CVODE is a solver for stiff and nonstiff ordinary differential equation (ODE) systems (initial value problem) given in explicit form y’ = f(t,y). The methods used in CVODE are variable-order, variable-step multistep methods. For nonstiff problems, CVODE includes the Adams-Moulton formulas, with the order varying between 1 and 12. For stiff problems, CVODE includes the Backward Differentiation Formulas (BDFs) in so-called fixed-leading coefficient form, with order varying between 1 and 5. For either choice of formula, the resulting nonlinear system is solved (approximately) at each integration step. For this, CVODE offers the choice of either functional iteration, suitable only for nonstiff systems, and various versions of Newton iteration. In the cases of a direct linear solver (dense or banded), the Newton iteration is a Modified Newton iteration, in that the Jacobian is fixed (and usually out of date). When using a Krylov method as the linear solver, the iteration is an Inexact Newton iteration, using the current Jacobian (through matrix-free products), in which the linear residual is nonzero but controlled. When used in conjunction with the serial NVECTOR module, CVODE provides both direct (dense and band) solvers and three preconditioned Krylov (iterative) solvers (GMRES, Bi-CGStab, and TFQMR). In the parallel version (CVODE used with a parallel NVECTOR module) only the Krylov linear solvers are available. An approximate diagonal Jacobian option is also available with both versions. For the serial version, there is a banded preconditioner module called CVBANDPRE for use with the Krylov solvers, while for the parallel version there is a preconditioner module called CVBBDPRE which provides a band-block-diagonal preconditioner. For use with Fortran applications, a set of Fortran/C interface routines, called FCVODE, is also supplied. These are written in C, but assume that the user calling program and all user-supplied routines are in Fortran.

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  1. Carrasco, Juan A.: Numerically stable methods for the computation of exit rates in Markov chains (2016)
  2. Hansen, M.A.; Sutherland, J.C.: Pseudotransient continuation for combustion simulation with detailed reaction mechanisms (2016)
  3. Kublik, Richard A.; Chopp, David L.: A locally adaptive time stepping algorithm for the solution to reaction diffusion equations on branched structures (2016)
  4. Loffeld, J.; Tokman, M.: Implementation of parallel adaptive-Krylov exponential solvers for stiff problems (2014)
  5. Scott, Joseph K.; Barton, Paul I.: Convex and concave relaxations for the parametric solutions of semi-explicit index-one differential-algebraic equations (2013)
  6. Scott, Joseph K.; Barton, Paul I.: Improved relaxations for the parametric solutions of ODEs using differential inequalities (2013)
  7. Scott, Joseph K.; Barton, Paul I.: Interval bounds on the solutions of semi-explicit index-one DAEs. II: Computation (2013)
  8. Chalupecký, Vladimír; Muntean, Adrian: Semi-discrete finite difference multiscale scheme for a concrete corrosion model: a priori estimates and convergence (2012)
  9. Abukhdeir, Nasser Mohieddin; Vlachos, Dionisios G.; Katsoulakis, Markos; Plexousakis, Michael: Long-time integration methods for mesoscopic models of pattern-forming systems (2011)
  10. Crunelli, Vincenzo; Errington, Adam C.; Hughes, Stuart W.; Tóth, Tibor I.: The thalamic low-threshold $Ca^2+$ potential: a key determinant of the local and global dynamics of the slow ($<1$ hz) sleep oscillation in thalamocortical networks (2011)
  11. Scott, Joseph K.; Stuber, Matthew D.; Barton, Paul I.: Generalized McCormick relaxations (2011)
  12. Tian, FuQiang; Gao, Long; Hu, HePing: A two-dimensional Richards equation solver based on CVODE for variably saturated soil water movement (2011)
  13. Safta, Cosmin; Ray, Jaideep; Najm, Habib N.: A high-order low-Mach number AMR construction for chemically reacting flows (2010)
  14. Dafilis, Mathew P.; Frascoli, Federico; Cadusch, Peter J.; Liley, David T.J.: Chaos and generalised multistability in a mesoscopic model of the electroencephalogram (2009)
  15. Varadhan, S.; Beaudoin, A.J.; Fressengeas, C.: Lattice incompatibility and strain-aging in single crystals (2009)
  16. Ropp, David L.; Shadid, John N.: Stability of operator splitting methods for systems with indefinite operators: Advection-diffusion-reaction systems (2008)
  17. Bridewell, Will; Langley, Pat; Todorovski, Ljupčo; Džeroski, Sašo: Inductive process modeling (2007)
  18. Moore, Peter K.: The impact of parameter selection on the performance of an automatic adaptive code for solving reaction-diffusion equations in three dimensions (2007)
  19. Tian, Fuqiang; Hu, Heping: Numerical model of Richard’s equation based on an ordinary differential equation solver (2007)
  20. Virányi, Zsanett; Tóth, Ágota; Horváth, Dezső: Modeling study of migration-driven lateral instability in autocatalytic systems (2007)

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