A problem-orientable numerical algorithm for modeling multi-dimensional radiative MHD flows in astrophysics—the hierarchical solution scenario

We present a hierarchical algorithm for the adaptation of numerical solvers in high energy astrophysics.This approach is based on clustering the entries of the global Jacobian in a hierarchical manner that enables employing a variety of solution procedures ranging from a purely explicit time-stepping up to fully implicit schemes.A gradual coupling of the radiative MHD equation with the radiative transfer equation in higher dimensions is possible.Using this approach, it is possible to follow the evolution of strongly time-dependent flows with low/high accuracies and with efficiency comparable to explicit methods, as well as searching quasi-stationary solutions for highly viscous flows.In particular, it is shown that the hierarchical approach is capable of modeling the formation of jets in active galactic nuclei and reproduce the corresponding spectral energy distribution with a reasonable accuracy.     

A parallel adaptive PM code with hierarchical particle reordering

We discuss the design and implementation of HYDRA_OMP a parallel implementation of the Smoothed Particle Hydrodynamics-Adaptive P³M (SPH-AP³M) code HYDRA. The code is designed primarily for conducting cosmological hydrodynamic simulations and is written in Fortran77+OpenMP. A number of optimizations for RISC processors and SMP-NUMA architectures have been implemented, the most important optimization being hierarchical reordering of particles within chaining cells, which greatly improves data locality thereby removing the cache misses typically associated with linked lists. Parallel scaling is good, with a minimum parallel scaling of 73% achieved on 32 nodes for a variety of modern SMP architectures. We give performance data in terms of the number of particle updates per second, which is a more useful performance metric than raw MFlops. A basic version of the code will be made available to the community in the near future.     

2-dimensional implicit hydrodynamics on adaptive grids

We present a numerical scheme for two-dimensional hydrodynamics computations using a 2D adaptive grid together with an implicit discretization. The combination of these techniques has offered favorable numerical properties applicable to a variety of one-dimensional astrophysical problems which motivated us to generalize this approach for two-dimensional applications. Due to the different topological nature of 2D grids compared to 1D problems, grid adaptivity has to avoid severe grid distortions which necessitates additional smoothing parameters to be included into the formulation of a 2D adaptive grid. The concept of adaptivity is described in detail and several test computations demonstrate the effectivity of smoothing. The coupled solution of this grid equation together with the equations of hydrodynamics is illustrated by computation of a 2D shock tube problem.     

An ADI-based adaptive mesh Poisson solver for the MHD code NIRVANA

I describe a Poisson solver for the adaptive mesh magnetohydrodynamics (MHD) code NIRVANA using ADI techniques (ADI: Alternative Direction Implicit). The solver is fit to the mesh refinement framework of the code and utilizes its special block-structured design. The key part of the method is an algorithm for the intelligent clustering of subgrids which permits the application of numerical methods based on dimensional operator splitting like ADI. Test problems show the convergence of this ansatz.     

JADAMILU: a software code for computing selected eigenvalues of large sparse symmetric matrices

A new software code for computing selected eigenvalues and associated eigenvectors of a real symmetric matrix is described. The eigenvalues are either the smallest or those closest to some specified target, which may be in the interior of the spectrum. The underlying algorithm combines the Jacobi-Davidson method with efficient multilevel incomplete LU (ILU) preconditioning. Key features are modest memory requirements and robust convergence to accurate solutions. Parameters needed for incomplete LU preconditioning are automatically computed and may be updated at run time depending on the convergence pattern. The software is easy to use by non-experts and its top level routines are written in FORTRAN 77. Its potentialities are demonstrated on a few applications taken from computational physics.

Program summary

Program title: JADAMILUCatalogue identifier: ADZT_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZT_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 101 359No. of bytes in distributed program, including test data, etc.: 7 493 144Distribution format: tar.gzProgramming language: Fortran 77Computer: Intel or AMD with g77 and pgf; Intel EM64T or Itanium with ifort; AMD Opteron with g77, pgf and ifort; Power (IBM) with xlf90.Operating system: Linux, AIXRAM: problem dependentWord size: real:8; integer: 4 or 8, according to user's choiceClassification: 4.8Nature of problem: Any physical problem requiring the computation of a few eigenvalues of a symmetric matrix.Solution method: Jacobi-Davidson combined with multilevel ILU preconditioning.Additional comments: We supply binaries rather than source code because JADAMILU uses the following external packages:

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MC64. This software is copyrighted software and not freely available. COPYRIGHT (c) 1999 Council for the Central Laboratory of the Research Councils.

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AMD. Copyright (c) 2004-2006 by Timothy A. Davis, Patrick R. Amestoy, and Iain S. Duff. All Rights Reserved. Source code is distributed by the authors under the GNU LGPL licence.

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BLAS. The reference BLAS is a freely-available software package. It is available from netlib via anonymous ftp and the World Wide Web.

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LAPACK. The complete LAPACK package or individual routines from LAPACK are freely available on netlib and can be obtained via the World Wide Web or anonymous ftp.

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For maximal benefit to the community, we added the sources we are proprietary of to the tar.gz file submitted for inclusion in the CPC library. However, as explained in the README file, users willing to compile the code instead of using binaries should first obtain the sources for the external packages mentioned above (email and/or web addresses are provided).

Running time: Problem dependent; the test examples provided with the code only take a few seconds to run; timing results for large scale problems are given in Section 5.     

A new MHD code with adaptive mesh refinement and parallelization for astrophysics

A new code, named MAP, is written in FORTRAN language for magnetohydrodynamics (MHD) simulations with the adaptive mesh refinement (AMR) and Message Passing Interface (MPI) parallelization. There are several optional numerical schemes for computing the MHD part, namely, modified Mac Cormack Scheme (MMC), Lax–Friedrichs scheme (LF), and weighted essentially non-oscillatory (WENO) scheme. All of them are second-order, two-step, component-wise schemes for hyperbolic conservative equations. The total variation diminishing (TVD) limiters and approximate Riemann solvers are also equipped. A high resolution can be achieved by the hierarchical block-structured AMR mesh. We use the extended generalized Lagrange multiplier (EGLM) MHD equations to reduce the non-divergence free error produced by the scheme in the magnetic induction equation. The numerical algorithms for the non-ideal terms, e.g., the resistivity and the thermal conduction, are also equipped in the code. The details of the AMR and MPI algorithms are described in the paper.     

Parallelization issues of a code for physically-based simulation of fabrics

The simulation of fabrics, clothes, and flexible materials is an essential topic in computer animation of realistic virtual humans and dynamic sceneries. New emerging technologies, as interactive digital TV and multimedia products, make necessary the development of powerful tools to perform real-time simulations. Parallelism is one of such tools. When analyzing computationally fabric simulations we found these codes belonging to the complex class of irregular applications. Frequently this kind of codes includes reduction operations in their core, so that an important fraction of the computational time is spent on such operations. In fabric simulators these operations appear when evaluating forces, giving rise to the equation system to be solved. For this reason, this paper discusses only this phase of the simulation. This paper analyzes and evaluates different irregular reduction parallelization techniques on ccNUMA shared memory machines, applied to a real, physically-based, fabric simulator we have developed. Several issues are taken into account in order to achieve high code performance, as exploitation of data access locality and parallelism, as well as careful use of memory resources (memory overhead). In this paper we use the concept of data affinity to develop various efficient algorithms for reduction parallelization exploiting data locality.     

Three-dimensional MHD high-resolution computations with CWENO employing adaptive mesh refinement

Until recently, numerical simulations of discontinuities in highly super-Alfvénic plasmas have been severely limited by comparatively crude resolution and accuracy. Significant progress in the numerical simulation of such plasmas was achieved with the recently implemented Central Weighted Essentially Non-Oscillatory (CWENO) scheme. Combining this technique with that of adaptive mesh refinement (AMR), we have developed a third-order numerical scheme, which is able to efficiently capture strong gradients on spatial scales being small compared to the overall scale of the plasma system considered. Here, we first describe important algorithmic aspects of the scheme as well as the physics included in it. Second, we present the results of various performance tests. And, third, we illustrate its application to ‘real world problems’ using the example of the dynamics of a Sedov-type explosion.     

A method for solving systems of non-linear differential equations with moving singularities

We present a method for solving a class of initial valued, coupled, non-linear differential equations with ‘moving singularities’ subject to some subsidiary conditions. We show that these types of singularities can be adequately treated by establishing certain ‘moving’ jump conditions across them. We show how a first integral of the differential equations, if available, can also be used for checking the accuracy of the numerical solution.     

On the performance of SPAI and ADI-like preconditioners for core collapse supernova simulations in one spatial dimension

The simulation of core collapse supernovæ calls for the time accurate solution of the (Euler) equations for inviscid hydrodynamics coupled with the equations for neutrino transport. The time evolution is carried out by evolving the Euler equations explicitly and the neutrino transport equations implicitly. Neutrino transport is modeled by the multi-group Boltzmann transport (MGBT) and the multi-group flux limited diffusion (MGFLD) equations. An implicit time stepping scheme for the MGBT and MGFLD equations yields Jacobian systems that necessitate scaling and preconditioning. Two types of preconditioners, namely, a sparse approximate inverse (SPAI) preconditioner and a preconditioner based on the alternating direction implicit iteration (ADI-like) have been found to be effective for the MGFLD and MGBT formulations. This paper compares these two preconditioners. The ADI-like preconditioner performs well with both MGBT and MGFLD systems. For the MGBT system tested, the SPAI preconditioner did not give competitive results. However, since the MGBT system in our experiments had a high condition number before scaling and since we used a sequential platform, care must be taken in evaluating these results.     

Towards a more consistent discretization scheme for adaptive implicit RHD computations

We present a new implicit numerical discretization for the equations of radiation hydrodynamics (RHD) which is based on a more geometrical representation of a finite volume scheme suitable for spherical systems. In particular, the motion of the grid points is directly included by appropriate volume changes. Several examples illustrate the accuracy gained by this improved difference scheme.     

A Fortran code for null geodesic solutions in the Lema?̂tre-Tolman-Bondi spacetime

This paper describes the Fortran 77 code SIMU, version 1.1, designed for numerical simulations of observational relations along the past null geodesic in the Lema?̂tre-Tolman-Bondi (LTB) spacetime. SIMU aims at finding scale invariant solutions of the average density, but due to its full modularity it can be easily adapted to any application which requires LTB's null geodesic solutions. In version 1.1 the numerical output can be read by the GNUPLOT plotting package to produce a fully graphical output, although other plotting routines can be easily adapted. Details of the code's subroutines are discussed, and an example of its output is shown.     

Trigonometrically-fitted method with the Fourier frequency spectrum for undamped Duffing equation

With non-linearities, the frequency spectrum of an undamped Duffing oscillator should be composed of odd multiples of the driving frequency which can be interpreted as resonance driving terms. It is expected that the frequency spectrum of the corresponding numerical solution with high accurateness should contain nearly the same components. Hence, to contain these Fourier components and to calculate the amplitudes of these components in a more accurate and efficient way is the key to develop a new numerical method with high stability, accuracy and efficiency for the Duffing equation. To explore the possibility of using trigonometrically-fitting technique to build a numerical method with resonance spectrum, we design four types of Numerov methods, in which the first one is the traditional Numerov method, which contains no Fourier component, the second one contains only the first resonance term, the third one contains the first two resonance terms, and the last one contains the first three resonance terms, and apply them to the well-known undamped Duffing equation with Dooren's parameters. The numerical results demonstrate that the Numerov method fitted with the Fourier components is much more stable, accurate and efficient than the one with no Fourier component. The accuracy of the fitted method with the first three Fourier components can attain 10⁻⁹ for a remarkable range of step sizes, including nearly infinite, except individual small range of instability, which is much higher than the one of the traditional Numerov method, with eight orders for step size of π/2.011.     

SIP-CESE MHD model of solar wind with adaptive mesh refinement of hexahedral meshes

Solar–interplanetary space involves many features, such as discontinuities and heliospheric current sheet, with spatial scales many orders of magnitude smaller than the system size. The scalable, massively parallel, block-based, adaptive-mesh refinement (AMR) promises to resolve different temporal and spatial scales on which solar-wind plasma occurs throughout the vast solar–interplanetary space with even less cells but can generate a good enough resolution. Here, we carry out the adaptive mesh refinement (AMR) implementation of our Solar–Interplanetary space–time conservation element and solution element (CESE) magnetohydrodynamic model (SIP-CESE MHD model) using a six-component grid system (Feng et al., 2007, 2010). The AMR realization of the SIP-CESE MHD model is naturalized directly in hexahedral meshes with the aid of the parallel AMR package PARAMESH available at http://sourceforge.net/projects/paramesh/. At the same time, the topology of the magnetic field expansion factor and the minimum angular separation (at the photosphere) between an open field foot point and its nearest coronal-hole boundary are merged into the model in order to determine the volumetric heating source terms. Our numerical results for the validation study of the solar-wind background of Carrington rotation 2060 show overall good agreements in the solar corona and in interplanetary space with the observations from the Solar and Heliospheric Observatory (SOHO) and spacecraft data from OMNI.     

Trigonometrically-fitted method for a periodic initial value problem with two frequencies

A second-order differential equation whose solution is periodic with two frequencies has important applications in many scientific fields. Nevertheless, it may exhibit ‘periodic stiffness’ for most of the available linear multi-step methods. The phenomena are similar to the popular Stömer-Cowell class of linear multi-step methods for one-frequency problems. According to the stability theory laid down by Lambert, ‘periodic stiffness’ appears in a two-frequency problem because the production of the step-length and the bigger angular frequency lies outside the interval of periodicity. On the other hand, for a two-frequency problem, even with a small step-length, the error in the numerical solution afforded by a P-stable trigonometrically-fitted method with one frequency would be too large for practical applications. In this paper we demonstrate that the interval of periodicity and the local truncation error of a linear multi-step method for a two-frequency problem can be greatly improved by a new trigonometric-fitting technique. A trigonometrically-fitted Numerov method with two frequencies is proposed and has been verified to be P-stable with vanishing local truncation error for a two-frequency test problem. Numerical results demonstrated that the proposed trigonometrically-fitted Numerov method with two frequencies has significant advantages over other types of Numerov methods for solving the ‘periodic stiffness’ problem.     

Addressing spatiotemporal and computational heterogeneity in structured adaptive mesh refinement

Structured adaptive mesh refinement (SAMR) techniques can provide accurate and cost- effective solutions to realistic scientific and engineering simulations modeling complex physical phenomena. However, the adaptive nature and inherent space–time heterogeneity of SAMR applications result in significant runtime management challenges. Moreover, certain SAMR applications involving reactive flows exhibit pointwise varying workloads and cannot be addressed by traditional parallelization approaches, which assume homogeneous loads. This paper presents hierarchical partitioning, bin-packing based load balancing, and Dispatch structured partitioning strategies to manage the spatiotemporal and computational heterogeneity in SAMR applications. Experimental evaluation of these schemes using 3-D Richtmyer–Meshkov compressible turbulence and 2-D reactive-diffusion kernels demonstrates the improvement in overall performance.     

Generalized dynamic programming: A variables freezing method

A variables freezing method is designed for widening the possibilities of numerical application of generalized dynamic programming algorithms. Reducing the size of the memory under certain conditions, the method surmounts the “dimensional curse.” This paper is the continuation of [1–3].     

Multithreaded transactions in scientific computing. The Growth06_v2 program

Writing a concurrent program can be more difficult than writing a sequential program. Programmer needs to think about synchronization, race conditions and shared variables. Transactions help reduce the inconvenience of using threads. A transaction is an abstraction, which allows programmers to group a sequence of actions on the program into a logical, higher-level computation unit. This paper presents a new version of the GROWTHGr and GROWTH06 programs.

New version program summary

Program title: GROWTH06_v2Catalogue identifier: ADVL_v2_1Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADVL_v2_1.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 65 255No. of bytes in distributed program, including test data, etc.: 865 985Distribution format: tar.gzProgramming language: Object PascalComputer: Pentium-based PCOperating system: Windows 9x, XP, NT, VistaRAM: more than 1 MBClassification: 4.3, 7.2, 6.2, 8, 14Catalogue identifier of previous version: ADVL_v2_0Journal reference of previous version: Comput. Phys. Comm. 175 (2006) 678Does the new version supersede the previous version?: YesNature of problem: The programs compute the RHEED intensities during the growth of thin epitaxial structures prepared using the molecular beam epitaxy (MBE). The computations are based on the use of kinematical diffraction theory.Solution method: Epitaxial growth of thin films is modelled by a set of non-linear differential equations [1]. The Runge-Kutta method with adaptive stepsize control was used for solving initial value problem for non-linear differential equations [2].Reasons for new version: According to the users' suggestions functionality of the program has been improved. Moreover, new use cases have been added which make the handling of the program easier and more efficient than the previous ones [3].Summary of revisions:

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The design pattern (See Fig. 2 of Ref. [3]) has been modified according to the scheme shown on Fig. 1.

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A graphical user interface (GUI) for the program has been reconstructed. Fig. 2 presents a hybrid diagram of a GUI that shows how onscreen objects connect to use cases.

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The program has been compiled with English/USA regional and language options.

Note: The figures mentioned above are contained in the program distribution file.Unusual features: The program is distributed in the form of source project GROWTH06_v2.dpr with associated files, and should be compiled using Borland Delphi compilers versions 6 or latter (including Borland Developer Studio 2006 and Code Gear compilers for Delphi).Additional comments: Two figures are included in the program distribution file. These are captioned

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Static classes model for Transaction design pattern.

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A model of a window that shows how onscreen objects connect to use cases.

Running time: The typical running time is machine and user-parameters dependent.References: [1] A. Daniluk, Comput. Phys. Comm. 170 (2005) 265.[2] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes in Pascal: The Art of Scientific Computing, first ed., Cambridge University Press, 1989.[3] M. Brzuszek, A. Daniluk, Comput. Phys. Comm. 175 (2006) 678.     

A Mathematica program for the two-step twelfth-order method with multi-derivative for the numerical solution of a one-dimensional Schrödinger equation

In this paper, we present the detailed Mathematica symbolic derivation and the program which is used to integrate a one-dimensional Schrödinger equation by a new two-step numerical method. We add the fourth- and sixth-order derivatives to raise the precision of the traditional Numerov's method from fourth order to twelfth order, and to expand the interval of periodicity from (0,6) to the one of (0,9.7954) and (9.94792,55.6062). In the program we use an efficient algorithm to calculate the first-order derivative and avoid unnecessarily repeated calculation resulting from the multi-derivatives. We use the well-known Woods-Saxon's potential to test our method. The numerical test shows that the new method is not only superior to the previous lower order ones in accuracy, but also in the efficiency. This program is specially applied to the problem where a high accuracy or a larger step size is required.

Program summary

Title of program: ShdEq.nbCatalogue number: ADTTProgram summary URL:http://cpc.cs.qub.ac.uk/summaries/ADTTProgram obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: noneComputer for which the program is designed and others on which it has been tested: The program has been designed for the microcomputer and been tested on the microcomputer.Computers: IBM PCOperating systems under which the program has been tested: Windows XPProgramming language used: Mathematica 4.2Memory required to execute with typical data: 51 712 bytesNo. of bytes in distributed program, including test data, etc.: 45 381No. of lines in distributed program, including test data, etc.: 7311Distribution format: tar gzip fileCPC Program Library subprograms used: noNature of physical problem: Numerical integration of one-dimensional or radial Schrödinger equation to find the eigenvalues for a bound states and phase shift for a continuum state.Method of solution: Using a two-step method twelfth-order method to integrate a Schrödinger equation numerically from both two ends and the connecting conditions at the matching point, an eigenvalue for a bound state or a resonant state with a given phase shift can be found.Restrictions on the complexity of the problem: The analytic form of the potential function and its high-order derivatives must be known.Typical running time: Less than one second.Unusual features of the program: Take advantage of the high-order derivatives of the potential function and efficient algorithm, the program can provide all the numerical solution of a given Schrödinger equation, either a bound or a resonant state, with a very high precision and within a very short CPU time. The program can apply to a very broad range of problems because the method has a very large interval of periodicity.References: [1] T.E. Simos, Proc. Roy. Soc. London A 441 (1993) 283.[2] Z. Wang, Y. Dai, An eighth-order two-step formula for the numerical integration of the one-dimensional Schrödinger equation, Numer. Math. J. Chinese Univ. 12 (2003) 146.[3] Z. Wang, Y. Dai, An twelfth-order four-step formula for the numerical integration of the one-dimensional Schrödinger equation, Internat. J. Modern Phys. C 14 (2003) 1087.