CURRENT METHODS FOR LARGE STIFF ODE SYSTEMS**Work performed under the auspices of U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract W-7405-Eng-48, and supported by DOE Office of Energy Research, Applied Mathematical Sciences Research Program.

Initial value problems for ODE systems are especially challenging when they are both large and stiff. Current methods range from explicit to fully implicit. Quasi-steady state, split-operator, and hybrid methods are often effective, but require much problem-specific setup and rarely allow for error control. General implicit methods with error control have the widest scope of applicability. They involve an algebraic system problem, which is usually solved with some type of Newton iteration, combined with a suitable linear algebraic system method. A wide variety of structured direct linear system algorithms and an even wider variety of iterative linear system algorithms have been studied for this purpose. Partitioning is often used effectively also. Research is in progress on matrix-free iteration methods for the algebraic system problem.

[1]  Some methods for separating stiff components in initial value problems , 1984 .

[2]  H. Mcdonald,et al.  On the structure and use of linearized block implicit schemes , 1980 .

[3]  T. Chan,et al.  Nonlinearly Preconditioned Krylov Subspace Methods for Discrete Newton Algorithms , 1984 .

[4]  Mohamed S. Kamel,et al.  Automatic Partitioning of Stiff Systems and Exploiting the Resulting Structure , 1979, TOMS.

[5]  J. F. Andrus,et al.  Numerical Solution of Systems of Ordinary Differential Equations Separated into Subsystems , 1979 .

[6]  R. F. Sincovec,et al.  Analysis of descriptor systems using numerical algorithms , 1981 .

[7]  Edward J. Kansa,et al.  An algorithm for multidimensional combusting flow problems , 1981 .

[8]  L. Oey.  Implicit schemes for differential equations , 1982 .

[9]  P. Brown,et al.  Matrix-free methods for stiff systems of ODE's , 1986 .

[10]  Richard E. Ewing,et al.  Incomplete Iteration for Time-Stepping a Galerkin Method for a Quasilinear Parabolic Problem , 1979 .

[11]  J. E. Dendy.  Alternating Direction Methods for Nonlinear Time-Dependent Problems , 1977 .

[12]  Arthur A. Mirin,et al.  FPPAC: A two-dimensional multispecies nonlinear Fokker-Planck package , 1984 .

[13]  R. F. Warming,et al.  Factored, a-stable, linear multistep methods: an alternative to the method of lines for multidimensions , 1979, SGNM.

[14]  The Livermore Regional Air Quality Model: I. Concept and Development , 1978 .

[15]  Zahari Zlatev,et al.  Classification of the systems of ordinary differential equations and practical aspects in the numerical integration of large systems , 1980, Comput. Chem..

[16]  Peter Deuflhard,et al.  An Advanced Simulation Package for Large Chemical Reaction Systems , 1985 .

[17]  Modular Simulation of Large Stiff Systems , 1981 .

[18]  I. Duff.  Direct Methods for Solving Sparse Systems of Linear Equations , 1984 .

[19]  G. Hall,et al.  A Single Code for the Solution of Stiff and Nonstiff ODE’s , 1985 .

[20]  A. Curtis.  Jacobian Matrix Properties and Their Impact on Choice of Software for Stiff ODE Systems , 1983 .

[22]  Y. A. Liu,et al.  The Occurrence and Numerical Solution of Physical and Chemical Systems Having Widely Varying Time Constants , 1974 .

[23]  P. Houwen,et al.  Preconditioning and Coarse Grid Corrections in the Solution of the Initial Value Problem for Nonlinear Partial Differential Equations , 1982 .

[24]  R. F. Warming,et al.  An extension ofA-stability to alternating direction implicit methods , 1979 .

[25]  H. H. Robertson,et al.  Numerical Integration of Systems of Stiff Ordinary Differential Equations with Special Structure , 1976 .

[26]  J. Lambert,et al.  Correction in the dominant space: a numerical technique for a certain class of stiff initial value problems , 1977 .

[27]  Robert K. Brayton,et al.  The Sparse Tableau Approach to Network Analysis and Design , 1971 .

[28]  Paul F. Dubois,et al.  An implicit scheme for calculating time- and frequency-dependent flux limited radiation diffusion in one dimension , 1984 .

[29]  Alan C. Hindmarsh,et al.  Experiments in numerical methods for a problem in combustion modeling , 1985 .

[30]  David S. Watkins,et al.  The Numerical Solution of Separably Stiff Systems by Precise Partitioning , 1983, TOMS.

[31]  E. Hofer,et al.  A Partially Implicit Method for Large Stiff Systems of ODEs with Only Few Equations Introducing Small Time-Constants , 1976 .

[32]  C. W. Gear,et al.  Iterative Solution of Linear Equations in ODE Codes , 1983 .

[33]  Achi Brandt,et al.  The multigrid method for semi-implicit hydrodynamics codes☆ , 1980 .

[34]  R. Balasubramanian,et al.  Hybrid simulation of multidimensional p.d.e.'s via solution of many one dimensional problems I. Theoretical basis and hybrid implementation , 1983 .

[35]  William R. Goodin,et al.  Numerical solution of the atmospheric diffusion equation for chemically reacting flows , 1982 .