BiCGStab Method Research Articles

MULTIGRID-CONJUGATE GRADIENT TYPE METHODS FOR REACTION–DIFFUSION SYSTEMS

We study multigrid methods in the context of continuation methods for reaction–diffusion systems, where the Bi-CGSTAB and GMRES methods are used as the relaxation scheme for the V-cycle, W-cycle and full approximation schemes, respectively. In particular, we apply the results of Brown and Walker [1997] to investigate how the GMRES method can be used to solve nearly singular systems that occur in continuation problems. We show that for the sake of switching branches safely, one would rather solve a perturbed problem near bifurcation points. We propose several multigrid-continuation algorithms for curve-tracking in nonlinear elliptic eigenvalue problems. Our numerical results show that the algorithms proposed have the advantage of being robust and easy to implement.

Adaptive moving mesh computations for reaction–diffusion systems

In this paper we describe an adaptive moving mesh technique and its application to reaction–diffusion models from chemistry. The method is based on a coordinate transformation between physical and computational coordinates. The transformation can be viewed as a solution of adaptive mesh partial differential equations (PDEs) which is derived from the minimization of a mesh-energy integral. For an efficient implementation we have used an approach in which the numerical solution of the physical PDEs and the adaptive PDEs are decoupled. Further, to avoid solving large nonlinear systems, a second-order implicit–explicit time-integration method in combination with the iterative method Bi-CGSTAB is applied in the method-of-lines procedure. Numerical examples are given in one and two space dimensions.

Ｌａｎｄａｕ‐Ｌｉｆｓｈｉｔｚ‐Ｇｉｌｂｅｒｔ方程式のＣｒａｎｋ‐Ｎｉｃｏｌｓｏｎ法による陰解法

The efficiency of implicit methods in comparison with the classical Runge-Kutta method is investigated when the former are applied to the Landau-Lifshitz-Gilbert equation. The Crank-Nicolson and the two-stage fourth-order implicit Runge-Kutta methods were chosen and implemented by means of Newton's method. Implicit solutions using Newton's method call for the solution of large-scale simultaneous algebraic equations. To decrease the computing time, only the nearest-neighbor interaction was taken into consideration in the coefficient matrix of the simultaneous algebraic equations, which were solved by using the preconditioned Bi-CGSTAB method. Through calculations of three-dimensional magnetization reversal in a fine magnetic particle, it was confirmed that the examined methods yield a much larger maximum usable time step than the classical Runge-Kutta method. It was found that the Crank-Nicolson method takes about a quarter of the computing time required by the classical Runge-Kutta method when a dissipation constant α of 0.01 and a cell size of 8 nm are used. The maximum usable time step yielded by the two-stage fourth-order Runge-Kutta method was found to be almost four times as large as that yielded by the Crank-Nicolson method, though the former method required a longer computing time than the latter in the present calculation.

An improvement in DS-BiCGstab( l) and its application for linear systems in lattice QCD

An improved algorithm of the DS-BiCGstab( l) method (IDSL) is proposed. The DS-BiCGstab( l) method (DSL) selects the value of l dynamically for the BiCGstab( l) method, but it does not always select a proper value for l in terms of its convergence. From this viewpoint, this research has focused on converging behavior with the BiCGstab( l) method, and the DSL has been improved. Several numerical results in lattice QCD show that this improvement leads dynamic selection of l to a value relevant to the convergence than the DSL without a substantial increase in the calculating costs.

BiCGStab(?) for Families of Shifted Linear Systems

We consider a seed system Ax = b together with a shifted linear system of the form (A + σI)x = b, σ ∈ C, A ∈ Cn×n, b ∈ Cn. We develop modifications of the BiCGStab(l) method which allow to solve the seed and the shifted system at the expense of just the matrix-vector multiplications needed to solve Ax = b via BiCGStab(l). On the shifted system, these modifications do not perform the corresponding BiCGStab(l)-method, but we show, that in the case that A is positive real and σ ≥ 0, the resulting method is still a well-smoothed variant of BiCG. Numerical examples from an application arising in quantum chromodynamics are given to illustrate the efficiency of the method developed.

PARALLEL COMPUTATION METHOD FOR PRESSURE FIELD IN INCOMPRESSIBLE FLOWS ON 3D CURVILINEAR COORDINATES

A parallel computational method has been proposed to calculate the pressure field in the threedimensional incompressible flows. The detailed numerical procedures are presented to solve the pressure Poisson equations which are derived with the CBP (Cell-Boundary Pressure) scheme on the collocated grid system. The C-HSMAC method, which is effective to reduce the numerical error related to the continuity of the fluid mass, is improved with the Bi-CGSTAB method and the reasonable treatment on the convergence. Then this method is parallelized with a domain decomposition method using overlapping grid points. As a result of the computations, the validity of the improved C-HSMAC method is confirmed. In addition, it is shown that the speed up of the parallel C-HSMAC method is sufficiently high in the large scale computations.

1506 反復型領域分割法を用いた定常流れ問題の安定化有限要素解析

We try to develop a method to solve stationary flow problems. An iterative domain decomposition method is applied to finite element approximations of stationary Navier-Stokes equations. BiCGSTAB method is used as the iterative solver of the reduced linear system in each step of the Newton iteration. Also, a stabilized finite element method is used in each step of the nonlinear iteration. It is shown that results of the domain decomposition method agree well with results of conventional finite element method.

A NUMERICAL STUDY OF SOLIDIFICATION IN THE PRESENCE OF A FREE SURFACE UNDER MICROGRAVITY CONDITIONS

A numerical study of the relative importance of Marangoni effects under microgravity conditions is presented. The mathematical formulation adopted is based on the enthalpy porosity method. One of the advantages of the fixed grid method is that a unique set of equations and boundary conditions is used for the whole domain, including both solid and liquid phases. The governing equations written in a vorticity-velocity formulation are discretized using a finite volume technique on a staggered grid. A fully implicit method has been adopted for the mass and momentum equations, while the temperature field is solved separately in order to evaluate the variation in the local liquid mass fraction. The resulting algebraic system of equations is solved using a preconditioned BI-CGStab method. Numerical results modelling the free surface, including the effects on it of Marangoni convection, are presented. The influence of the presence of argon in the gap above the free surface is investigated. During the numerical simulations presented in this paper 161 2 41 and 641 2 161 uniform meshes on the whole computational domain for values of Marangoni number ( Ma ) up to 16,120 and Rayleigh number ( Ra ) of 5 have been used.

LEFTMOST EIGENVALUE OF REAL AND COMPLEX SPARSE MATRICES ON PARALLEL COMPUTER USING APPROXIMATE INVERSE PRECONDITIONING

An efficient parallel approach for the computation of the eigenvalue of smallest absolute magnitude of sparse real and complex matrices is provided. The proposed strategy tries to improve the efficiency of the reverse power method. At each inverse power iteration the linear system is solved either by the conjugate gradient scheme (symmetric case) or by the Bi-CGSTAB method (symmetric case). Both solvers are preconditioned employing the approximate inverse factorization and thus are easily parallelized. The satisfactory speed-ups obtained on the CRAY T3E supercomputer show the high degree of parallelization reached by the proposed algorithm.

Efficient Poisson Solver for Semiconductor Device Modeling Using the Multi-Grid Preconditioned BiCGSTAB Method

This paper presents the performance results of an efficient algorithm for solving the three dimensional Poisson equation. The multi-grid method exploits the efficient oscillatory error reduction of basic iterative methods by smoothing on a set of progressively coarsened grids. When used as a preconditioner for BiCGSTAB method, a computationally demanding solver can be shown to be effective for large scale simulations. Varying the number of grids used and the level of overrelaxation as well as exploring the benefits of semicoarsening in the multi-grid preconditioner reveals the underlying strengths of this combined scheme. The convergence properties of the developed solver are tested on a 3D split-gate silicon on insulator (SOI) device.

GPBiCG( m,ℓ): A hybrid of BiCGSTAB and GPBiCG methods with efficiency and robustness

In this paper a new iterative method which is a hybrid of the BiCGSTAB and GPBiCG methods is proposed. The present method is an extension of the BiCGSTAB2 method. In the BiCGSTAB2 method which is a hybrid type method, two parameters of the BiCGSTAB method are adopted at even iteration step, and those of the GPBiCG method are used at odd iteration step. In the present GPBiCG( m,ℓ) method, parameters of the BiCGSTAB method are adopted at consecutive m iteration steps, and afterwards those of the GPBiCG method are also utilized in succession at ℓ iteration steps. We will examine the convergence property of the GPBiCG( m,ℓ) method and investigate its effectiveness through numerical experiments.

Parallel implementation of the Bi-CGSTAB method with block red–black Gauss–Seidel preconditioner applied to the Hermite collocation discretization of partial differential equations

We describe herein the parallel implementation of the Bi-CGSTAB method with a block red–black Gauss–Seidel (RBGS) preconditioner applied to the systems of linear algebraic equations that arise from the Hermite collocation discretization of partial differential equations in two spatial dimensions. The method is implemented on the Cray T3E, a parallel processing supercomputer. Speedup results are discussed.

Convergence acceleration by varying time‐step size using Bi‐CGSTAB method for turbulent flow computation

AbstractA design of varying step size approach both in time span and spatial coordinate systems to achieve fast convergence is demonstrated in this study. This method is based on the concept of minimization of residuals by the Bi‐CGSTAB algorithm, so that the convergence can be enforced by varying the time‐step size. The numerical results show that the time‐step size determined by the proposed method improves the convergence rate for turbulent computations using advanced turbulence models in low Reynolds‐number form, and the degree of improvement increases with the degree of the complexity of the turbulence models. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 454–474, 2001.

Implementations of multi-fluid hydrodynamic simulations on distributed memory computer with a fully parallelizable preconditioned Bi-CGSTAB method

This paper reports recent work in implementing a multi-phase fluid dynamic code on a super computer system with distributed memory architecture. In the numerical model, the dynamics of different fluids is formulated based on a unified framework for general continuum. The computational code was constructed so that it is easily implemented on a distributed memory machine. A projection procedure for solving the flows of various compressibilities results in a pressure Poisson equation which has large discontinuities and singularities in the coefficients and the right-hand side terms. A preconditioned Bi-CGSTAB method is used to solve the Poisson equation. A tridiagonal factorization preconditioner is adopted to utilize the large scale pipeline structure of the hardware. To get perfect parallelism, we made use of a Jacobi splitting to invert the tri-diagonal system in the partitioning direction. The resulting preconditioner is tested with a problem that has large jumps and singularities. A satisfying vector processor utilization and a parallel speed-up of the fluid code are achieved.

A preconditioned Krylov method for solution of the multi-dimensional, two fluid hydrodynamics equations

A preconditioned Krylov method is introduced for reducing the computational burden in the solution of the multi-dimensional, two fluid hydrodynamics equations. The BILU3D preconditioned BICGSTAB method was applied to the linearized continuity equation in the inner iteration of the fully implicit solution of the mass, energy, and momentum equations in the EPRI code VIPRE-02. A nuclear reactor thermal-hydraulics test problem was performed with 104 multi-dimensional flow channels in the vessel. For the simulation of a typical pressurized water Reactor steam line break transient, the overall execution time was reduced by more than 50% compared to the existing solution techniques which utilize stationary alternate direction implicit (ADI) iterative methods for the inner iteration.

Hermite collocation solution of partial differential equations via preconditioned Krylov methods

We are concerned with the numerical solution of partial differential equations (PDEs) in two spatial dimensions discretized via Hermite collocation. To efficiently solve the resulting systems of linear algebraic equations, we choose a Krylov subspace method. We implement two such methods: Bi-CGSTAB [1] and GMRES [2]. In addition, we utilize two different preconditioners: one based on the Gauss–Seidel method with a block red-black ordering (RBGS); the other based upon a block incomplete LU factorization (ILU). Our results suggest that, at least in the context of Hermite collocation, the RBGS preconditioner is superior to the ILU preconditioner and that the Bi-CGSTAB method is superior to GMRES. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:120–136, 2001

Hermite collocation solution of partial differential equations via preconditioned Krylov methods

We are concerned with the numerical solution of partial differential equations (PDEs) in two spatial dimensions discretized via Hermite collocation. To efficiently solve the resulting systems of linear algebraic equations, we choose a Krylov subspace method. We implement two such methods: Bi-CGSTAB [1] and GMRES [2]. In addition, we utilize two different preconditioners: one based on the Gauss–Seidel method with a block red-black ordering (RBGS); the other based upon a block incomplete LU factorization (ILU). Our results suggest that, at least in the context of Hermite collocation, the RBGS preconditioner is superior to the ILU preconditioner and that the Bi-CGSTAB method is superior to GMRES. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:120–136, 2001

Efficient preconditioning of linear systems arising from the discretization of hyperbolic conservation laws

In this paper, we describe a novel formulation of a preconditioned BiCGSTAB algorithm for the solution of ill-conditioned linear systems Ax=b. The developed extension enables the control of the residual r m =b−Ax m of the approximate solution x m independent of the specific left, right or two-sided preconditioning technique considered. Thereby, the presented modification does not require any additional computational effort and can be introduced directly into existing computer codes. Furthermore, the proceeding is not restricted to the BiCGSTAB method, hence the strategy can serve as a guideline to extend similar Krylov sub-space methods in the same manner. Based on the presented algorithm, we study the behavior of different preconditioning techniques. We introduce a new physically motivated approach within an implicit finite volume scheme for the system of the Euler equations of gas dynamics which is a typical representative of hyperbolic conservation laws. Thereupon a great variety of realistic flow problems are considered in order to give reliable statements concerning the efficiency and performance of modern preconditioning techniques.

Two-dimensional fluid simulation of plasma reactors for the immobilization of krypton

Direct current glow discharge plasma sources are being developed as reactors for immobilization of radioactive krypton gas recovered from the off-gas stream of a nuclear fuel reprocessing plant. In order to design these reactors, two-dimensional simulations have been performed using a fluid model coupled with Poisson's equation. Nonlinear governing equations were expressed by a fully implicit scheme and the equations were solved using the Newton–Krylov method. We investigated algorithm performance of two methods from the Krylov projection techniques, namely the Bi-CGSTAB method and the transpose-free quasi-minimal residual (TFQMR) method. The results showed that the Bi-CGSTAB method converged considerably faster than the TFQMR method. Direct current glow discharge plasmas in three types of krypton immobilization apparatus were simulated to study the differences in electrode geometry. Simulation results agreed qualitatively with measurements of electron density.

Comparisons of the ILU(0), Point-SSOR, and SPAI Preconditioners on the CRAY-T3E for Nonsymmetric Sparse Linear Systems Arising from PDEs on Structured Grids

When solving general sparse linear systems, multicoloring techniques can yield a parallelism of order N, where N is the dimension of the matrix. However, this strategy often suffers from the deterioration of the rate of convergence, as in the preconditioned conjugate gradient method. Another technique that does not suffer from this deterioration of the rate of convergence is the wavefront technique, which exploits parallelism that is available in the original matrix. The rate of convergence remains the same, but the maximum parallelism is limited by the length of the wavefronts, which are often nonuniform. Another popular technique is the sparse approximate inverse (SPAI) technique, capturing the inverse in the sparse form, which is very nice for parallel computation. In this paper, the author compares these two approaches of ILU(0), SPAI, and point-SSOR preconditioners using the BICGSTAB method as the outer iterative method on the CRAY-T3E. It was found that for the problems tested, ILU(0) with multicoloring outperforms the other alternatives.