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Application of Numerical Methods to Selected Model Equations
Published in Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar, Computational Fluid Mechanics and Heat Transfer, 2020
Dale A. Anderson, John C. Tannehill, Richard H. Pletcher, Munipalli Ramakanth, Vijaya Shankar
There is another useful family of iterative methods that has been widely used in solving large systems of algebraic equations. These are loosely related to the conjugate gradient method, which, in its original form, was a direct method. The methods are also known in some circles as Krylov methods or Krylov subspace methods after the Russian scientist A. N. Krylov. The methods solve a system of the form [A]u = C by repeatedly performing matrix–vector multiplications involving [A]. The conjugate gradient method (Hestenes and Stiefel, 1952) is the “original” Krylov subspace iterative method. The conjugate gradient method requires that [A] be symmetric and positive definite, which limits its use for many equations arising in CFD. However, a modification known as the generalized minimal residual method (GMRES) (Saad and Schultz, 1986) is often used for the more general systems of equations arising in CFD. To provide a background for the development of GMRES, the conjugate gradient method will be outlined first.
Surface Plasmon-Polariton Gap Waveguide and its Applications
Published in Sergey I Bozhevolnyi, Plasmonic, 2019
The volume integral region V in (2) represents the entire space, and εr(x) is the permittivity, where εr(x) = ε1 in the metal of the straight SPGW and the screen and εr (x) = ε0 in the surrounding free space. An expression for the incident Gaussian beam can be found in the literature10. In the numerical technique employed in this chapter, the entire region must be a rectangular parallelepiped of size Bx × By × (l + Bz) including the free-space regions shown in Fig. 3 and this region is divided into small discretized cubes of size δ × δ × δ. Under these conditions, the volume integral equation (1) with (2) is discretized by the method of moments using roof-top functions as basis and testing functions. The resultant system of linear equations is then solved by iteration using the generalized minimized residual method (GMRES)11 with a fast Fourier transform (FFT). The numerical expressions are omitted here since they can be found in the literature8–11.
Matrix-Free Iterative Solution Procedure for Finite Element Problems
Published in João Pedro A. Bastos, Nelson Sadowski, Magnetic Materials and 3D Finite Element Modeling, 2017
João Pedro A. Bastos, Nelson Sadowski
A slight modification of the procedure enables its application in non-stationary solvers such as the conjugate gradient method (CG), the biconjugate gradient method (BiCG) and the stabilized version (BiCGstab), the quasiminimal residual method (QMR), or the generalized minimal residual method (GMRES). Algorithmic descriptions of these solvers are provided in [13, 23, 24]. In all the Krylov subspace methods mentioned, the system matrix A is only referenced in the context of a matrix-vector product of the form q(i)=Ap(i) $$ q^{(i)}=Ap^{(i)} $$
An overset generalised minimal residual method for the multi-solver paradigm
Published in International Journal of Computational Fluid Dynamics, 2020
Dylan Jude, Jayanarayanan Sitaraman, Vinod Lakshminarayan, James Baeder
The works of both Galbraith and Brazell used a flexible Generalised Minimal Residual Method (GMRES) for the linear solver of the Newton or quasi-Newton linearisation of the Navier–Stokes equations. GMRES is just one of many methods that can be used to solve a linear system. Traditional methods in CFD use Gauss–Seidel sweeps or approximate factorisation methods to iteratively approximate the inversion of the linear system. The Lower–Upper Symmetric Gauss–Seidel (Jameson and Yoon 1987) and Diagonalised Alternating Direction Implicit (Pulliam and Steger 1980) schemes are examples of two popular approximate methods for handling the implicit system resulting from the Euler or Navier–Stokes equations on structured grids. Although GMRES has generally been popular in unstructured solvers like FUN3D (NASA) (Anderson, Rausch, and Bonhaus 1996) or mStrand (CREATE A/V) (Lakshminarayan et al. 2017), the linear solver can also help with the convergence of structured grid based codes using large, high-order stencils in areas of large gradients. The GMRES method by itself can take as many iterations as there are unknowns to achieve convergence, which is clearly intractable. Therefore, a suitable preconditioning scheme is always necessary within GMRES to achieve improved convergence. In this context, it is worth noting that both Galbraith and Brazell used the exact same preconditioners for all of their overset meshes. In a multi-solver context, this is not often feasible since the preconditioners implemented within each code depends on the specific needs of that solver.
Consistent pCMFD Acceleration Schemes of the Three-Dimensional Transport Code PROTEUS-MOC
Published in Nuclear Science and Engineering, 2019
Guangchun Zhang, Albert Hsieh, Won Sik Yang, Yeon Sang Jung
Using the 2-D C5G7 benchmark problem, three different solution schemes for WG equations were tested with single- and two-level pCMFD accelerations: fully converged GMRES iteration, single GMRES iteration, and single transport sweep. A 2-norm residual error criterion of 1.0 × 10−7 was used for the fission source convergence of transport calculations. For pCMFD acceleration calculations, a residual error criterion of 1.0 × 10−9 was used for the fission source convergence. For the fully converged GMRES inner iteration, an error criterion of 0.01 was used for the GMRES residual convergence. The maximum number of GMRES iterations was set to 60 and the maximum subspace dimension was set to 30. For the single-level pCMFD calculation, a quarter-pin mesh was used as the coarse mesh. For the two-level pCMFD calculation, quarter-pin and single-assembly meshes were used as the first- and second-level coarse meshes, respectively. All the calculations in this subsection were performed with one CPU.
Dose Rate Evaluation for the ES-3100 Package with HEU Content Using MCNP, ADVANTG, Monaco, and MAVRIC
Published in Nuclear Technology, 2018
The following supporting computer codes are used by MCNP, ADVANTG, Monaco, and MAVRIC: ORIGEN-S (Ref. 4) is a general-purpose point depletion and decay code. Given an initial isotopic distribution, materials are decayed to provide time-dependent, energy-grouped photon and neutron sources. ORIGEN cases were run using SCALE 6.1.1 on a Y-12 cluster.Denovo4 is a 3-D block-parallel discrete ordinates transport SN code developed at ORNL as part of SCALE. The Denovo code is used to generate adjoint scalar fluxes for the CADIS method or the FW-CADIS method in MAVRIC. The Denovo code is fast, positive, and robust. The phase-space shape of the forward and adjoint fluxes, as opposed to a highly accurate solution, is the most important quality for Monte Carlo weight window generation. Denovo uses an orthogonal, nonuniform mesh that is ideal for CADIS applications because of the speed and robustness of the calculations on this mesh type. Denovo uses the highly robust Generalized Minimum Residual (GMRES) Krylov method to solve the SN equations in each group. GMRES has been shown to be more robust and efficient than traditional source (fixed-point) iteration. The Denovo calculations driven by ADVANTG for ADVANTG/MCNP are performed on a Y-12 cluster using parallel processors. The ENDF/B-VII.0 cross-section libraries for photons and neutrons used for ADVANTG in MCNP calculations are 27n19g and 200n47g. The Denovo calculations for the MAVRIC code are performed on a Y-12 cluster using a single processor using 200n47g in the ENDF/B-VII.0 cross-section libraries for both photons and neutrons.