科學(xué)計算常用的基礎(chǔ)并行求解軟件庫介紹王彥棡2009年12月

1、 科學(xué)計算常用的基礎(chǔ)并行求解軟件庫介紹科學(xué)計算常用的基礎(chǔ)并行求解軟件庫介紹 王彥棡 2009年12月18日背景介紹背景介紹 深騰深騰70007000是世界上規(guī)模最大的一個結(jié)點無盤啟動的是世界上規(guī)模最大的一個結(jié)點無盤啟動的機群系統(tǒng)，成功實現(xiàn)了基于機群系統(tǒng)，成功實現(xiàn)了基于14281428個無盤結(jié)點的機群系統(tǒng)。個無盤結(jié)點的機群系統(tǒng)。 深騰深騰70007000是國內(nèi)第一個實際性能突破每秒百萬億次是國內(nèi)第一個實際性能突破每秒百萬億次的異構(gòu)機群系統(tǒng)，成功實現(xiàn)了的異構(gòu)機群系統(tǒng)，成功實現(xiàn)了12401240個個2 2路薄結(jié)點和路薄結(jié)點和3838個個1616路厚結(jié)點的協(xié)同計算，實際路厚結(jié)點的協(xié)同計算，實際Linp

2、ackLinpack性能突破每秒性能突破每秒106.5106.5萬億次。萬億次。背景介紹背景介紹JaguarJaguar（rankrank 1 1）petscpsplineScalapacksprngsundialssuperlusuperlu_distTrilinosumfpackarpackaztecfftpack fftwgslhypre libsci metis Parmetismumps 背景介紹背景介紹JUGENE(rank 4)JUGENE(rank 4)、JuRoPa(rank 13)JuRoPa(rank 13)、JUMPJUMPNAG Parallel Library、Sc

3、aLAPACK、ARPACK、PARPACK、PETSc、MUMPS、SPRNG、ParMETIS、hypre、sundials背景介紹背景介紹Alabama Supercomputer AuthorityAlabama Supercomputer Authoritydeal.II, METIS, Octave, PDE2D, PETSc, R, SCSL, SLATEC, TrilinosEcole Polytechnique Fdrale de Lausanne Ecole Polytechnique Fdrale de Lausanne ARPACK,AZTEC,MUMPS, PETSC

4、,BLACS,ScaLAPACK, SPRNG, FFTW, NAG Fortran 90 library, LAPACK/BLAS from MKL背景介紹背景介紹Trilinospetsc sundialshypretaoslepcadicAztecBlockSolve95gslMUMPSParMetispARMSspaispoolesfftwSuperLU_distsprngarpackparpack類別類別工具工具功能功能數(shù)值軟件 Trilinos解決大規(guī)模、復(fù)雜物理工程和科學(xué)應(yīng)用的面向?qū)ο蟮能浖蚣芟麻_發(fā)并行解決算法和數(shù)學(xué)庫。PETSc用于在分布式存儲環(huán)境高效求解偏微分方程組及相關(guān)問

5、題。SUNDIALS用于求解非線性微分/代數(shù)方程。Hypre用于大規(guī)模并行計算機上求解大型稀疏線性方程組。ScaLAPACK可擴展線性代數(shù)程序包。SLEPc可并行求解大規(guī)模稀疏矩陣特征問題。SuperLU大型稀疏矩陣直接求解器。TAO在高性能機器上求解大規(guī)模最優(yōu)化問題。軟件介紹軟件介紹計算問題計算問題方法方法 算法算法函數(shù)庫函數(shù)庫線性方程組直接求解法LU FactorizationScaLAPACK(dense)SuperLU (sparse)Cholesky FactorizationScaLAPACKLDLT (Tridiagonal matrices)ScaLAPACKQR Factor

6、izationScaLAPACKQR with column pivotingScaLAPACKLQ factorizationScaLAPACK軟件介紹軟件介紹計算問題計算問題方法方法 算法算法函數(shù)庫函數(shù)庫線性方程組迭代方法Conjugate GradientAztecOO (Trilinos)PETScGMRESAztecOOPETScHypreCG SquaredAztecOOPETScBi-CG StabAztecOOPETScQuasi-Minimal Residual (QMR)AztecOOTranspose Free QMRAztecOOPETSc軟件介紹軟件介紹計算問題計算問

7、題方法方法 算法算法函數(shù)庫函數(shù)庫線性方程組迭代方法SYMMLQPETScPrecondition CGAztecOOPETScHypreRichardson PETScBlock Jacobi PreconditionerAztecOOPETScHyprePoint Jocobi PreconditionerAztecOOLeast Squares Polynomials PETSc軟件介紹軟件介紹軟件介紹軟件介紹計算問題計算問題方法方法 算法算法函數(shù)庫函數(shù)庫線性方程組迭代方法SOR PreconditioningPETScOverlapping Additive SchwartzPETScA

8、pproximate InverseHypreSparse LU preconditionerAztecOOPETScHypreIncomplete LU (ILU) preconditionerAztecOOLeast Squares Polynomials PETSc多重網(wǎng)格MG PreconditionerPETScHypreAlgebraic MGHypreSemi-coarsening Hypre計算問題計算問題方法方法算法算法函數(shù)庫函數(shù)庫線性最小二乘最小二乘ScaLAPACK最小范數(shù)ScaLAPACK極小范數(shù)最小二乘ScaLAPACK特征值問題對稱特征值For A=AH or A=

9、ATScaLAPACK (dense)SLEPc (sparse)Anasazi(sparse)奇異值問題奇異值分解ScaLAPACK (dense)SLEPc (sparse)Anasazi(sparse)廣義特征值特征值問題ScaLAPACK (dense)SLEPc (sparse)Anasazi(sparse) m m i in nx| | | | b Ax | | | |2 m m i in nx| | | | x | | | |2 m m i in nx| | | | x | | | |2 m m i in nx| | | | b Ax | | | |2 Az z A UVTA U

10、VH Az BzABz zBAz z軟件介紹軟件介紹計算問題計算問題方法方法算法算法函數(shù)庫函數(shù)庫非線性方程組Newton BasedLine SearchPETScNOX&LOCATrust RegionsPETScNOX&LOCAPseudo-Transient ContinuationPETScNOX&LOCAMatrix FreePETScNOX&LOCA軟件介紹軟件介紹計算問題計算問題方法方法算法算法函數(shù)庫函數(shù)庫非線性最優(yōu)化問題Newton BasedNewtonMOOCHOTAOFinite-Difference NewtonMOOCHOTAOQuasi-NewtonMOOCHOTA

11、ONon-linear Interior PointMOOCHOTAOCGStandard Non-linear CGMOOCHOTAOLimited Memory BFGSMOOCHOGradient ProjectionsTAO軟件介紹軟件介紹計算問題計算問題方法方法算法算法函數(shù)庫函數(shù)庫常微分方程組IntegrationAdam-Moulton(Variable coefficient forms)CVODE (SUNDIALS)CVODESBackward Differential FormulaDirect and Iterative SolversCVODECVODES非線性代數(shù)方程

12、Inexact NewtonLine SearchKINSOL (SUNDIALS)差分代數(shù)方程Backward Differential FormulaDirect and Iterative SolversIDA (SUNDIALS)軟件介紹軟件介紹Trilinos Trilinos受到美國政府ASC、LDRD（Laboratory Directed Research and Development）等計劃聯(lián)合資助，是Sandia國家實驗室實施的一個大型數(shù)值軟件項目。其目的是要在一個為解決大規(guī)模、復(fù)雜物理工程和科學(xué)應(yīng)用的面向?qū)ο蟮能浖蚣芟麻_發(fā)并行解決算法和數(shù)學(xué)庫。 自2001年啟動至今，

13、Trilinos已演變到第10版，目前仍處于活躍開發(fā)中。它廣泛采用面向?qū)ο蠹夹g(shù)，大部分代碼用C+編寫，底層關(guān)鍵部分則用FORTRAN（主要是BLAS和LAPACK程序）、C（ML）實現(xiàn)。Trilinos能在串行、并行系統(tǒng)上求解線性、非線性和特征問題，提供一致的數(shù)值應(yīng)用程序接口（APIs, Application Programming Interfaces）以方便數(shù)值軟件協(xié)作。特別地，它已經(jīng)被成功的移植到目前世界最快的計算機Roadrunner上。ObjectivePackage(s)Linear algebra objectsEpetra, Jpetra, Tpetra, KokkosKry

14、lov solversAztecOO, Belos, KomplexILU-type preconditionersAztecOO, IFPACKMultilevel preconditionersML, CLAPSEigenvalue problemsAnasaziBlock preconditionersMerosDirect sparse linear solversAmesosDirect dense solversEpetra, Teuchos, PlirisAbstract interfacesThyraNonlinear system solversNOX, LOCATime I

15、ntegrators/DAEsRythmosC+ utilities, (some) I/OTeuchos, EpetraExt, KokkosTrilinos TutorialDidasko“Skins”PyTrilinos, WebTrilinos, Star-P, Stratimikos, ForTrilinosOptimizationMOOCHO, AristosArchetype packageNewPackageOther new in 7.0 (8.0)Galeri, Isorropia, Moertel, RTOp, Aristos, RBGenAztecOO, Belos,

16、KomplexIFPACK, ML, CLAPSEpetra, Jpetra, TpetraThyraTeuchos, EpetraExt, KokkosEpetra, Teuchos, Pliris, AmesosNOX, LOCAMOOCHO, Aristos, RythmosPyTrilinos, WebTrilinos, Star-P, Stratimikos, ForTrilinos, Didasko, NewPackageGaleri, Isorropia, Moertel, RTOp, Aristos, RBGen, Sacado, StokhosTrilinosTrilinos

17、的應(yīng)用的應(yīng)用流體力學(xué)，如不可壓縮湍流問題、線性可壓縮流問題、三流體力學(xué)，如不可壓縮湍流問題、線性可壓縮流問題、三維海洋流的分叉性分析問題、與時間相關(guān)的熱輻射傳維海洋流的分叉性分析問題、與時間相關(guān)的熱輻射傳遞問題、風(fēng)洞流問題、大振幅穩(wěn)態(tài)水波旋流問題等。遞問題、風(fēng)洞流問題、大振幅穩(wěn)態(tài)水波旋流問題等。Gregory Larson et al., Application of single-level, pointwise algebraic, and smoothed aggregation multigrid methods to direct numerical simulations of i

18、ncompressible turbulent flows, Comput Visual Sci, 11(2008), 2740.H. C. Elman et al., A parallel block multi-level preconditioner for the 3D incompressible Navier-Stokes equations, Journal of Computational Physics, 187 (2003), 504523.Howard Elman et al., A taxonomy and comparison of parallel block mu

19、lti-level preconditioners for the incompressible Navier-Stokes equations, Journal of Computational Physics, 227 (2008), 17901808.Dave A. May and Louis Moresi, Preconditioned iterative methods for Stokes flow problems arising in computational geodynamics, Physics of the Earth and Planetary Interiors,

20、 171 (2008), 3347.David K. Gartling and Clark R. Dohrmann, Quadratic finite elements and incompressible viscous flows, Comput. Methods Appl. Mech. Engrg., 195 (2006), 16921708. Trilinos的應(yīng)用的應(yīng)用電磁學(xué)電磁學(xué)Peter Arbenz et al., On a parallel multilevel preconditioned Maxwell eigensolver, Parallel Computing, 3

21、2 (2006), 157165.T. Vejchodsky et al., Modular hp-FEM system HERMES and its application to Maxwells equations, Mathematics and Computers in Simulation, 76 (2007), 223228.半導(dǎo)體工藝技術(shù)，如共振隧穿二極管中的電子轉(zhuǎn)半導(dǎo)體工藝技術(shù)，如共振隧穿二極管中的電子轉(zhuǎn)移問題、具輻射損害性的雙極型結(jié)型晶體管大移問題、具輻射損害性的雙極型結(jié)型晶體管大尺度瞬間靈敏度分析問題。尺度瞬間靈敏度分析問題。M. S. Lasater et al., Pa

22、rallel Parameter Study of the Wigner- Poisson Equations for RTDs, Computers and Mathematics with Applications, 51 (2006), 1677- -1688.E. T. Phipps et al., Large-Scale Transient Sensitivity Analysis of a Radiation- Damaged Bipolar Junction Transistor via Automatic Differentiation.Trilinos的應(yīng)用的應(yīng)用醫(yī)學(xué)，如心電

23、學(xué)中心肌組織作用位的擴展問題。醫(yī)學(xué)，如心電學(xué)中心肌組織作用位的擴展問題。L. Gerardo-Giorda et al., A model-based block-triangular preconditioner for the Bidomain system in electrocardiology, Journal of Computational Physics, 228 (2009), 36253639. 材料學(xué)，如表面壓力下金屬納米線的共振性質(zhì)問題、石英材料學(xué)，如表面壓力下金屬納米線的共振性質(zhì)問題、石英晶體的高頻共振問題。晶體的高頻共振問題。Harold S. Park and P

24、atrick A. Klein, Surface stress effects on the resonant properties of metal nanowires: The importance of finite deformation kinematics and the impact of the residual surface stress, Journal of the Mechanics and Physics of Solids, 56 (2008), 31443166.Ji Wang et al., Parallel finite element analysis o

25、f high frequency vibrations of quartz crystal resonators on LINUX cluster, Acta Mechanica Solida Sinica, Vol. 21, No. 6, December, 2008.Trilinos的應(yīng)用的應(yīng)用大氣學(xué)，如海洋氣候模型等。大氣學(xué)，如海洋氣候模型等。Katherine J. Evans et al., A Scalable and Adaptable Solution Framework within Components of the Community Climate System Mod

26、el, ICCS 2009, Part II, LNCS 5545, pp. 332341, 2009.Arie de Niet et al., A tailored solver for bifurcation analysis of ocean-climate models, Journal of Computational Physics, 227 (2007), 654679.其它，如核物理學(xué)。其它，如核物理學(xué)。M. Rizea et al., Finite difference approach for the two-dimensional Schringer equation w

27、ith application to scission-neutron emission, Computer Physics Communications, 179 (2008), 466478. PETSc 可擴展可移植科學(xué)計算工具箱PETSc (Portable, Extensible Toolkit for Scientific Computation) 是美國能源部DOE2000支持開發(fā)的20多個ACTS工具箱之一，是由Argonne國家實驗室開發(fā)的可移植可擴展科學(xué)計算工具箱，主要用于在分布式存儲環(huán)境高效求解偏微分方程組及相關(guān)問題。PETSc所有消息傳遞通信均采用MPI標(biāo)準(zhǔn)實現(xiàn)。 PE

28、TSc 包含許多并行線性和非線性方程求解器，這些求解器是使用C ，C+，F(xiàn)ortran77/90和現(xiàn)在的Python編寫的，而且PETSc支持有助于有限差分方法的并行分布式陣列。PETSc用C語言開發(fā)，遵循面向?qū)ο笤O(shè)計的基本特征，用戶基于PETSc對象可以靈活開發(fā)應(yīng)用程序。PETSc支持Fortran 77/90、C和C+編寫的串行和并行代碼。 PETSc最新版本為Petsc-3.0.0。PETSc接口接口Chaco - a graph partitioning package. FFTW - Fastest Fourier Transform in the West, developed a

29、t MIT by Matteo Frigo and Steven G. Johnson. Hypre - the LLNL preconditioner library. MUMPS - MUltifrontal Massively Parallel sparse direct Solver. ParMeTiS - parallel graph partitionerpARMS - A Package for the Parallel Iterative Solution of General Large Sparse Linear System, by Zhongze Li and Yous

30、ef Saad. ScaLAPACK - Scalable LAPACK. SPAI - for parallel sparse approximate inverse preconditioning. SPOOLES - SParse Object Oriented Linear Equations Solve developed by Cleve Ashcraft. SPRNG - The Scalable Parallel Random Number Generators Library. Sundial/CVODE - the LLNL SUite of Nonlinear and D

31、Ifferential/ALgebraic equation Solvers. SuperLU, SuperLU_Dist - robust and efficient sequential and parallel direct sparse solves. Zoltan - Parallel Partitioning, Load Balancing and Data-Management Services. PETSc結(jié)構(gòu)結(jié)構(gòu)Computation and Communication KernelsMPI, MPI-IO, BLAS, LAPACKProfiling InterfaceAp

32、plication CodesMatrices, Vectors, IndicesGridManagementLinear SolversPreconditioners + Krylov MethodsNonlinear SolversODE IntegratorsVisualizationInterfacePETSc結(jié)構(gòu)結(jié)構(gòu)PETSc codeUser codeApplicationInitializationFunctionEvaluationJacobianEvaluationPost-ProcessingPCPETScMain RoutineLinear Solvers (KSP)No

33、nlinear Solvers (SNES)Timestepping Solvers (TS)TAO TAO (The Toolkit for Advanced Optimization)是美國能源部DOE2000 支持開發(fā)的20 多個ACTS 工具箱之一，是2001年由Argonne 國家實驗室開發(fā)的高級最優(yōu)化工具箱。TAO的核心開發(fā)人員是Argonne國家實驗室數(shù)學(xué)與計算機科學(xué)部的Steve Benson ，Lois Curfman McInnes ，Jorge Mor，Jason Sarich等人。 研發(fā)TAO的主要目的是在高性能機器上求解大規(guī)模最優(yōu)化問題，采用面向?qū)ο蟮木幊碳夹g(shù)，充分利

34、用底層工具箱所提供的支持（并行稀疏矩陣數(shù)據(jù)結(jié)構(gòu)、預(yù)條件子、解法器等）。在此基礎(chǔ)上進行開發(fā)，不必重寫這些代碼，可以提高開發(fā)效率，節(jié)省開發(fā)時間，目標(biāo)是設(shè)計出移植性好、性能高、擴展性好、獨立于體系結(jié)構(gòu)的并行優(yōu)化軟件。TAOSLEPc SLEPc(Scalable Library for Eigenvalue Problem Computations) 由西班牙Politecnica de Valencia大學(xué)的高性能網(wǎng)絡(luò)設(shè)計與計算小組的成員開發(fā)，主要負(fù)責(zé)人是Jose E. Roman and Andrs Toms。SLEPc是一個并行求解大規(guī)模稀疏矩陣特征問題的軟件庫。它建立在PETSc基礎(chǔ)之上，從

35、軟件結(jié)構(gòu)到語法標(biāo)準(zhǔn)都與PETSc完全一致，可以理解為PETSc的功能擴展。 SLEPc提供了多個軟件包的接口，其中包括ARPACK, BLZPACK, PLANSO, TRLAN等，這些軟件包都是可選擇安裝的，并不影響SLEPc的使用。SLEPc全面支持Fortran語言、C語言以及C+，可以在絕大多數(shù)UNIX系統(tǒng)上運行。PETSc、TAO、SLEPc的應(yīng)用的應(yīng)用流體力學(xué)，如可壓縮非粘性流問題、地下水或地表水的流流體力學(xué)，如可壓縮非粘性流問題、地下水或地表水的流體問題、粘彈流體流問題、維多利亞水螅水母引起的體問題、粘彈流體流問題、維多利亞水螅水母引起的流體流動模擬、冰川的形變和流變等。流體流動

36、模擬、冰川的形變和流變等。Mehmet Sahin and Helen J. Wilson, A semi-staggered dilation-free finite volume method for the numerical solution of viscoelastic fluid flows on all-hexahedral elements, J. Non-Newtonian Fluid Mech., 147 (2007), 7991.C.M. Klaij et al., Pseudo-time stepping methods for space-time disconti

37、nuous Galerkin discretizations of the compressible Navier-Stokes equations, Journal of Computational Physics, 219 (2006), 622643.Laslo T. Diosady and David L. Darmofal, Preconditioning methods for discontinuous Galerkin solutions of the Navier-Stokes equations, Journal of Computational Physics, 228

38、(2009), 39173935. Feng-Nan Hwang and Xiao-Chuan Cai, A parallel nonlinear additive Schwarz preconditioned inexact Newton algorithm for incompressible Navier-Stokes equations, Journal of Computational Physics, 204 (2005), 666691. PETSc、TAO、SLEPc的應(yīng)用的應(yīng)用醫(yī)學(xué)，如三維超聲心電圖的研究、三維心肌圖像恢復(fù)模擬醫(yī)學(xué)，如三維超聲心電圖的研究、三維心肌圖像恢復(fù)模擬

39、問題、腦模型、計算生物學(xué)。問題、腦模型、計算生物學(xué)。L. Carracciuolo et al., Towards a parallel component for imaging in PETSc programming environment: A case study in 3-D echocardiography, Parallel Computing, 32 (2006), 6783.P. Colli Franzone et al., Simulating patterns of excitation, repolarization and action potential dura

40、tion with cardiac Bidomain and Monodomain models, Mathematical Biosciences, 197 (2005), 3566. Prashanth Dumpuri et al., An atlas-based method to compensate for brain shift: Preliminary results, Medical Image Analysis, 11 (2007), 128145.Joe Pitt-Francis et al., Chaste: A test-driven approach to softw

41、are development for biological modeling, Computer Physics Communications, 40th Anniversary Issue.PETSc、TAO、SLEPc的應(yīng)用的應(yīng)用動力學(xué)，如地球動力學(xué)模擬、煙囪中由熱化學(xué)對流引起的動力學(xué)，如地球動力學(xué)模擬、煙囪中由熱化學(xué)對流引起的灰塵定向凝固模擬、回轉(zhuǎn)動力學(xué)粒子模擬、在微流控灰塵定向凝固模擬、回轉(zhuǎn)動力學(xué)粒子模擬、在微流控芯片中粒子流動和運動現(xiàn)象的電動學(xué)模擬、化學(xué)工程芯片中粒子流動和運動現(xiàn)象的電動學(xué)模擬、化學(xué)工程中的分子動力學(xué)模擬、中的分子動力學(xué)模擬、Bose Hubbard 模型（氣態(tài)模型

42、（氣態(tài)BoseEinstein冷凝物動力學(xué)）、磁流體動力學(xué)等。冷凝物動力學(xué)）、磁流體動力學(xué)等。R.F. Katz et al., Numerical simulation of geodynamic processes with the Portable Extensible Toolkit for Scientific Computation, Physics of the Earth and Planetary Interiors, 163 (2007), 5268.煙囪：Richard F. Katz and M. Grae Worster, Simulation of directio

43、nal solidification, thermochemical convection, and chimney formation in a Hele-Shaw cell, Journal of Computational Physics, 227 (2008), 98239840. Y. Nishimura et al., A finite element Poisson solver for gyrokinetic particle simulations in a global field aligned mesh, Journal of Computational Physics

44、, 214 (2006), 657671.Hypre 高性能預(yù)條件子Hypre（High Performance Preconditioners）由美國加州大學(xué)（UC）和勞倫斯-利弗莫爾國家實驗室（LLNL）應(yīng)用科學(xué)計算中心（CASC）開發(fā)。開發(fā)Hypre軟件包的動機起因于美國能源部LLNL在研究國防、環(huán)境、能源和生物科學(xué)中的物理現(xiàn)象時開發(fā)的一些模擬代碼。該軟件包主要用于大規(guī)模并行計算機上求解大型稀疏線性方程組，主要目的是為用戶提供高級并行預(yù)條件子。Hypre具有功能強大性、易用性、適應(yīng)性和互動性等特點。HypreSolverStructSStructFEIIJJacobi SMG PFMG

45、BoomerAMG ParaSails Euclid PILUT PCG GMRES HypreHypre流體力學(xué)，如地下儲水層復(fù)雜流模擬、正壓大氣模型（天流體力學(xué)，如地下儲水層復(fù)雜流模擬、正壓大氣模型（天氣預(yù)測）、大氣、海洋模型、可變粘性氣預(yù)測）、大氣、海洋模型、可變粘性Stokes流問題流問題等。等。Eric Chnier et al, A collocated finite volume scheme to solve free convection for general non-conforming grids, Journal of Computational Physics, 2

46、28 (2009), 22962311.C. Burstedde et al., Parallel scalable adjoint-based adaptive solution of variable-viscosity Stokes flow problems, Comput. Methods Appl. Mech. Engrg., 198 (2009), 16911700.M. Oevermann et al, A sharp interface finite volume method for elliptic equations on Cartesian grids, Journa

47、l of Computational Physics, 228 (2009), 51845206.Hypre動力學(xué)，如水動力學(xué)、生物流體動力學(xué)（如血流動力學(xué)、動力學(xué)，如水動力學(xué)、生物流體動力學(xué)（如血流動力學(xué)、肌肉壁和心瓣的彈性結(jié)構(gòu)動力學(xué)）、低磁雷諾數(shù)下的自肌肉壁和心瓣的彈性結(jié)構(gòu)動力學(xué)）、低磁雷諾數(shù)下的自由表面流的磁流體動力學(xué)等。由表面流的磁流體動力學(xué)等。Boyce E. Griffith et al., An adaptive, formally second order accurate version of the immersed boundary method, Journal of

48、Computational Physics, 223 (2007), 1049.其他，如電離輻射的傳輸問題、中性粒子傳輸模型、具有其他，如電離輻射的傳輸問題、中性粒子傳輸模型、具有限單元矩陣的核空間的計算問題等。限單元矩陣的核空間的計算問題等。P. N. Brown et al., Fully implicit solution of large-scale non-equilibrium radiation diffusion with high order time integration, Journal of Computational Physics, 204 (2005), 760783.SUNDIALS SUNDIALS (Suite of Nonlinear