3.1. Introduction
CVODE is part of a software family called SUNDIALS: SUite of Nonlinear and DIfferential/ALgebraic equation Solvers [69]. This suite consists of CVODE, ARKODE, KINSOL, and IDA, and variants of these with sensitivity analysis capabilities.
3.1.1. Historical Background
Fortran solvers for ODE initial value problems are widespread and heavily used. Two solvers that have been written at LLNL in the past are VODE [21] and VODPK [27]. VODE is a general purpose solver that includes methods for both stiff and nonstiff systems, and in the stiff case uses direct methods (full or banded) for the solution of the linear systems that arise at each implicit step. Externally, VODE is very similar to the well known solver LSODE [96]. VODPK is a variant of VODE that uses a preconditioned Krylov (iterative) method, namely GMRES, for the solution of the linear systems. VODPK is a powerful tool for large stiff systems because it combines established methods for stiff integration, nonlinear iteration, and Krylov (linear) iteration with a problem-specific treatment of the dominant source of stiffness, in the form of the user-supplied preconditioner matrix [22]. The capabilities of both VODE and VODPK have been combined in the C-language package CVODE [38].
At present, CVODE may utilize a variety of Krylov methods provided in SUNDIALS that can be used in conjuction with Newton iteration: these include the GMRES (Generalized Minimal RESidual) [102], FGMRES (Flexible Generalized Minimum RESidual) [101], Bi-CGStab (Bi-Conjugate Gradient Stabilized) [127], TFQMR (Transpose-Free Quasi-Minimal Residual) [55], and PCG (Preconditioned Conjugate Gradient) [64] linear iterative methods. As Krylov methods, these require almost no matrix storage for solving the Newton equations as compared to direct methods. However, the algorithms allow for a user-supplied preconditioner matrix, and for most problems preconditioning is essential for an efficient solution. For very large stiff ODE systems, the Krylov methods are preferable over direct linear solver methods, and are often the only feasible choice. Among the Krylov methods in SUNDIALS, we recommend GMRES as the best overall choice. However, users are encouraged to compare all options, especially if encountering convergence failures with GMRES. Bi-CGStab and TFQMR have an advantage in storage requirements, in that the number of workspace vectors they require is fixed, while that number for GMRES depends on the desired Krylov subspace size. FGMRES has an advantage in that it is designed to support preconditioners that vary between iterations (e.g. iterative methods). PCG exhibits rapid convergence and minimal workspace vectors, but only works for symmetric linear systems.
In the process of translating the VODE and VODPK algorithms into C, the overall CVODE organization has been changed considerably. One key feature of the CVODE organization is that the linear system solvers comprise a layer of code modules that is separated from the integration algorithm, allowing for easy modification and expansion of the linear solver array. A second key feature is a separate module devoted to vector operations; this facilitated the extension to multiprosessor environments with minimal impacts on the rest of the solver, resulting in PVODE [30], the parallel variant of CVODE.
Around 2002, the functionality of CVODE and PVODE were combined
into one single code, simply called CVODE. Development of this
version of CVODE was concurrent with a redesign of the vector
operations module across the SUNDIALS suite. The key feature of the
N_Vector
module is that it is written in terms of abstract vector
operations with the actual vector kernels attached by a particular
implementation (such as serial or parallel) of N_Vector
. This allows
writing the SUNDIALS solvers in a manner independent of the actual
N_Vector
implementation (which can be user-supplied), as well as
allowing more than one N_Vector
module linked into an executable
file. SUNDIALS (and thus CVODE) is supplied with a wide range of different
N_Vector
implementations, including: serial, MPI-parallel, both OpenMP and
Pthreads thread-parallel N_Vector
implementations, a Hypre parallel
implementation, a PETSc implementation, and various GPU-enabled
implementations.
3.1.2. Changes to SUNDIALS in release X.Y.Z
New Features
Added CMake infrastructure that enables externally maintained addons/plugins to be optionally built with SUNDIALS. See Contributing for details.
Bug Fixes
Updated the CMake variable HIP_PLATFORM
default to amd
as the previous
default, hcc
, is no longer recognized in ROCm 5.7.0 or newer. The new
default is also valid in older version of ROCm (at least back to version 4.3.1).
Fixed a bug in the HIP execution policies where WARP_SIZE
would not be set
with ROCm 6.0.0 or newer.
Changed the CMake version compatibility mode for SUNDIALS to AnyNewerVersion
instead of SameMajorVersion
. This fixes the issue seen
here.
Fixed a bug in some Fortran examples where c_null_ptr
was passed as an argument
to a function pointer instead of c_null_funptr
. This caused compilation issues
with the Cray Fortran compiler.
Fixed a bug where MRIStepEvolve()
would not handle a recoverable error
produced from evolving the inner stepper.
Added support for Kokkos Kernels v4.
For changes in prior versions of SUNDIALS see §15.
3.1.3. Reading this User Guide
This user guide is a combination of general usage instructions. Specific example programs are provided as a separate document. We expect that some readers will want to concentrate on the general instructions, while others will refer mostly to the examples, and the organization is intended to accommodate both styles.
There are different possible levels of usage of CVODE. The most casual user, with a small IVP problem only, can get by with reading §3.2.1, then §3.4 through §3.4.3.9 only, and looking at examples in [71].
In a different direction, a more expert user with an IVP problem may
want to (a) use a package preconditioner
(§3.4.5), (b) supply his/her own Jacobian
or preconditioner routines
(§3.4.4.6), (c) do multiple runs of
problems of the same size (§3.4.3.13), (d)
supply a new N_Vector
module (§8), (e)
supply new SUNLinearSolver
and/or SUNMatrix
modules
(§9 and §10),
or even (f) supply new SUNNonlinearSolver
modules
(§11).
The structure of this document is as follows:
In §3.2, we give short descriptions of the numerical methods implemented by CVODE for the solution of initial value problems for systems of ODEs, and continue with short descriptions of preconditioning (§3.2.3), stability limit detection (§3.2.4), and rootfinding (§3.2.5).
The following chapter describes the software organization of the CVODE solver (§3.3).
§3.4 is the main usage document for CVODE for C applications. It includes a complete description of the user interface for the integration of ODE initial value problems.
In §1.9, we describe the use of CVODE with Fortran applications.
§8 gives a brief overview of the generic
N_Vector
module shared among the various components of SUNDIALS, and details on theN_Vector
implementations provided with SUNDIALS.§9 gives a brief overview of the generic
SUNMatrix
module shared among the various components of SUNDIALS, and details on theSUNMatrix
implementations provided with SUNDIALS: a dense implementation (§9.9), a banded implementation (§9.12) and a sparse implementation (§9.14).§10 gives a brief overview of the generic
SUNLinearSolver
module shared among the various components of SUNDIALS. This chapter contains details on theSUNLinearSolver
implementations provided with SUNDIALS. The chapter also contains details on theSUNLinearSolver
implementations provided with SUNDIALS that interface with external linear solver libraries.§11 describes the
SUNNonlinearSolver
API and nonlinear solver implementations shared among the various components of SUNDIALS.Finally, in the appendices, we provide detailed instructions for the installation of CVODE, within the structure of SUNDIALS (§1.1), as well as a list of all the constants used for input to and output from CVODE functions (§3.5).
Finally, the reader should be aware of the following notational
conventions in this user guide: program listings and identifiers (such
as CVodeInit()
) within textual explanations are hyperlinked
to their definitions directly; fields
in C structures (such as content) appear in italics; and packages
or modules, such as CVLS, are written in all capitals.
3.1.4. SUNDIALS License and Notices
All SUNDIALS packages are released open source, under the BSD 3-Clause license for more details see the LICENSE and NOTICE files provided with all SUNDIALS packages.
3.1.5. Acknowledgments
We wish to acknowledge the contributions to previous versions of the CVODE and PVODE codes and their user guides by Scott D. Cohen [37] and George D. Byrne [29].