Alan C. Hindmarsh
Abstract
SUNDIALS is a suite of advanced computational codes for solving large-scale problems that can be modeled as a system of nonlinear algebraic equations, or as initial-value problems in ordinary differential or differential-algebraic equations. The basic versions of these codes are called KINSOL, CVODE, and IDA, respectively. The codes are written in ANSI standard C and are suitable for either serial or parallel machine environments. Common and notable features of these codes include inexact Newton-Krylov methods for solving large-scale nonlinear systems; linear multistep methods for time-dependent problems; a highly modular structure to allow incorporation of different preconditioning and/or linear solver methods; and clear interfaces allowing for users to provide their own data structures underneath the solvers. We describe the current capabilities of the codes, along with some of the algorithms and heuristics used to achieve efficiency and robustness. We also describe how the codes stem from previous and widely used Fortran 77 solvers, and how the codes have been augmented with forward and adjoint methods for carrying out first-order sensitivity analysis with respect to model parameters or initial conditions.
- Brenan, K. E., Campbell, S. L., and Petzold, L. R. 1996. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. SIAM Press, Philadelphia, PA.Google Scholar
- Brown, P. N. 1987. A local convergence theory for combined inexact-Newton/finite difference projection methods. SIAM J. Numer. Anal. 24, 2, 407--434. Google Scholar
- Brown, P. N., Byrne, G. D., and Hindmarsh, A. C. 1989. VODE, a variable-coefficient ODE solver. SIAM J. Sci. Stat. Comput. 10, 1038--1051. Google Scholar
- Brown, P. N. and Hindmarsh, A. C. 1989. Reduced storage matrix methods in stiff ODE systems. J. Appl. Math. Comp. 31, 49--91.Google Scholar
- Brown, P. N., Hindmarsh, A. C., and Petzold, L. R. 1994. Using Krylov methods in the solution of large-scale differential-algebraic systems. SIAM J. Sci. Comput. 15, 1467--1488. Google Scholar
- Brown, P. N., Hindmarsh, A. C., and Petzold, L. R. 1998. Consistent initial condition calculation for differential-algebraic systems. SIAM J. Sci. Comput. 19, 1495--1512. Google Scholar
- Brown, P. N. and Saad, Y. 1990. Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Stat. Comput. 11, 450--481. Google Scholar
- Brown, P. N. and Woodward, C. S. 2001. Preconditioning strategies for fully implicit radiation diffusion with material-energy transfer. SIAM J. Sci. Comput. 23, 2, 499--516. Google Scholar
- Byrne, G. D. 1992. Pragmatic experiments with Krylov methods in the stiff ODE setting. In Computational Ordinary Differential Equations, J. Cash and I. Gladwell, Eds. Oxford University Press, Oxford, U.K., 323--356.Google Scholar
- Byrne, G. D. and Hindmarsh, A. C. 1975. A polyalgorithm for the numerical solution of ordinary differential equations. ACM Trans. Math. Softw. 1, 71--96. Google Scholar
- Byrne, G. D. and Hindmarsh, A. C. 1998. User documentation for PVODE, an ODE solver for parallel computers. Tech. rep. UCRL-ID-130884. Lawrence Livermore National Laboratory, Livermore, CA.Google Scholar
- Byrne, G. D. and Hindmarsh, A. C. 1999. PVODE, an ODE solver for parallel computers. Intl. J. High Perf. Comput. Apps. 13, 4, 254--365. Google Scholar
- Cao, Y., Li, S., Petzold, L. R., and Serban, R. 2003. Adjoint sensitivity analysis for differential-algebraic equations: The adjoint DAE system and its numerical solution. SIAM J. Sci. Comput. 24, 3, 1076--1089. Google Scholar
- Cohen, S. D. and Hindmarsh, A. C. 1994. CVODE user guide. Tech. rep. UCRL-MA-118618. Lawrence Livermore National Laboratory, Livermore, CA.Google Scholar
- Cohen, S. D. and Hindmarsh, A. C. 1996. CVODE, a stiff/nonstiff ODE solver in C. Comput. Phys. 10, 2, 138--143. Google Scholar
- Collier, A. M., Hindmarsh, A. C., Serban, R., and Woodward, C. S. 2004. User documentation for KINSOL v2.2.1. LLNL Tech. rep. UCRL-SM-208116. Lawrence Livermore National Laboratory, Livermore, CA.Google Scholar
- Curtis, A. R., Powell, M. J. D., and Reid, J. K. 1974. On the estimation of sparse Jacobian matrices. J. Inst. Math. Applic. 13, 117--119.Google Scholar
- Dembo, R. S., Eisenstat, S. C., and Steihaug, T. 1982. Inexact Newton methods. SIAM J. Numer. Anal. 19, 400--408.Google Scholar
- Dennis, J. E. and Schnabel, R. B. 1996. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Press, Philadelphia, PA. Google Scholar
- Eisenstat, S. C. and Walker, H. F. 1996. Choosing the forcing terms in an inexact Newton method. SIAM J. Sci. Comput. 17, 16--32. Google Scholar
- Feehery, W. F., Tolsma, J. E., and Barton, P. I. 1997. Efficient sensitivity analysis of large-scale differential-algebraic systems. Appl. Num. Math. 25, 1, 41--54. Google Scholar
- Hairer, E. and Wanner, G. 1991. Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, Germany.Google Scholar
- Hiebert, K. L. and Shampine, L. F. 1980. Implicitly defined output points for solutions of ODEs. Tech. rep. SAND80-0180. Sandia National Laboratories, Albuquerque, NM.Google Scholar
- Hindmarsh, A. C. 1992. Detecting stability barriers in BDF solvers. In Computational Ordinary Differential Equations, J. Cash and I. Gladwell, Eds. Oxford University Press, Oxford, U.K., 87--96.Google Scholar
- Hindmarsh, A. C. 1995. Avoiding BDF stability barriers in the MOL solution of advection-dominated problems. Appl. Num. Math. 17, 311--318. Google Scholar
- Hindmarsh, A. C. 2000. The PVODE and IDA algorithms. Tech. rep. UCRL-ID-141558. Lawrence Livermore National Laboratory, Livermore, CA.Google Scholar
- Hindmarsh, A. C. and Serban, R. 2004a. User documentation for CVODE v2.2.1. LLNL Tech. rep. UCRL-SM-208108. Lawrence Livermore National Laboratory, Livermore, CA.Google Scholar
- Hindmarsh, A. C. and Serban, R. 2004b. User documentation for CVODES v2.2.1. LLNL Tech. rep. UCRL-SM-208111. Lawrence Livermore National Laboratory, Livermore, CA.Google Scholar
- Hindmarsh, A. C. and Serban, R. 2004c. User documentation for IDA v2.2.1. LLNL Tech. rep. UCRL-SM-208112. Lawrence Livermore National Laboratory, Livermore, CA.Google Scholar
- Jackson, K. R. and Sacks-Davis, R. 1980. An alternative implementation of variable step-size multistep formulas for stiff ODEs. ACM Trans. Math. Softw. 6, 295--318. Google Scholar
- Jones, J. E. and Woodward, C. S. 2001. Newton-Krylov-multigrid solvers for large-scale, highly heterogeneous, variably saturated flow problems. Adv. Water Resourc. 24, 763-- 774.Google Scholar
- Kelley, C. T. 1995. Iterative Methods for Solving Linear and Nonlinear Equations. SIAM Press, Philadelphia, PA.Google Scholar
- Lee, S. L., Hindmarsh, A. C., and Brown, P. N. 2000. User documentation for SensPVODE, a variant of PVODE for sensitivity analysis. Tech. Rep. UCRL-MA-140211. Lawrence Livermore National Laboratory, Livermore, CA.Google Scholar
- Lee, S. L., Woodward, C. S., and Graziani, F. 2003. Analyzing radiation diffusion using time-dependent sensitivity-based techniques. J. Comp. Phys. 192, 1, 211--230. Google Scholar
- Maly, T. and Petzold, L. R. 1996. Numerical methods and software for sensitivity analysis of differential-algebraic systems. Appl. Num. Math. 20, 57--79. Google Scholar
- Radhakrishnan, K. and Hindmarsh, A. C. 1993. Description and use of LSODE, the Livermore solver for ordinary differential equations. Tech. rep. UCRL-ID-113855. Lawrence Livermore National Laboratory, Livermore, CA.Google Scholar
- Rognlien, T. D., Xu, X. Q., and Hindmarsh, A. C. 2002. Application of parallel implicit methods to edge-plasma numerical simulations. J. Comp. Phys. 175, 249--268. Google Scholar
- Saad, Y. and Schultz, M. H. 1986. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comp. 7, 856--869. Google Scholar
- Serban, R. and Hindmarsh, A. C. 2005. CVODES, the Sensitivity-enabled ODE solver in SUNDIALS. In Proceedings of the 2005 ASME International Design Engineering Technical Conference (Long Beach, CA, Sep. 24--28). Also published as 2005 LLNL Tech. rep. UCRL-PROC-21030, Lawrence Livermore National Laboratory, Livermore, CA.Google Scholar
- Woodward, C. S. 1998. A Newton-Krylov-multigrid solver for variably saturated flow problems. In Proceedings of the Twelfth International Conference on Computational Methods in Water Resources, vol. 2. Computational Mechanics Publications, Southampton, U.K., 609--616.Google Scholar
- Woodward, C. S., Grant, K. E., and Maxwell, R. 2002. Applications of sensitivity analysis to uncertainty quantification for variably saturated flow. In Computational Methods in Water Resources, S. M. Hassanizadeh, R. J. Schotting, W. G. Gray, and G. F. Pinder, Eds. Elsevier, Amsterdam, The Netherlands, 73--80.Google Scholar
Index Terms
- SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers
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Wolfgang Schreiner
The simulation of many physical phenomena, from turbulences in fusion reactors to water flows in porous media, requires the solution of systems of nonlinear and differential algebraic equations with thousands of unknowns. For the efficient computation of these solutions, the Lawrence Livermore National Laboratory (LLNL) has, starting in 1993, developed the suite of nonlinear and differential/algebraic equation solvers (SUNDIALS), which this paper describes. SUNDIALS is an open-source C-based package that encapsulates LLNL's accumulated expertise in solving real-world problems of the kind mentioned above. The package is structured into various modules, with the goal of maximizing its reusability: users can choose and configure combinations of algorithms and data structures, and even provide their own data structures and integrate their own solvers. The main part of the paper (sections 2 to 4) explains the algorithms underlying the solvers, the facilities for preconditioning systems to obtain acceptable efficiency, and the capabilities for analyzing the sensitivity of solutions to the initial model parameters. The rest (sections 5 to 6) presents the general organization of the code within the suite, and the approach for using it. The paper intentionally does not discuss the rationale of the algorithms and the heuristics they apply (appropriate references are given), nor does it show the details of how to write actual code (a user manual with code examples is available). It does, however, serve its core purpose: to show the big picture of a complex software suite that solves an important class of problems. Online Computing Reviews Service
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