Optimal Successive Overrelaxation Research Articles

A comparative study on the acceleration techniques for solving finite difference discretization poisson’s equation in the PIC/MCC Method

A wide variety of plasma phenomena have been investigated during the past decades using the particle-in-cell/Monte Carlo collisions (PIC/MCC) method. As an important component of the PIC/MCC method, solving Poisson’s equation is crucial for the accuracy and efficiency of calculations. Different acceleration techniques for solving finite difference discretization Poisson’s equation are investigated and compared, including direct method, iterative method, multigrid (MG) method, parallel computing and inherited initial value. The charge density distribution with a known analytical solution is used to validate the algorithm and code. The optimal relaxation factor for the successive over-relaxation (SOR) method in 2D Poisson’s equation with unequal grid node numbers in different dimensions is derived, which is only related to the dimension with the largest grid number. Although there will be a ‘more optimal’ relaxation factor deviated from in some simulation cases, selecting the optimal relaxation factor derived always leads to a not slow solving speed. However, when SOR is used in MG for smoothing, the optimal relaxation factor will shift to 0.5–1.2 from the theoretical optimal value derived with the increase of MG levels. By comparing the convergence order under different relaxation factors and MG levels, the suitable MG level is proposed as log2[min(N x , N y )]−2. Combining the optimal SOR relaxation factor, MG, parallel computing and inherited initial values, the computational cost may decrease by 5 orders of magnitude than that by the simple Gaussian elimination (GE). Based on the optimal acceleration techniques mentioned above, a benchmark simulation case electron cyclotron drift instability (ECDI) in magnetized plasmas was run to further validate the developed PIC/MCC code. The distributions of electric field in the x-direction, electron density and electron temperature are all consistent with the literatures. This paper provides a reference for the acceleration strategy selection for solving Poisson’s equation quickly in plasma simulations.

A Revisit to CMFD Schemes: Fourier Analysis and Enhancement

The coarse-mesh finite difference (CMFD) scheme is a very effective nonlinear diffusion acceleration method for neutron transport calculations. CMFD can become unstable and fail to converge when the computational cell optical thickness is relatively large in k-eigenvalue problems or diffusive fixed-source problems. Some variants and fixups have been developed to enhance the stability of CMFD, including the partial current-based CMFD (pCMFD), optimally diffusive CMFD (odCMFD), and linear prolongation-based CMFD (lpCMFD). Linearized Fourier analysis has proven to be a very reliable and accurate tool to investigate the convergence rate and stability of such coupled high-order transport/low-order diffusion iterative schemes. It is shown in this paper that the use of different transport solvers in Fourier analysis may have some potential implications on the development of stabilizing techniques, which is exemplified by the odCMFD scheme. A modification to the artificial diffusion coefficients of odCMFD is proposed to improve its stability. In addition, two explicit expressions are presented to calculate local optimal successive overrelaxation (SOR) factors for lpCMFD to further enhance its acceleration performance for fixed-source problems and k-eigenvalue problems, respectively.

A Feasible Approach to Determine the Optimal Relaxation Parameters in Each Iteration for the SOR Method

The paper presents a dynamic and feasible approach to the successive over-relaxation (SOR) method for solving large scale linear system through iteration. Based on the maximal orthogonal projection technique, the optimal relaxation parameter is obtained by minimizing a derived merit function in terms of right-hand side vector, the coefficient matrix and the previous step values of unknown variables. At each iterative step, we can quickly determine the optimal relaxation value in a preferred interval. When the theoretical optimal value is hard to be achieved, the new method provides an alternative choice of the relaxation parameter at each iteration. Numerical examples confirm that the dynamic optimal successive over-relaxation (DOSOR) method outperforms the classical SOR method.

A practical asymptotical optimal SOR method

This paper presents a practical asymptotical optimal successive over-relaxation (SOR) method for solving the large sparse linear system. Based on two optimization models, asymptotically optimal relaxation factors are given, which are computed by solving the low-order polynomial equations in each iteration. The coefficients of the two polynomials are determined by the residual vector and the coefficient matrix A of the real linear system. The numerical examples show that the new methods are more feasible and effective than the classical SOR method.

Parallelization of Linear Iterative Methods for Solving the 3-D Pressure Poisson Equation Using Various Programming Languages

In this paper we explored parallelization of PPE (pressure Poisson equation) solvers with the aim of decreasing total arithmetic and run-time expenditure in incompressible CFD. For the current work, Jacobi and successive overrelaxation (SOR) iterations are used to solve this 3-D PPE, and they are parallelized with OpenMP employing programming languages Fortran, C and C++. Results show that parallelization greatly improves the serial calculation speed in general. Also it is found that 1) speedup increases sub-linearly with increasing number of cores; 2) parallelization efficiency first increases, and later decreases, as the problem size becomes larger; 3) Fortran is approximately 10% faster than C, and C is approximately 10% faster than C++ for serial cases; for parallel runs, Fortran shows little difference in execution time with C, but still is approximately 10% faster than C++. Comparisons are made in the context of having the same algorithm and code structure, and the same optimization level in compilation. Moreover, we found that optimal SOR is nearly two orders of magnitude faster than Jacobi iteration, both in serial and parallel modes, even for a very simple approach to SOR parallelization.

Comparison of Iterative Methods for the Solution of Compressible-Fluid Reynolds Equation

This study presents an efficacy comparison of iterative solution methods for solving the compressible-fluid Reynolds equation in modeling air- or gas-lubricated bearings. A direct fixed-point iterative (DFI) method and Newton’s method are employed to transform the Reynolds equation in a form that can be solved iteratively. The iterative solution methods examined are the Gauss–Seidel method, the successive over-relaxation (SOR) method, the preconditioned conjugate gradient (PCG) method, and the multigrid method. The overall solution time is affected by both the transformation method and the iterative method applied. In this study, Newton’s method shows its effectiveness over the straightforward DFI method when the same iterative method is used. It is demonstrated that the use of an optimal relaxation factor is of vital importance for the efficiency of the SOR method. The multigrid method is an order faster than the PCG and optimal SOR methods. Also, the multigrid and PCG methods involve an extended coding work and are less flexible in dealing with gridwork and boundary conditions. Consequently, a compromise has to be made in terms of ease of use as well as programming effort for the solution of the compressible-fluid Reynolds equation.

A Novel Iterative Scheme for the Very Fast and Accurate Solution of Non-LTE Radiative Transfer Problems

view Abstract Citations (97) References (22) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS A Novel Iterative Scheme for the Very Fast and Accurate Solution of Non-LTE Radiative Transfer Problems Trujillo Bueno, J. ; Fabiani Bendicho, P. Abstract Iterative schemes based on Gauss-Seidel (G-S) and optimal successive over-relaxation (SOR) iteration are shown to provide a dramatic increase in the speed with which non-LTE radiation transfer (RT) problems can be solved. The convergence rates of these new RT methods are identical to those of upper triangular nonlocal approximate operator splitting techniques, but the computing time per iteration and the memory requirements are similar to those of a local operator splitting method. In addition to these properties, both methods are particularly suitable for multidimensional geometry, since they neither require the actual construction of nonlocal approximate operators nor the application of any matrix inversion procedure. Compared with the currently used Jacobi technique, which is based on the optimal local approximate operator (see Olson, Auer, & Buchler 1986), the G-S method presented here is faster by a factor 2. It gives excellent smoothing of the high-frequency error components, which makes it the iterative scheme of choice for multigrid radiative transfer. This G-S method can also be suitably combined with standard acceleration techniques to achieve even higher performance. Although the convergence rate of the optimal SOR scheme developed here for solving non-LTE RT problems is much higher than G-S, the computing time per iteration is also minimal, i.e., virtually identical to that of a local operator splitting method. While the conventional optimal local operator scheme provides the converged solution after a total CPU time (measured in arbitrary units) approximately equal to the number n of points per decade of optical depth, the time needed by this new method based on the optimal SOR iterations is only √n/2√2. This method is competitive with those that result from combining the above-mentioned Jacobi and G-S schemes with the best acceleration techniques. Contrary to what happens with the local operator splitting strategy currently in use, these novel methods remain effective even under extreme non-LTE conditions in very fine grids. Publication: The Astrophysical Journal Pub Date: December 1995 DOI: 10.1086/176612 Bibcode: 1995ApJ...455..646T Keywords: METHODS: NUMERICAL; RADIATIVE TRANSFER; STARS: ATMOSPHERES full text sources ADS |

Best Cyclic Repartitioning for Optimal Successive Overrelaxation Convergence

In this paper, the successive overrelaxation (SOR) method for the solution of a linear system whose matrix coefficient A is block p-cyclic consistently ordered is discussed. In recent works, many researchers considered some “natural” assumptions on the spectrum $\sigma (J_p )$ of the block Jacobi matrix $J_p $ associated with A and answered the following question: What is the repartitioning of A into a block q-cyclic form $(2\leqq q\leqq p)$ which yields the best optimal SOR method for the solution of the given system? In this paper, the same question is answered in the most general case considered so far, that is, under the assumption $\sigma (J_p^p ) \subset [ - \alpha ^p ,\beta ^p ] \subset \mathbb{R}\backslash \{ [1,\infty )\} ,\alpha ,\beta \geqq 0$. It is also shown that the results in all previous works are recovered as particular subcases of the case considered here.

Optimal successive overrelaxation iterative methods for P-cyclic matrices

We consider linear systems whose associated block Jacobi matricesJ p are weakly cyclic of indexp. In a recent paper, Pierce, Hadjidimos and Plemmons [13] proved that the block two-cyclic successive overrelaxation (SOR) iterative method is numerically more effective than the blockq-cyclic SOR-method, 2 2). Finally, as a counterpart to the result of Pierce, Hadjidimos and Plemmons, we show that, under this more general assumption, the two-cyclic SOR method is not always superior to theq-cyclic SOR method, 2<q?p.

Optimal block iterative schemes for certain large, sparse, and nonsymmetric linear systems

The problem of determining fast iterative solutions of certain large, sparse, and nonsymmetric linear systems, arising in applications, is addressed here. Several iterative schemes, from the accelerated overrelaxation family, are considered. Different geometrical algorithms are used for the explicit determination of the optimal factors. Direct comparisons of the spectral radii of the resulting optimal schemes reveal that the optimal extrapolated accelerated Gauss-Seidel (EAGS) is always asymptotically faster than the optimal successive overrelaxation, while the optimal EAGS and extrapolated Gauss-Seidel strongly compete. Application of the collocation method on simple boundary-value problems is used to demonstrate our results numerically.

Multigrid Solution of an Elliptic Boundary Value Problem from Tropical Cyclone Theory

Abstract We consider the multigrid solution of the transverse circulation equation for a tropical cyclone. First we develop a standard multigrid scheme (SMG) which cycles between different levels of discretization (grids) to efficiently reduce the error in the solution on all scales. Whereas relaxation is inefficient as a solution method, it is used within the multigrid approach as a smoother to reduce the high-wavenumber errors on each grid. The added cost of the coarse grids is small because they contain relatively few points. The efficiency of the SMG scheme is compared to more conventional methods. Gauss-Seidel and successive-over-relaxation (SOR). Results show that the SMG scheme solves to the level of truncation error 26 times faster than an optimal SOR method. In the SMG scheme an unbounded domain is approximated with a wall at a finite radius which leads to significant errors in the numerical solution. To better simulate an unbounded domain, we develop a second scheme (LRMG) which naturally combin...

Extrapolation of S. O. R. iterations

In the paper, the system of $n$ linear algebraic equations $Ax=b$ with 2-cyclic matrix is considered. Methods are derived which converge to the solution $x$ faster than the optimal successive overrelaxation iterative method.