Gauss-Seidel Scheme Research Articles

A lightweight, conservation-moment-based implicit gas kinetic Lax–Wendroff scheme for all Knudsen number isothermal gas flows

This work presents an efficient implicit gas kinetic Lax–Wendroff scheme for steady isothermal gas flows in all Knudsen number (Kn) regimes. In the scheme, the discrete velocity Bhatnagar–Gross–Krook model equations (DVE) and the associated conservation moment equations (CME) are coupled and solved by matrix-free implicit schemes. Thanks to obtaining the fluxes of the CME by multiplying the fluxes of the DVE with a projection matrix and utilizing the equilibrium distribution functions at the new time predicted by the CME, both the complicated macro fluxes reconstruction of the CME and the calculation of the Jacobian matrix of the equilibrium distribution functions are not needed in the scheme, which makes the scheme lightweight. Moreover, to enhance the accuracy of the predicted equilibrium distribution functions at the new time to improve the convergence, a symmetric Gauss–Seidel scheme with inner iterations is used to solve the CME system. Due to the coupling between the DVE and the CME, the highly efficient implicit scheme for the CME drives the DVE system to converge quickly for the continuum and near-continuum flows. Furthermore, to verify the accuracy and high efficiency of the proposed implicit scheme, comparison studies of several two-dimensional isothermal rarefied gas flow cases simulated by the present implicit scheme and the explicit gas kinetic Lax–Wendroff scheme are also provided. The numerical results show that the present implicit scheme can be as accurate as its explicit counterpart with one to two orders times speed-up in all Kn number flow regimes.

A compressible flow solver for turbomachinery of the real gases with strongly variable properties

Accurate prediction of real gas flows with strongly variable properties is an essential prerequisite for turbomachinery performance analysis. This paper presents a density-based solver for real gas flows in turbomachinery. The solver employs a third-order discretization scheme to improve computational accuracy. Real gas equations of state and look-up table are utilized in the solver to account for the strong nonlinear variations of thermophysical properties. The preconditioning method is adopted for simulations, including low-Mach flow regions, to address convergence difficulties. The implicit Time Consistent Preconditioned Gauss-Seidel Scheme (TCPGS) is employed for the time-stepping. The solver is validated by transonic compressor cascade wind tunnel experiments. Numerical test cases are adopted to evaluate the solver performance, including NASA Rotor 67, NASA Rotor 37, real gas flows in the Laval nozzle, and compressor seal. It has been proved that the numerical test results are consistent with published technical data. The advantages of this solver include numerical stability, computational efficiency, and physical accuracy, which indicate its applicability in the turbomachinery design process for real gases.

Estimation of Nuclear Mass Formulas Coefficients Using Least-Squares Method Based on Gauss-Seidel Scheme: A Comparative Study Between Three Models

This paper presents the analysis and implementation of the least-squares method based on the Gauss-Seidel scheme for solving nuclear mass formulas. The least-squares method leads to the solution of the system by iterations. The main advantages of the discussed method are simplicity and high accuracy. Moreover, the method enables us to process large data quickly in practice. To demonstrate the effectiveness of the method, implementation using the FORTRAN language is carried out. The steps of the algorithm are detailed. Using 2331 nuclear masses with Z ≥ 8 and N ≥ 8, it was shown that the performance of the liquid drop mass formula with six parameters improved in terms of root mean square (r.m.s. deviation equals 1.28 MeV), compared to the formula of liquid drop mass with six parameters without microscopic energy, deformation energy and congruence energy (r.m.s. deviation equals 2.65 MeV). The nuclear liquid drop model is revisited to make explicit the role of the microscopic corrections (shell and pairing). Deformation energy and the congruence energy estimate have been used to obtain the best fit. It is shown that the performance of the new approach is improved by a model of eight parameters, compared to the previous model of six parameters. The obtained r.m.s. result for the new liquid drop model in terms of masses is equal to 1.05 MeV.

Implicit block data-parallel relaxation scheme of Navier–Stokes equations using graphics processing units

The granularity of computational fluid dynamics (CFD) generally refers to the point granularity parallelization as a unit of the grid when graphics processing units (GPUs) are utilized as the computing carrier. In commonly deployed implicit time advancement schemes, the parallel dimensionality must be reduced, resulting in the time advancement procedure becoming the only highly time-consuming step in the whole CFD computing procedures. In this paper, a block data-parallel lower-upper relaxation (BDPLUR) scheme based on Jacobi iteration and Roe's flux scheme is proposed and then implemented on a GPU. Numerical experiments are carried out and show that the convergence speed of the BDPLUR scheme, especially when implemented on a GPU, is approximately ten times higher than that of the original data-parallel lower-upper relaxation scheme and more than 100 times higher than that of the lower-upper symmetric Gauss–Seidel scheme. Moreover, the influence of different Courant–Friedrichs–Lewy numbers on the convergence time is discussed, and different viscous matrices are compared. Standard cases are adopted to verify the effectiveness of the BDPLUR scheme.

Least-squares fitting applied to nuclear mass formulas. Solution by the Gauss–Seidel method

In this paper, a numerical method optimizing the coefficients of the semi empirical mass formula or those of similar mass formulas is presented. The optimization is based on the least-squares adjustments method and leads to the resolution of a linear system which is solved by iterations according to the Gauss–Seidel scheme. The steps of the algorithm are given in detail. In practice, the method is very simple to implement and is able to treat large data in a very fast way. In fact, although this method has been illustrated here by specific examples, it can be applied without difficulty to any experimental or statistical data of the same type, i.e. those leading to linear system characterized by symmetric and positive-definite matrices.

A theoretical and numerical analysis of a Dirichlet-Neumann domain decomposition method for diffusion problems in heterogeneous media

Problems with localized nonhomogeneous material properties present well-known challenges for numerical simulations. In particular, such problems may feature large differences in length scales, causing difficulties with meshing and preconditioning. These difficulties are increased if the region of localized dynamics changes in time. Overlapping domain decomposition methods, which split the problem at the continuous level, show promise due to their ease of implementation and computational efficiency. Accordingly, the present work aims to further develop the mathematical theory of such methods at both the continuous and discrete levels. For the continuous formulation of the problem, we provide a full convergence analysis. For the discrete problem, we show how the described method may be interpreted as a Gauss-Seidel scheme or as a Neumann series approximation, establishing a convergence criterion in terms of the spectral radius of the system. We then provide a spectral scaling argument and provide numerical evidence for its justification.

Multigrid Method for Optimal Control Problem Constrained by Stochastic Stokes Equations with Noise

Optimal control problems governed by stochastic partial differential equations have become an important field in applied mathematics. In this article, we investigate one such important optimization problem, that is, the stochastic Stokes control problem with forcing term perturbed by noise. A multigrid scheme with three-factor coarsening to solve the corresponding discretized control problem is presented. On staggered grids, a three-factor coarsening strategy helps in simplifying the inter-grid transfer operators and reduction in computation (CPU time). For smoothing, a distributive Gauss–Seidel scheme with a line search strategy is employed. To validate the proposed multigrid staggered grid framework, numerical results are presented with white noise at the end.

A partially proximal S-ADMM for separable convex optimization with linear constraints

A classical approach to solving two-block separable convex optimization could be the symmetric alternating direction method of multipliers (S-ADMM). However, its convergence may not be guaranteed for a general multi-block case without additional assumptions. Bai et al. proposed a variant of S-ADMM entitled the generalized symmetric ADMM (GS-ADMM), in which the variables are regrouped into two groups firstly. The two groups of variables are updated in a Gauss-Seidel scheme, while the variables within each group are updated in a Jacobi scheme and the Lagrangian multipliers are updated two times. In order to derive its convergence property, the authors add a special proximal term to each subproblem. In this paper, inspired by the partial PPA block-wise ADMM (PPBADMM) [32] proposed by Shen et al., we propose a partially proximal S-ADMM (PPSADMM). In PPSADMM, the special proximal term is only added to the subproblems in the first group as PPBADMM. We perform an extension step on all variables with a fixed step size at the end of each iteration. Without stringent assumptions, we establish the global convergence result and the O(1/t) convergence rate in the ergodic sense for PPSADMM. Its numerical performance is justified on two types of problems.

Extrapolated refinement of generalized gauss- seidel scheme for solving system of linear equations

In this paper, Extrapolated Refinement of Generalized Gauss-Seidel scheme for solving system of linear equations has been presented. In order to accelerate the rate of convergence of the scheme, the one-parameter family of splitting procedure has been introduced and the convergence of the method is well established. To validate the proposed method, three numerical examples were considered. Comparisons were made among Refinement of Generalized Jacobi, Generalized Gauss-Seidel, Refinement of Generalized Gauss-Seidel and Extrapolated Refinement of Generalized Gauss-Seidel schemes with respect to the number of iterations to converge, computational running time and storage capacity. The numerical results presented in tables show that the Extrapolated Refinement of Generalized Gauss-Seidel scheme is more efficient than the other three schemes considered for comparison.

Interaction between laser-driven high-velocity metal granule and gas

We apply laser-driven launching technique to the interaction between high-velocity metal granule and gas to investigate high speed gas-solid two-phase flow transport. Optimization of target design and adjustment of laser parameters were used to control the laser-driven process. Diagnostic tools with high resolving power of space and time as well as higt accuracy were set up to capture the high-velocity granule. The unsteady flow field of the granule was simulated by solving the the 3-D Reynolds averaged Navier-Stokes equations and six degree of freedom ballistic equation. With the implicit Gauss-Seidel scheme, the code was advanced in time. The results indicated that the laser-driven launching technique was effective and the shadow photographs of high-velocity granule were taken successfully. The aerodynamic drag of high-velocity granule was computed with numerical simulation.

Gas-Kinetic Harmonic Balance Method for Unsteady Supersonic/Hypersonic Flows with Oscillating Motions

In this paper, a novel gas-kinetic harmonic balance method (HBM) is developed for unsteady flows with oscillating motions. Particularly, the emphasis here is on supersonic and hypersonic flows. Because of the presence of an additional source term in the harmonic balance equations, usually very small time steps are needed for explicit algorithms, and some factorization-based implicit algorithms may fail to converge in the case of higher frequencies. To achieve better convergence performance and robustness, a Jacobian-free Newton–Krylov method is adopted. By constructing a simplified preconditioning matrix, the lower–upper symmetric Gauss–Seidel scheme is employed as a preconditioner, so that the computational cost and memory requirement can be reduced. Considering that conventional spatial discretization schemes may encounter numerical difficulties and low robustness in high-Mach-number flows, a more general gas-kinetic scheme (GKS) is extended for the harmonic balance equations. As the HBM solves a mathematically steady problem, the present GKS focuses on evaluation of the time-independent fluxes at each cell interface and each subtime level. The gas-kinetic HBM is applied to a supersonic oscillating flat plate cascade and an oscillating panel in hypersonic flow. By comparison with other methods, the advantages of the developed method are demonstrated.

An Efficient Preconditioner for 3-D Finite Difference Modeling of the Electromagnetic Diffusion Process in the Frequency Domain

Krylov subspace solvers for frequency-domain electromagnetic forward modeling problems converge remarkably more slowly as the period increases. In this article, we present an efficient four-color cellblock Gauss Seidel (GS) preconditioner for finite-difference (FD) electromagnetic modeling in geophysical applications. Rather than updating the FD electromagnetic (EM) equation edge by edge, as in a traditional GS scheme, we renew six edge components attached to one node simultaneously (i.e., in cellblock manner) effectively enforcing a local divergence free condition for currents. To improve implementation efficiency, we reorder the nodes on the FD grid into four colors so that nodes in each color are uncoupled, allowing the use of highly parallel vectorized algorithms. The four-color cellblock GS preconditioner is implemented in the MATLAB code, in conjunction with a BiCGstab solver. It is compared, in terms of iteration number and computing time, with other three commonly used preconditioners [GS, symmetric successive overrelaxation (SSOR) and incomplete lower and upper triangular matrix decomposition (ILU)] on three models—two synthetic and one modified from the version of real data inversion. The comparison indicates that the proposed algorithm is extremely stable and efficient compared with the other three preconditioners tested, over a range of periods (1-1000 s). Especially at long periods, the improvement of our proposed algorithm is substantial. In addition, a parallel implementation of the cellblock GS preconditioner is straightforward due to the independence of nodes in each color.

A conservation-moment-based implicit finite volume lattice Boltzmann method for steady nearly incompressible flows

This paper presents an efficient, low memory cost, implicit finite volume lattice Boltzmann method (FVLBM) based on conservation moments acceleration for steady nearly incompressible flows. In the proposed scheme, not as the conventional implicit schemes, both the micro lattice Boltzmann equations (LBE) and the associated conservation moment equations are solved by the matrix-free, lower-upper symmetric Gauss-Seidel scheme (LUSGS) and the conservation moment equations are used to predict equilibrium distribution functions at the new time, which eliminates the storage of the Jacobian matrix of the collision term in the implicit LBE system and provides a driving force for the fast convergence of the LBE. Moreover, by utilizing the projection matrix and the collision invariant, we can construct the fluxes of the moment equations efficiently from the fluxes of the LBE and avoid the time-consuming reconstruction procedure for obtaining the fluxes of the moment equations. To demonstrate the accuracy and high efficiency of the proposed scheme, comparison studies of simulation results of several two-dimensional testing cases by the present scheme and an explicit FVLBM are conducted and numerical results show that the proposed implicit scheme can be as accurate as its explicit counterpart with 1 or 2 orders times speedup.

A linearly convergent doubly stochastic Gauss–Seidel algorithm for solving linear equations and a certain class of over-parameterized optimization problems

Consider the classical problem of solving a general linear system of equations $Ax=b$. It is well known that the (successively over relaxed) Gauss-Seidel scheme and many of its variants may not converge when $A$ is neither diagonally dominant nor symmetric positive definite. Can we have a linearly convergent G-S type algorithm that works for {\it any} $A$? In this paper we answer this question affirmatively by proposing a doubly stochastic G-S algorithm that is provably linearly convergent (in the mean square error sense) for any feasible linear system of equations. The key in the algorithm design is to introduce a {\it nonuniform double stochastic} scheme for picking the equation and the variable in each update step as well as a stepsize rule. These techniques also generalize to certain iterative alternating projection algorithms for solving the linear feasibility problem $A x\le b$ with an arbitrary $A$, as well as high-dimensional minimization problems for training over-parameterized models in machine learning. Our results demonstrate that a carefully designed randomization scheme can make an otherwise divergent G-S algorithm converge.

Numerical simulation of multiple steady and unsteady flow modes in a medium-gap spherical Couette flow

We study the multiple steady and unsteady flow modes in a medium-gap spherical Couette flow (SCF) by solving the three-dimensional incompressible Navier–Stokes equations. We have used an artificial compressibility method with an implicit line Gauss–Seidel scheme. The simulations are performed in SCF with only the inner sphere rotating. A medium-gap clearance ratio, $$\sigma =\left( R_{2}-R_{1}\right) /R_{1}=0.25,$$ has been used to investigate various flow states in a range of Reynolds numbers, $${Re}\in [400,6500]$$ . First, we compute the 0-vortex basic flow directly from the Stokes flow as an initial condition. This flow exists up to $${Re}=4900$$ after which it evolves into spiral 0-vortex flows with wavenumber $$s_p=3,4$$ in the range $${Re} \in [4900,6000]$$ , and then the flows become turbulent when $${Re}>6000$$ . Second, we obtain the steady 1-vortex flow by using the 1-vortex flow at $${Re} =700$$ for $$\sigma =0.18$$ as the initial conditions and found that it exists for $${Re} \in [480,4300]$$ . The 1-vortex flow becomes wavy 1-vortex in the range $${Re} \in [4400,5000]$$ . Further increasing the Reynolds number, we obtain new spiral waves of wavenumber $$s_p=3$$ for $${Re}\in [5000, 6000]$$ . The flow becomes turbulent when $${Re}>6000$$ . Third, we obtain the steady 2-vortex flow by using the 2-vortex flow at $${Re} =900$$ for $$\sigma =0.18$$ as the initial conditions and found that it exists for $${Re} \in [700,1900]$$ . With increasing Reynolds number the 2-vortex flow becomes partially wavy 2-vortex in the small range $${Re} \in [1900,2100]$$ . We obtain distorted spiral wavy 2-vortex in the range $${Re} \in [4000,5000]$$ . when $${Re}>6000$$ the flow evolves into spiral 0-vortex flow and becomes turbulent. The present flow scenarios with increasing Re agree well with the experimental results and further we obtain new flow states for the 1-vortex and 2-vortex flows.

High order Gauss-Seidel schemes for charged particle dynamics

Gauss-Seidel projection methods are designed for achieving desirable long-term computational efficiency and reliability in micromagnetics simulations. While conventional Gauss-Seidel schemes are explicit, easy to use and furnish a better stability as compared to Euler's method, their order of accuracy is only one. This paper proposes an improved Gauss-Seidel methodology for particle simulations of magnetized plasmas. A novel new class of high order schemes are implemented via composition strategies. The new algorithms acquired are not only explicit and symmetric, but also volume-preserving together with their adjoint schemes. They are highly favorable for long-term computations. The new high order schemes are then utilized for simulating charged particle motions under the Lorentz force. Our experiments indicate a remarkable satisfaction of the energy preservation and angular momentum conservation of the numerical methods in multi-scale plasma dynamics computations.

Improvement of computational efficiency of circular function-based gas kinetic scheme by using Jacobian-free Newton–Krylov method

The gas-kinetic BGK scheme is a promising method for simulation of inviscid and viscous flows. Different from conventional Navier–Stokes (N-S) solvers, it evaluates the inviscid and viscous fluxes simultaneously by reconstructing the local solution of BGK Boltzmann equation at the cell interface. Due to its inherently superior dissipation property, it usually gives accurate and robust numerical results. However, a notable drawback of the BGK-type scheme is the low computational efficiency. Recently, aimed at reducing the computational effort, a circular function-based BGK (CBGK) scheme was developed. Nevertheless, it is still time consuming because the original scheme used an explicit way in the time integration as in most existing BGK schemes and the convergence speed can be slow. To improve the computational efficiency, an implicit CBGK scheme is developed in this paper by incorporating the Jacobian-free Newton–Krylov (JFNK) method into the scheme. Particularly, the generalized minimal residual (GMRES) approach is employed to iteratively solve the large linear equation system. With the help of the Jacobian-free approach, a faster convergence speed can be achieved without explicitly computing and storing the flux Jacobian, which is usually a large and sparse matrix. In order to reduce the number of GMRES iterations, the preconditioning is also adopted and the Lower-Upper symmetric Gauss-Seidel (LUSGS) scheme is employed as a preconditioner. For validation of the present CBGK-JFNK method, several two-dimensional inviscid and viscous test cases are investigated. The numerical results show that the needed computational time is significantly reduced as compared with the original explicit CBGK scheme and the CBGK-LUSGS method.

Verification and validation of a high-order strand grid method for two-dimensional turbulent flows

Abstract In this paper, we construct a novel hybrid of two one-dimensional schemes in order to leverage several advantages for solving two-dimensional turbulent flows. Building upon previous work by the authors and others, we combine one-dimensional flux correction along body surfaces along with high-order summation-by-parts finite differences normal to surfaces. A new semi-implicit multigrid solution method is presented that capitalizes on the unique directional properties of each scheme, utilizing an explicit multigrid scheme along the surface direction, and an implicit Gauss-Seidel scheme along the strand direction. Turbulence closure is achieved with a robust version of the Spalart-Allmaras turbulence model that accommodates negative values of the turbulence working variable. The hybrid scheme exhibits fourth-order convergence using the method of manufactured solutions. Fundamental validation studies of the turbulent flux correction method are conducted in two dimensions, using the NASA-Langley turbulence resource as a means for comparison. Results are presented that demonstrate improvements in accuracy with minimal computational and algorithmic overhead over traditional second-order algorithms.

IHDG: An Iterative HDG Framework for Partial Differential Equations

We present a scalable iterative solver for high-order hybridized discontinuous Galerkin (HDG) discretizations of linear partial differential equations. It is an interplay between domain decomposition methods and HDG discretizations, and hence inheriting advances from both sides. In particular, the method can be viewed as a Gauss-Seidel approach that requires only independent element-by-element and face-by-face local solves in each iteration. As such, it is well-suited for current and future computing systems with massive concurrencies. Unlike conventional Gauss-Seidel schemes which are purely algebraic, the convergence of iHDG, thanks to the built-in HDG numerical flux, does not depend on the ordering of unknowns. We rigorously show the convergence of the proposed method for the transport equation, the linearized shallow water equation and the convection-diffusion equation. For the transport equation, the method is convergent regardless of mesh size $h$ and solution order $p$, and furthermore the convergence rate is independent of the solution order. For the linearized shallow water and the convection-diffusion equations we show that the convergence is conditional on both $h$ and $p$. Extensive steady and time-dependent numerical results for the 2D and 3D transport equations, the linearized shallow water equation, and the convection-diffusion equation are presented to verify the theoretical findings.

Unsteady aerodynamic characteristics investigation of rotor airfoil under variational freestream velocity

Numerical investigations based on CFD method have been accomplished to illustrate the unsteady aerodynamic characteristics of the SC1095 airfoil under conditions of coupled freestream velocity/pitching oscillation. In order to simulate the variational freestream velocity of airfoil, the moving-embedded grid method is employed to calculate the fluctuating velocity. The unsteady RANS equations coupling with the dual time-stepping approach are chosen as the governing equations to predict flowfield of airfoil under variational freestream velocity condition. The implicit Lower–Upper Symmetric Gauss–Seidel scheme and Spalart–Allmaras turbulence model are employed to improve the calculation efficiency and accuracy, respectively. Based on the present CFD method, the unsteady aerodynamic characteristics of the airfoil with variational freestream velocity are simulated under conditions of fixed angle and variational angle. The simulated results indicate that the peaks of lift coefficient, drag coefficient and pitching moment coefficient of airfoil are enlarged with the increasing of fluctuating velocity, and the start of airflow reattachment process on airfoil is delayed with the increasing of fluctuating velocities and reduced frequencies due to the effects of the airflow hysteresis and compressibility. Additionally, it is illustrated in the case of different phase angles that these peaks would be reduced by increasing phase angle.