discretization of the Hartree-Fock equations in operational form leads to unsymmetric positive definite and indefinite linear equations. To solve these equations a combination of the multigrid method and the Orthomin method with Gauss-Seidel relaxation as preconditioner is used. The differential equations are approximated to the sixth order and the solution is extrapolated to the eighth order. The method is fully parallelized. The largest molecule treated is CuH.

An improved upwind finite volume relaxation method for high speed viscous flows

An improved upwind relaxation algorithm for the Navier-Stokes equations is presented, and results are given from the application of the method to two test problems, including (1) a shock/boundary layer interaction on a flat plate ( M ∞ = 2.0) and (2) a high-speed inlet ( M ∞ = 5.0). The technique is restricted to high-speed (i.e., supersonic/hypersonic) viscous flows. The new algorithm depends on a partitioning of the global domain into regions or sub-domains, where a relatively thin “elliptic” region is identified near each solid wall boundary, and the remainder of the flowfield is identified to be a single larger “hyperbolic” (i.e., hyperbolic/parabolic in the streamwise direction) region. A direct solution procedure by lower/upper factorization is applied to the elliptic region(s), the results of which are then coupled to a standard line Gauss-Seidel relaxation sweep across the entire domain in the primary flow direction. In the first test problem, the new algorithm reduced total run times as much as 75% when compared to the standard alternating forward/backward vertical line Gauss-Seidel (VLGS) algorithm, whereas in the second test problem, a total savings as high as 20% was achieved. Essentially all of this improvement occurred only after the initial transient in the solution was overcome. However, in the second test problem, a significant improvement in the computational performance of the standard forward/backward VLGS algorithm was noted when overcoming the initial transient simply by converting from the use of conserved variables to primitive variables in the spatial discretization and linearizatlon of all terms.

Local solution acceleration method for the Euler and Navier-Stokes equations

The solution of the compressible Euler and Navier-Stokes equations via an upwind finite volume scheme is obtained. For the inviscid fluxes, the monotone upstream-centered scheme for conservation laws (MUSCL) has been incorporated into a Riemann solver. The MUSCL scheme is used for the unfactored implicit equations that are solved by a Newton form, and relaxation is performed via Gauss-Seidel relaxation technique. The solution on the fine grid is obtained by iterating first on a sequence of coarser grids and then interpolating the solution up to the next refined grid. Since the distribution of the numerical error is nonuniform, the local solution of the equations can be obtained in regions where the numerical errors are large. The construction of the partial meshes, in which the iterations will be continued, is determined by an adaptive procedure taking into account some convergence criteria. Reduction of the computational work units for two-dimensional problems is obtained via the local adaptive mesh solution which is expected to be more effective in three-dimensional complex flow computations.

Optimal Multilevel Iterative Methods for Adaptive Grids

Many elliptic partial differential equations can be solved numerically with near optimal efficiency through the uses of adaptive refinement and multigrid solution techniques. This paper presents a more unified approach to the combined process of adaptive refinement and multigrid solution which can be used with high order finite elements. Refinement is achieved by the bisection of pairs of triangles, corresponding to the addition of one or more basis functions to the approximation space. An approximation of the resulting change in the solution is used as an error indicator. The multigrid iteration uses red-black Gauss-Seidel relaxation with local black relaxations. The grid transfers use the change between the nodal and hierarchical bases. This multigrid iteration requires only $O(N)$ operations, even for highly nonuniform grids, and is defined for any finite element space. The full multigrid method is an optimal blending of the processes of adaptive refinement and multigrid iteration. To minimize the number of operations required, the duration of the refinement phase is based on increasing the dimension of the approximation space by the largest possible factor, given the error reduction of the multigrid iteration. The algorithm (i) uses only $O(N)$ operations, (ii) solves the discrete system to the accuracy of the discretization error, and (iii) achieves optimal convergence of the discretization error in the presence of singularities. Numerical experiments confirm this for linear, quadratic, and cubic elements.

Temporal and acoustic accuracy of an implicit upwind method for ducted flows

An implicit, time-accurate method for ducted-flow aeroacoustics is presented. The method is unfactored, using line Gauss-Seidel relaxation and multiple axial sweeps for convergence of each time step.

Waveform-relaxation-based circuit simulation on the Victor V256 parallel processor

Present-day circuit-analysis tools permit designers to verify performance for circuits consisting of up to 10,000 transistors. However, current designs often exceed several tens of thousands and even hundreds of thousands of transistors. The gap between the number of transistors that can be simulated and the number per design inhibits proper analysis prior to manufacturing, yet incomplete analysis often overlooks design flaws and forces redesign, resulting in increased costs and longer development times. This gap is expected to widen in the foreseeable future. To help close the ever-increasing simulation/design gap, we have developed an experimental parallel circuit simulator, WR_V256, for the Victor V256 distributed-memory parallel processor. WR_V256 has been used to analyze circuits from fewer than 300 to more than 180,000 MOSFETs. WR_V256 was originally based on a Gauss-Seidel relaxation algorithm, which was later replaced with a bounded-chaotic one in order to achieve good parallel speedups for a wider variety of circuits. At this time, speedups of up to 190 have been observed for large circuits.

Accuracy of an unstructured-grid upwind-Euler algorithm for the ONERA M6 wing

Improved algorithms for the solution of the three-dimensional, time-dependent Euler equations are presented for aerodynamic analysis involving unstructured dynamic meshes. The improvements have been developed recently to the spatial and temporal discretizations used by unstructured-grid flow solvers. The spatial discretization involves a flux-split approach that is naturally dissipative and captures shock waves sharply with at most one grid point within the shock structure. The temporal discretization involves either an explicit time-integration scheme using a multistage Runge-Kutta procedure or an implicit time-integration scheme using a Gauss-Seidel relaxation procedure, which is computationally efficient for either steady or unsteady flow problems. With the implicit Gauss-Seidel procedure, very large time steps may be used for rapid convergence to steady state, and the step size for unsteady cases may be selected for temporal accuracy rather than for numerical stability. Steady flow results are presented for both the NACA 0012 airfoil and the Office National d'Etudes et de Recherches Aerospatiales M6 wing to demonstrate applications of the new Euler solvers. The paper presents a description of the Euler solvers along with results and comparisons that assess the capability.

Damped, direction-dependent multigrid for hypersonic flow computations

A nonlinear multigrid technique with improved robustness is developed for the solution of the steady Euler equations. The system of nonlinear equations is discretized by an upwind finite volume method. Collective symmetric point Gauss-Seidel relaxation is applied as the standard smoothing technique. In case of failure of the point relaxation, a switch is made to a local evolution technique. The novel robustness improvements to the nonlinear multigrid method are a local damping of the restricted defect, a global upwind prolongation of the correction and a global upwind restriction of the defect. The defect damping operator is derived from a two-grid convergence analysis. The upwind prolongation operator is made such that it is consistent with the upwind finite volume discretization. It makes efficient use of the P-variant of Osher's approximate Riemann solver. The upwind restriction operator is an approximate adjoint of the upwind prolongation operator. Satisfactory convergence results are shown for the computation of a hypersonic launch and reentry flow around a blunt forebody with canopy. For the test cases considered, it appears that the improved multigrid method performs significantly better than a standard nonlinear multigrid method. For all test cases considered it appears that the most significant improvement comes from the upwind prolongation, rather than from the upwind restriction and the defect damping.

A deferred-correction multigrid algorithm based on a new smoother for the Navier-Stokes equations

An algorithm which brings together the techniques of multigrid and deferred correction through their common relationship with imperfect Newton iteration manages to combine the ease of calculation of a low-order with the accuracy of a high-order difference approximation of any given differential-equation problem. A stable explicit Gauss-Seidel relaxation algorithm for the ψ-ζ Navier-Stokes equations based on an appropriate kind of “upwinding” of ψ- as well as ζ-derivatives, especially developed for use as a multigrid smoother in this context, is presented and the complete algorithm is tested on the standard conservative second-order discretization of the driven-cavity problem.

Implicit flux-split schemes for the Euler equations

Recent progress in the development of implicit algorithms for the Euler equations using the flux-vector splitting method is described. Comparisons of the relative efficiency of relaxation and spatially-split approximately factored methods on a vector processor for two-dimensional flows are made. For transonic flows, the higher convergence rate per iteration of the Gauss-Seidel relaxation algorithms, which are only partially vectorizable, is amply compensated for by the faster computational rate per iteration of the approximately factored algorithm. For supersonic flows, the fully-upwind line-relaxation method is more efficient since the numerical domain of dependence is more closely matched to the physical domain of dependence. A hybrid three-dimensional algorithm using relaxation in one coordinate direction and approximate factorization in the cross-flow plane is developed and applied to a forebody shape at supersonic speeds and a swept, tapered wing at transonic speeds.

A Note on the Use of Symmetric Line Gauss–Seidel for the Steady Upwind Differenced Euler Equations

Symmetric Line Gauss–Seidel (SLGS) relaxation, when used to compute steady solutions to the upwind differenced Euler equations of gas dynamics, is shown to be unstable. The instability occurs for the long waves. If SLGS is used in a multigrid scheme, stability is restored. However, the use of an unstable relaxation scheme will not provide a robust multigrid code. Damped Symmetric Point Gauss–Seidel relaxation is stable and provides similar multigrid convergence rates at much lower cost. However, it fails if the flow is aligned with the grid over a substantial part of the computational domain. Damped Alternating Direction Line Jacobi relaxation can overcome this problem.

Multigrid and defect correction for the steady Navier-Stokes equations

Theoretical and experimental convergence results are presented for nonlinear multigrid and iterative defect correction applied to finite volume discretizations of the full, steady, 2D, compressible Navier-Stokes equations. Iterative defect correction is introduced for circumventing the difficulty in solving Navier-Stokes equations discretized with a second- or higher-order accurate convective part. By Fourier analysis applied to a model equation, an optimal choice is made for the operator to be inverted in the defect correction iteration. As a smoothing technique for the multigrid method, collective symmetric point Gauss-Seidel relaxation is applied with as the basic solution technique: exact Newton iteration applied to a continuously differentiable, first-order upwind discretization of the full Navier-Stokes equations. For non-smooth flow problems, the convergence results obtained are already competitive with those of well-established Navier-Stokes methods. For smooth flow problems, the present method performs better than any standard method. Here, first-order discretization error accuracy is attained in a single multigrid cycle, and second-order accuracy in only one defect correction cycle. The method contributes to the state of the art in efficiently computing compressible viscous flows.

Implicit methods for the Navier-Stokes equations

Numerical solutions of the Navier-Stokes equations using explicit schemes can be obtained at the expense of efficiency. Conventional implicit methods which often achieve fast convergence rates suffer high cost per iteration. A new implicit scheme based on lower-upper factorization and symmetric Gauss-Seidel relaxation offers very low cost per iteration as well as fast convergence. High efficiency is achieved by accomplishing the complete vectorizability of the algorithm on oblique planes of sweep in three dimensions.

EMOTA: an event-driven MOS timing simulator for VLSI circuits

A novel new event-driven MOS timing simulator for VLSI circuits, EMOTA, is present- ed. The traditional event-driven simulation scheme used in logic simulation is modified for use in this circuit level timing simulator. The fast per- formance is due to the result of mixing the new derived event-driven algorithm and its associated time-wheel control mechanism, together with the unidirectional nonlinear Gauss-Seidel relaxation technique to decouple the circuit equations. EMOTA allows both interactive and batch simu- lation modes and its input format is the same as SPICE. The simulation speed of EMOTA is shown to be more than 300 times faster than SPICE2G.5 for circuits of several hundred tran- sistors and the simulated waveforms have accept- able accuracy.

Euler flow solutions for transonic shock wave–boundary layer interaction

AbstractSteady 2D Euler flow computations have been performed for a wind tunnel section, designed for research on transonic shock wave–boundary layer interaction. For the discretization of the steady Euler equations, an upwind finite volume technique has been applied. The solution method used is collective, symmetric point Gauss–Seidel relaxation, accelerated by non‐linear multigrid. Initial finest grid solutions have been obtained by nested iteration. Automatic grid adaptation has been applied for obtaining sharp shocks. An indication is given of the mathematical quality of four different boundary conditions for the outlet flow. Two transonic flow solutions with shock are presented: a choked and a non‐choked flow. Both flow solutions show good shock capturing. A comparison is made with experimental results.

Recursive and/or iterative estimation of the two-dimensional velocity field and reconstruction of three-dimensional motion

In this article an approach is presented for estimating the two-dimensional velocity field from a sequence of images as well as a method for reconstructing the 3-D motion and structure from the velocity field. We suppose that the 2-D velocity vector is the projection of the 3-D velocity vector on the image plane and that the surfaces of the 3-D objects are locally planar and rigid, toobtain spatial relations on the velocity field. If the projection is perspective, we use the “perspective” velocity vector. The spatial relations are used for a linear etimation of the velocity field using local measurements. If the velocity vector has to be estimated on a 2-D region, measurement of the spatial and temporal gradients of the intensity is needed; for edge-based estimation the normal velocity component on an edge is used. In the case of moving edges a Kalman filter is proposed, and in the case of two-dimensional regions a Gauss-Seidel relaxation algorithm, which is iterative and non-recursive, is proposed. We also present a recursive estimator which may be useful in predictive motion-compensating coding. Concerning the reconstruction of the 3-D motion parameters, we demonstrate that, if the “perspective” velocity vector is known at three points, the 3-D motion parameters and the orientation of the plane defined by the three points can be reconstructed. We also demonstrate that, if the “perspective” velocity vector is known at four points, the 3-D motion parameters and the relative depths can be reconstructed by solving two systems of three linear equations and a system of four linear equations.

Defect correction and multigrid for an efficient and accurate computation of airfoil flows

Results are presented for an efficient solution method for second-order accurate discretizations of the 2D steady Euler equations. The solution method is based on iterative defect correction. Several schemes are considered for the computation of the second-order defect. In each defect correction cycle, the solution is computed by non-linear multigrid iteration, in which collective symmetric Gauss-Seidel relaxation is used as the smoothing procedure. A finite volume Osher discretization is applied throughout. The computational method does not require any tuning of parameters. The airfoil flow solutions obtained show a good resolution of all flow phenomena and are obtained at low computational costs. The rate of convergence is grid-independent. The method contributes to the state of the art in efficiently computing flows with discontinuities.

Computation of vortical interaction for a sharp-edged double-delta wing

An implicit finite-difference scheme is used to compute the three-dimensional incompressible laminar vortical flow around a sharp-edged double-delta wing with an aspect ratio of 2.06. fiy adding a time derivative of the pressure to the continuity equation, the unsteady incompressible Navier-Stokes equations can be integrated like a conventional parabolic time-dependent system of equations. The flux-difference split scheme combines approximate factorization in crossflow planes with a symmetric planar Gauss-Seidel relaxation in the remaining spatial direction. Up to second-order spatial accuracy is achieved by using an upwind differencing similar to a total variation diminishing scheme. Computations are performed at Re = 1.4 X IO6 and a = 20 deg. Numerical results indicate that the first-order-accurate scheme is unable to capture the wing vortex, while the second-brder-accurate scheme has successfully simulated the vortical interaction between the strake and wing vortices.

Relaxation Methods for Minimum Cost Ordinary and Generalized Network Flow Problems

We propose a new class of algorithms for linear cost network flow problems with and without gains. These algorithms are based on iterative improvement of a dual cost and operate in a manner that is reminiscent of coordinate ascent and Gauss-Seidel relaxation methods. We compare our coded implementations of these methods with mature state-of-the-art primal simplex and primal-dual codes, and find them to be several times faster on standard benchmark problems, and faster by an order of magnitude on large, randomly generated problems. Our experiments indicate that the speedup factor increases with problem dimension.