On the equivalence between the Scheduled Relaxation Jacobi method and Richardson's non-stationary method [CL]


The Scheduled Relaxation Jacobi (SRJ) method is an extension of the classical Jacobi iterative method to solve linear systems of equations ($Au=b$) associated with elliptic problems. It inherits its robustness and accelerates its convergence rate computing a set of $P$ relaxation factors that result from a minimization problem. In a typical SRJ scheme, the former set of factors is employed in cycles of $M$ consecutive iterations until a prescribed tolerance is reached. We present the analytic form for the optimal set of relaxation factors for the case in which all of them are different, and find that the resulting algorithm is equivalent to a non-stationary generalized Richardson’s method. Our method to estimate the weights has the advantage that the explicit computation of the maximum and minimum eigenvalues of the matrix $A$ is replaced by the (much easier) calculation of the maximum and minimum frequencies derived from a von Neumann analysis. This set of weights is also optimal for the general problem, resulting in the fastest convergence of all possible SRJ schemes for a given grid structure. We also show that with the set of weights computed for the optimal SRJ scheme for a fixed cycle size it is possible to estimate numerically the optimal value of the parameter $\omega$ in the Successive Overtaxation (SOR) method in some cases. Finally, we demonstrate with practical examples that our method also works very well for Poisson-like problems in which a high-order discretization of the Laplacian operator is employed. This is of interest since the former discretizations do not yield consistently ordered $A$ matrices. Furthermore, the optimal SRJ schemes here deduced, are advantageous over existing SOR implementations for high-order discretizations of the Laplacian operator in as much as they do not need to resort to multi-coloring schemes for their parallel implementation. (abridged)

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J. Adsuara, I. Cordero-Carrion, P. Cerda-Duran, et. al.
Thu, 14 Jul 16

Comments: 28 pages, 5 figures, submitted to JCP

Hybrid Riemann Solvers for Large Systems of Conservation Laws [CL]


In this paper we present a new family of approximate Riemann solvers for the numerical approximation of solutions of hyperbolic conservation laws. They are approximate, also referred to as incomplete, in the sense that the solvers avoid computing the characteristic decomposition of the flux Jacobian. Instead, they require only an estimate of the globally fastest wave speeds in both directions. Thus, this family of solvers is particularly efficient for large systems of conservation laws, i.e. with many different propagation speeds, and when no explicit expression for the eigensystem is available. Even though only fastest wave speeds are needed as input values, the new family of Riemann solvers reproduces all waves with less dissipation than HLL, which has the same prerequisites, requiring only one additional flux evaluation.

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B. Schmidtmann, M. Astrakhantceva and M. Torrilhon
Tue, 28 Jun 16

Comments: 9 pages

The IDSA and the homogeneous sphere: Issues and possible improvements [CL]


In this paper, we are concerned with the study of the Isotropic Diffusion Source Approximation (IDSA) (Baxter et al., Phys. Rev. E 73, 046118, 2006) of radiative transfer. After having recalled well-known limits of the radiative transfer equation, we present the IDSA and adapt it to the case of the homogeneous sphere. We then show that for this example the IDSA suffers from severe numerical difficulties. We argue that these difficulties originate in the min-max switch coupling mechanism used in the IDSA. To overcome this problem we reformulate the IDSA to avoid the problematic coupling. This allows us to access the modeling error of the IDSA for the homogeneous sphere test case. The IDSA is shown to overestimate the streaming component, hence we propose a new version of the IDSA which is numerically shown to be more accurate than the old one.
Analytical results and numerical tests are provided to support the accuracy of the new proposed approximation.

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J. Michaud
Tue, 14 Jun 16

Comments: 25 pages, 8 figures, accepted for publication in DCDS-S

On the kernel and particle consistency in smoothed particle hydrodynamics [CL]


The problem of consistency of smoothed particle hydrodynamics (SPH) has demanded considerable attention in the past few years due to the ever increasing number of applications of the method in many areas of science and engineering. A loss of consistency leads to an inevitable loss of approximation accuracy. In this paper, we revisit the issue of SPH kernel and particle consistency and demonstrate that SPH has a limiting second-order convergence rate. Numerical experiments with suitably chosen test functions validate this conclusion. In particular, we find that when using the root mean square error as a model evaluation statistics, well-known corrective SPH schemes, which were thought to converge to second, or even higher order, are actually first-order accurate, or at best close to second order. We also find that observing the joint limit when $N\to\infty$, $h\to 0$, and $n\to\infty$, as was recently proposed by Zhu et al., where $N$ is the total number of particles, $h$ is the smoothing length, and $n$ is the number of neighbor particles, standard SPH restores full $C^{0}$ particle consistency for both the estimates of the function and its derivatives and becomes insensitive to particle disorder.

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L. Sigalotti, J. Klapp, O. Rendon, et. al.
Wed, 18 May 16

Comments: 27 pages, 10 figures. Submitted to Journal of Applied Numerical Mathematics

A parallel code for multiprecision computations of the Lane-Emden differential equation [SSA]


We compute multiprecision solutions of the Lane-Emden equation. This differential equation arises when introducing the well-known polytropic model into the equation of hydrostatic equilibrium for a nondistorted star. Since such multiprecision computations are time-consuming, we apply to this problem parallel programming techniques and thus the execution time of the computations is drastically reduced.

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V. Geroyannis and V. Karageorgopoulos
Thu, 28 Apr 16

Comments: 8 pages

Scheduled Relaxation Jacobi method: improvements and applications [CL]


Elliptic partial differential equations (ePDEs) appear in a wide variety of areas of mathematics, physics and engineering. Typically, ePDEs must be solved numerically, which sets an ever growing demand for efficient and highly parallel algorithms to tackle their computational solution. The Scheduled Relaxation Jacobi (SRJ) is a promising class of methods, atypical for combining simplicity and efficiency, that has been recently introduced for solving linear Poisson-like ePDEs. The SRJ methodology relies on computing the appropriate parameters of a multilevel approach with the goal of minimizing the number of iterations needed to cut down the residuals below specified tolerances. The efficiency in the reduction of the residual increases with the number of levels employed in the algorithm. Applying the original methodology to compute the algorithm parameters with more than 5 levels notably hinders obtaining optimal SRJ schemes, as the mixed (non-linear) algebraic-differential equations from which they result become notably stiff. Here we present a new methodology for obtaining the parameters of SRJ schemes that overcomes the limitations of the original algorithm and provide parameters for SRJ schemes with up to 15 levels and resolutions of up to $2^{15}$ points per dimension, allowing for acceleration factors larger than several hundreds with respect to the Jacobi method for typical resolutions and, in some high resolution cases, close to 1000. Furthermore, we extend the original algorithm to apply it to certain systems of non-linear ePDEs.

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J. Adsuara, I. Cordero-Carrion, P. Cerda-Duran, et. al.
Tue, 17 Nov 15

Comments: 37 pages, 8 figures, submitted to JCP

Efficient conservative ADER schemes based on WENO reconstruction and space-time predictor in primitive variables [CL]


We present a new version of conservative ADER-WENO finite volume schemes, in which both the high order spatial reconstruction as well as the time evolution of the reconstruction polynomials in the local space-time predictor stage are performed in primitive variables, rather than in conserved ones. Since the underlying finite volume scheme is still written in terms of cell averages of the conserved quantities, our new approach performs the spatial WENO reconstruction twice: the first WENO reconstruction is carried out on the known cell averages of the conservative variables. The WENO polynomials are then used at the cell centers to compute point values of the conserved variables, which are converted into point values of the primitive variables. A second WENO reconstruction is performed on the point values of the primitive variables to obtain piecewise high order reconstruction polynomials of the primitive variables. The reconstruction polynomials are subsequently evolved in time with a novel space-time finite element predictor that is directly applied to the governing PDE written in primitive form. We have verified the validity of the new approach over the classical Euler equations of gas dynamics, the special relativistic hydrodynamics (RHD) and ideal magnetohydrodynamics (RMHD) equations, as well as the Baer-Nunziato model for compressible two-phase flows. In all cases we have noticed that the new ADER schemes provide less oscillatory solutions when compared to ADER finite volume schemes based on the reconstruction in conserved variables, especially for the RMHD and the Baer-Nunziato equations. For the RHD and RMHD equations, the accuracy is improved and the CPU time is reduced by about 25%. We recommend to use this version of ADER as the standard one in the relativistic framework. The new approach can be extended to ADER-DG schemes on space-time adaptive grids.

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O. Zanotti and M. Dumbser
Tue, 17 Nov 15

Comments: N/A