Devoted to computational sciences, this journal publishes pioneering methods and applications that bring about the solution of complex problems, or even make such solutions possible at all. Since visualization has become an important scientific tool, especially in the analysis of complex situations, it is treated in close connection with the other areas covered by the journal.

The journal "Computing and Visualization in Science" is dedicated to computational sciences, publishing pioneering methods and applications that solve complex problems or make such solutions possible. As visualization has become an essential scientific tool, particularly in analyzing complex situations, it is closely integrated with other areas covered by the journal.

《Computing and Visualization in Science》期刊致力于计算科学，发表开创性的方法和应用，以解决复杂问题，甚至使这些解决方案成为可能。随着可视化成为一种重要的科学工具，尤其是在复杂情况的分析中，期刊将其与其他领域紧密结合。

Computing and Visualization in Science

Bolten Matthias

Friedhoff Stephanie

Hahne Jens

Schops Sebastian

MGRITを用いた電気機械の並列インタイムシミュレーション【JST・京大機械翻訳】

使用MGRIT的电气机械并行时间模拟【JST·京大机器翻译】

使用MGRIT的电气机械并行时间仿真【JST·京大机器翻译】

Daniel Ruprecht

Robert Speck Jülich

Sebastian Schöps

Editorial

社论

Abstract We develop new error estimates for the one-dimensional advection equation, considering general space-time discretization schemes based on Runge–Kutta methods and finite difference discretizations. We then derive conditions on the number of points per wavelength for a given error tolerance from these new estimates. Our analysis also shows the existence of synergistic space-time discretization methods that permit to gain one order of accuracy at a given CFL number. Our new error estimates can be used to analyze the choice of space-time discretizations considered when testing Parallel-in-Time methods.

Martin J. Gander

Thibaut Lunet

Toward error estimates for general space-time discretizations of the advection equation

关于对流方程一般时空离散化的误差估计

Abstract The Poisson–Boltzmann equation (PBE) is a nonlinear elliptic parametrized partial differential equation that arises in biomolecular modeling and is a fundamental tool for structural biology. It is used to calculate electrostatic potentials around an ensemble of fixed charges immersed in an ionic solution. Efficient numerical computation of the PBE yields a high number of degrees of freedom in the resultant algebraic system of equations, ranging from several hundred thousand to millions. Coupled with the fact that in most cases the PBE requires to be solved multiple times for a large number of system configurations, for example, in Brownian dynamics simulations or in the computation of similarity indices for protein interaction analysis, this poses great computational challenges to conventional numerical techniques. To accelerate such onerous computations, we suggest to apply the reduced basis method (RBM) and the (discrete) empirical interpolation method ((D)EIM) to the PBE with a special focus on simulations of complex biomolecular systems, which greatly reduces this computational complexity by constructing a reduced order model (ROM) of typically low dimension. In this study, we employ a simple version of the PBE for proof of concept and discretize the linearized PBE (LPBE) with a centered finite difference scheme. The resultant linear system is solved by the aggregation-based algebraic multigrid (AGMG) method at different samples of ionic strength on a three-dimensional Cartesian grid. The discretized LPBE, which we call the high-fidelity full order model (FOM), yields solution as accurate as other LPBE solvers. We then apply the RBM to the FOM. DEIM is applied to the Dirichlet boundary conditions which are nonaffine in the parameter (ionic strength), to reduce the complexity of the ROM. From the numerical results, we notice that the RBM reduces the model order from $${\mathcal {N}} = 2\times 10^{6}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo>×</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>6</mml:mn></mml:msup></mml:mrow></mml:math> to $$N = 6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:math> at an accuracy of $${\mathcal {O}}(10^{-9})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>9</mml:mn></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:math> and reduces the runtime by a factor of approximately 7600. DEIM, on the other hand, is also used in the offline-online phase of solving the ROM for different values of parameters which provides a speed-up of 20 for a single iteration of the greedy algorithm.

Cleophas Kweyu

Lihong Feng

Matthias Stein

Peter Benner

Fast solution of the linearized Poisson–Boltzmann equation with nonaffine parametrized boundary conditions using the reduced basis method

使用缩减基方法快速求解具有非仿射参数化边界条件的线性化泊松-玻尔兹曼方程

Abstract We apply the multigrid-reduction-in-time (MGRIT) algorithm to an eddy current simulation of a two-dimensional induction machine supplied by a pulse-width-modulation signal. To resolve the fast-switching excitations, small time steps are needed, such that parallelization in time becomes highly relevant for reducing the simulation time. The MGRIT algorithm is an iterative method that allows calculating multiple time steps simultaneously by using a time-grid hierarchy. It is particularly well suited for introducing time parallelism in the simulation of electrical machines using existing application codes, as MGRIT is a non-intrusive approach that essentially uses the same time integrator as a traditional time-stepping algorithm. However, the key difficulty when using time-stepping routines of existing application codes for the MGRIT algorithm is that the cost of the time integrator on coarse time grids must be less expensive than on the fine grid to allow for speedup over sequential time stepping on the fine grid. To overcome this difficulty, we consider reducing the costs of the coarse-level problems by adding spatial coarsening. We investigate effects of spatial coarsening on MGRIT convergence when applied to two numerical models of an induction machine, one with linear material laws and a full nonlinear model. Parallel results demonstrate significant speedup in the simulation time compared to sequential time stepping, even for moderate numbers of processors.

Matthias Bolten

Stephanie Friedhoff

Jens Hahne

Parallel-in-time simulation of an electrical machine using MGRIT

使用MGRIT的电机并行时间仿真

This paper describes a space-time parallel algorithm with space-time adaptive mesh refinement (AMR). AMR with subcycling is added to multigrid reduction-in-time (MGRIT) in order to provide solution efficient adaptive grids with a reduction in work performed on coarser grids. This algorithm is achieved by integrating two software libraries: XBraid (Parallel time integration with multigrid. https://computation.llnl.gov/projects/parallel-timeintegration-multigrid ) and Chombo (Chombo software package for AMR applications—design document, 2014). The former is a parallel time integration library using multigrid and the latter is a massively parallel structured AMR library. Employing this adaptive space-time parallel algorithm is Chord (Comput Fluids 123:202–217, 2015), a computational fluid dynamics (CFD) application code for solving compressible fluid dynamics problems. For the same solution accuracy, speedups are demonstrated from the use of space-time parallelization over the time-sequential integration on Couette flow and Stokes' second problem. On a transient Couette flow case, at least a $$1.5\times $$ speedup is achieved, and with a time periodic problem, a speedup of up to $$13.7\times $$ over the time-sequential case is obtained. In both cases, the speedup is achieved by adding processors and exploring additional parallelization in time. The numerical experiments show the algorithm is promising for CFD applications that can take advantage of the time parallelism. Future work will focus on improving the parallel performance and providing more tests with complex fluid dynamics to demonstrate the full potential of the algorithm.

Joshua Christopher

Robert D. Falgout

Jacob B. Schroder

Stephen M. Guzik

Xinfeng Gao

A space-time parallel algorithm with adaptive mesh refinement for computational fluid dynamics

一种用于计算流体动力学的自适应网格细化时空并行算法

ISSN	1432-9360, 1433-0369
h-index	48

Publication Information	Publisher: Springer Verlag，Publishing cycle: ，Journal Type: journal，Open Access Journals: No
Basic data	Year of publication: 2015，Proportion of original research papers: 0.00%，Self Citation Rate:N.A.， Gold OA Rate: 0.00%

Computing and Visualization in Science

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MGRITを用いた電気機械の並列インタイムシミュレーション【JST・京大機械翻訳】

MGRITを用いた電気機械の並列インタイムシミュレーション【JST・京大機械翻訳】

MGRITを用いた電気機械の並列インタイムシミュレーション【JST・京大機械翻訳】

MGRITを用いた電気機械の並列インタイムシミュレーション【JST・京大機械翻訳】

MGRITを用いた電気機械の並列インタイムシミュレーション【JST・京大機械翻訳】

Editorial

Toward error estimates for general space-time discretizations of the advection equation

Fast solution of the linearized Poisson–Boltzmann equation with nonaffine parametrized boundary conditions using the reduced basis method

Parallel-in-time simulation of an electrical machine using MGRIT

A space-time parallel algorithm with adaptive mesh refinement for computational fluid dynamics