Lower-upper Symmetric Gauss-Seidel Research Articles

Computation for Unsteady Incompressible Flow on Unstructured Grid.

The artificial compressibility algorithm for incompressible flows is applied to an unstructured grid method. It is based on a finite volume cell-vertex upwind technique and uses a time-marching procedure to efficiently solve the continuity equations as well as the momentum equations. The Lower-Upper Symmetric Gauss-Seidel(LU-SGS)approximate factorization scheme is implemented with the Navier-Stokes solver on arbitrary three-dimensional unstructured grid. Numerical simulation is performed for steady and unsteady incompressible flow problems, 2-D wake of cylinder, 3-D stenosed tube and branched tube.

Fast, block lower-upper symmetric Gauss-Seidel scheme for arbitrary grids

Fast, block lower-upper symmetric Gauss-Seidel scheme for arbitrary grids

Numerical prediction of enhanced heat flux due to shock‐on‐shock interaction in hypersonic nonequilibrium flow

This paper describes the numerical solutions of type‐IV shock‐on‐shock interactions in hypersonic thermochemical nonequilibrium air flows around blunt bodies. The Navier‐Stokes equation solver for a chemically reacting and vibrationally relaxing gas mixture was applied to the present problem, where the concepts of the Advection Upstream Splitting Method (AUSM) and the Lower‐Upper Symmetric Gauss‐Seidel (LU‐SGS) method were basically employed along with the two‐temperature thermochemical model of Park. The aerodynamic heating with or without the shock‐on‐shock interaction to a sphere and circular cylinders are simulated for a hypersonic nonequilibrium flow. The numerical results show that typical type‐IV shock‐on‐shock interaction pattern with a supersonic jet structure is also formed in a high‐enthalpy thermochemical nonequilibrium flow, and that the contribution of convective heat flux in the translational/rotational mode to the total heat flux is dominant. Furthermore, the inherent unsteadiness of nonequilibrium type‐IV shock‐on‐shock interaction flowfield is discussed briefly.

Validation of Unstructured Grid Method for Incompressible Flows using Artificial Compressibility Method.

The artificial compressibility algorithm for incompressible flows is applied to an unstructed grid method. It is based on a finite volume cell-vertex upwind technique and uses a time-marching procedure to efficiently solve the continuity equation as well as the momentum equations. The Lower-Upper Symmetric Gauss-Seidel (LU-SGS) approximate factorization scheme is implemented with the Navier-Stokes solver on arbitrary three-dimensional unstructured grid. The LU-SGS algorithm provides a significant enhancement of the convergence rate. The accuracy of the method is validated by applying the method to incompressible flow problems of a 2-D cavity and 3-D duct.

A Fast, Matrix-free Implicit Method for Compressible Flows on Unstructured Grids

A fast, matrix-free implicit method has been developed to solve the three-dimensional compressible Euler and Navier–Stokes equations on unstructured meshes. An approximate system of linear equations arising from the Newton linearization is solved by the GMRES (generalized minimum residual) algorithm with a LU-SGS (lower–upper symmetric Gauss–Seidel) preconditioner. A remarkable feature of the present GMRES+LU-SGS method is that the storage of the Jacobian matrix can be completely eliminated by approximating the Jacobian with numerical fluxes, resulting in a matrix-free implicit method. The method developed has been used to compute the compressible flows around 3D complex aerodynamic configurations for a wide range of flow conditions, from subsonic to supersonic. The numerical results obtained indicate that the use of the GMRES+LU-SGS method leads to a significant increase in performance over the best current implicit methods, GMRES+ILU and LU-SGS, while maintaining memory requirements similar to its explicit counterpart. An overall speedup factor from eight to more than one order of magnitude for all test cases in comparison with the explicit method is demonstrated.

Numerical Simulation of Hypersonic Shock-Induced Combustion Ramjets

Hypersonic air-breathing propulsion utilizing shock-induced combustion ramjets is investigated. Twodimensional geometries are simulated with planar and axisymmetric configurations, as well as external and mixed-compression configurations. The lower-upper Symmetric Gauss-Seidel scheme combined with a symmetric shock-capturing total variation diminishing scheme are used to solve the Euler equations, with nonequilibrium chemical reactions. The finite rate chemistry model includes 13 species (H2, O2, H, O, OH, H2O, HO2, H2O2, N, NO, HNO, N2, and NO2) and 33 reactions. The numerical method has been verified by comparison with H2/air induction delay times, analytical solutions to wedge problems, and exothermic blunt body flows. Results obtained with an inviscid, chemically nonequilibrium numerical approach and with realistic geometries demonstrate that shock-induced combustion can be used as a viable means of hypersonic propulsion. As part of the combustor design, it has also been numerically demonstrated that a minimum-entrop y, or Chapman-Jouguet condition exists for oblique-detonation waves generated by wedges in nonequilibrium chemically reacting H2/air flowfields.

Reordering of Hybrid Unstructured Grids for Lower-Upper Symmetric Gauss-Seidel Computations

This note is devoted to the three-dimensional hybrid grid implementation of the LU-SGS method. The original method was based on sweeps through grid node numbers and was applied to two-dimensional Euler computations. Unfortunately, in its original formulation, the method cannot be vectorized; in addition the method is not compatible with commonly used edge-based data structure. To fit the method to the three-dimensional computations on a vector supercomputer, grid reordering is necessary. It was applied to the cell-centered finite volume method

Reordering of hybrid unstructured grids for lower-upper symmetric Gauss-Seidel computations

Reordering of hybrid unstructured grids for lower-upper symmetric Gauss-Seidel computations

Parallelization of a three-dimensional flow solver for Euler rotorcraft aerodynamics predictions

An approach for parallelizing the three-dimensional Euler/Navier-Stokes rotorcraft computational fluid dynamics flow solver transonic unsteady rotor Navier-Stokes (TURNS) is introduced. Parallelization is performed using a domain decomposition technique that is developed for distributed-memory parallel architectures. Communication between the subdomains on each processor is performed via message passing in the form of message passing interface subroutine calls. The most difficult portion of the TURNS algorithm to implement efficiently in parallel is the implicit time step using the lower-upper symmetric Gauss-Seidel (LU-SGS) algorithm. Two modifications of LUSGS are proposed to improve the parallel performance. First, a previously introduced Jacobi-like method called data-parallel lower upper relaxation (DP-LUR) is used. Second, a new hybrid method is introduced that combines the Jacobi sweeping approach in DP-LUR for interprocessor communications and the symmetric Gauss-Seidel algorithm in LU-SGS for on-processor computations. The parallelized TURNS code with the modified implicit operator is implemented on two distributed-memory multiprocessor, the IBM SP2 and Thinking Machines CM-5, and used to compute the three-dimensional quasisteady and unsteady flowfield of a helicopter rotor in forward flight. Good parallel speedups with a low percentage of communication are exhibited by the code. The proposed hybrid algorithm requires less CPU time than DP-LUR while maintaining comparable parallel speedups and communication costs. Execution rates found on the IBM SP2 are impressive; on 114 processors of the SP2, the solution time of both quasisteady and unsteady calculations is reduced by a factor of about 12 over a single processor of the Cray C-90.

Lower-upper symmetric Gauss-Seidel scheme exhibiting asymmetric vortices over slender bodies

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Data-parallel lower-upper relaxation method for the Navier-Stokes equations

The lower-upper symmetric Gauss-Seidel method is modified for the simulation of viscous flows on massively parallel computers. The resulting diagonal data-parallel lower-upper relaxation (DP-LUR) method is shown to have good convergence properties on many problems. However, the convergence rate decreases on the high cell aspect ratio grids required to simulate high Reynolds number flows. Therefore, the diagonal approximation is relaxed, and a full matrix version of the DP-LUR method is derived. The full matrix method retains the data-parallel properties of the original and reduces the sensitivity of the convergence rate to the aspect ratio of the computational grid. Both methods are implemented on the Thinking Machines CM-5, and a large fraction of the peak theoretical performance of the machine is obtained. The low memory use and high parallel efficiency of the methods make them attractive for large-scale simulation of viscous flows.

Matrix dissipation for central difference schemes with combustion

The effect of artificial viscosity is investigated for problems related to supersonic combustion. An implicit lower-upper symmetric Gauss-Seidel finite volume method is employed for solving the full, compressible, two-dimensional averaged Navier-Stokes and species transport equations. For the right-hand side discretization central differences are used. Therefore some kind of artificial viscosity is necessary to reduce oscillations near shock waves and to enable convergence to machine accuracy. In comparison to the standard second- and fourth-order scalar dissipation a matrix dissipation reduces the amount of artificial viscosity by scaling each equation individually. For the necessary absolute flux Jacobian matrix a new decomposition is presented keeping the additional cost moderate. With an appropriate sensor the scheme also gets total variation diminishing properties. Calculations using different dissipation models are presented and advantages using a matrix dissipation are shown.

Comparison of nonequilibrium solution algorithms applied to chemically stiff hypersonic flows

Three solution algorithms, explicit under-relaxation, point implicit, and lower-upper symmetric Gauss-Seidel, are used to compute nonequilibrium flow around the Apollo 4 return capsule at the 62-km altitude point in its descent trajectory. By varying the Mach number, the efficiency and robustness of the solution algorithms were tested for different levels of chemical stiffness. The performance of the solution algorithms degraded as the Mach number and stiffness of the flow increased. At Mach 15 and 30, the lower-upper symmetric Gauss-Seidel method produces an eight order of magnitude drop in the energy residual in one-third to one-half the Cray C-90 computer time as compared to the point implicit and explicit under-relaxation methods. The explicit under-relaxation algorithm experienced convergence difficulties at Mach 30 and above. At Mach 40 the performance of the lower-upper symmetric Gauss-Seidel algorithm deteriorates to the point that it is out performed by the point implicit method. The effects of the viscous terms are investigated. Grid dependency questions are explored.

Comparison of Relaxation Methods for the Euler Equations.

The performance of four relaxation methods for the compressible Euler equations is examined. The Euler equations are cast into the Euler implicit delta form. Four spatial discretization methods, the central difference (CD), total variation diminishing of Yee and Harten (TVD), flux difference splitting of Roe (FDS) and flux vector splitting of Van Leer (FVS), are applied to the spatial derivatives of the right-hand side. Flux splitting Gauss Seidel (FSGS), approximate flux splitting Gauss Seidel (AFSGS-I and AFSGS-II), and lower-upper symmetric Gauss Seidel (LU-SGS) methods are tested for the solution of the implicit equations. Numerical results are presented for a two-dimensional flow about a circular arc bump. Compatibility and efficiency of the numerical solutions are analyzed for all combinations between relaxation methods and spatial discretization methods. Steady-state solutions are independent of the relaxation methods. The AFSGS-II method gives the best efficiency for all spatial discretization methods.

Data-parallel lower-upper relaxation method for reacting flows

The implicit lower-upper symmetric Gauss-Seidel (LU-SGS) method of Yoon and Jameson is modified for use on massively parallel computers. The method has been implemented on the Thinking Machines CM-5 and the MasPar MP-1 and MP-2, where large percentages of the theoretical peak floating point performance are obtained. It is shown that the new data-parallel LU relaxation method has better convergence properties than the original method for two different inviscid compressible flow simulations. The convergence is also improved for five-species reacting air computations. The performance of the method on various partitions of the CM-5 and on the MasPar computers is discussed. The new method shows promise for the efficient simulation of very large perfect gas and reacting flows.

Multigrid convergence of an implicit symmetric relaxation scheme

The multigrid method has been applied to an existing three-dimensional compressible Euler solver to accelerate the convergence of the implicit symmetric relaxation scheme. This lower-upper symmetric Gauss-Seidel implicit scheme is shown to be an effective multigrid driver in three dimensions. A grid refinement study is performed including the effects of large cell aspect ratio meshes. Performance figures of the present multigrid code on Cray computers including the new C90 are presented. A reduction of three orders of magnitude in the residual for a three-dimensional transonic inviscid flow using 920 k grid points is obtained in less than 4 min on a Cray C90.

Particle Deposition on a Truncated Cylinder in a Supersonic Flow at Low Pressure

Deposition of particles, suspended in supersonic gas flows, onto the impaction plane of a truncated cylinder in low pressure has been studied numerically. The analysis of axisymmetric compressible Navier-Stokes equations was carried out, for a gas phase, by using the lower-upper symmetric Gauss-Seidel (LU-SGS) scheme. In addition, the Lagrangian trajectory model was used for a particle phase. The dilute gas-particle flow was assumed to be coupled in one direction. The effect of freestream pressure on the process of particle deposition was investigated. The local characteristics of deposition were also demonstrated in terms of particle size and freestream pressure. It is found that the particles with low total deposition efficiencies can be deposited rather uniformly on the impaction plane when the effect of thermophoresis is considered.

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.