The ARKode infrastructure provides adaptive-step time integration modules for stiff, nonstiff and mixed stiff/nonstiff systems of ordinary differential equations (ODEs). ARKode itself is structured to support a wide range of one-step (but multi-stage) methods, allowing for rapid development of parallel implementations of state-of-the-art time integration methods. At present, ARKode is packaged with two time-stepping modules, ARKStep and ERKStep.
ARKStep supports ODE systems posed in split, linearly-implicit form,
where \(t\) is the independent variable, \(y\) is the set of dependent variables (in \(\mathbb{R}^N\)), \(M\) is a user-specified, nonsingular operator from \(\mathbb{R}^N\) to \(\mathbb{R}^N\), and the right-hand side function is partitioned into up to two components:
Either of these operators may be disabled, allowing for fully explicit, fully implicit, or combination implicit-explicit (ImEx) time integration.
The algorithms used in ARKStep are adaptive- and fixed-step additive Runge Kutta methods. Such methods are defined through combining two complementary Runge-Kutta methods: one explicit (ERK) and the other diagonally implicit (DIRK). Through appropriately partitioning the ODE right-hand side into explicit and implicit components (1), such methods have the potential to enable accurate and efficient time integration of stiff, nonstiff, and mixed stiff/nonstiff systems of ordinary differential equations. A key feature allowing for high efficiency of these methods is that only the components in \(f_I(t,y)\) must be solved implicitly, allowing for splittings tuned for use with optimal implicit solver algorithms.
This framework allows for significant freedom over the constitutive methods used for each component, and ARKode is packaged with a wide array of built-in methods for use. These built-in Butcher tables include adaptive explicit methods of orders 2-8, adaptive implicit methods of orders 2-5, and adaptive ImEx methods of orders 3-5.
ERKStep focuses specifically on problems posed in explicit form,
allowing for increased computational efficiency and memory savings. The algorithms used in ERKStep are adaptive- and fixed-step explicit Runge Kutta methods. As with ARKStep, the ERKStep module is packaged with adaptive explicit methods of orders 2-8.
For problems that include nonzero implicit term \(f_I(t,y)\), the resulting implicit system (assumed nonlinear, unless specified otherwise) is solved approximately at each integration step, using a modified Newton method, inexact Newton method, or an accelerated fixed-point solver. For the Newton-based methods and the serial or threaded NVECTOR modules in SUNDIALS, ARKode may use a variety of linear solvers provided with SUNDIALS, including both direct (dense, band, or sparse) and preconditioned Krylov iterative (GMRES [SS1986], BiCGStab [V1992], TFQMR [F1993], FGMRES [S1993], or PCG [HS1952]) linear solvers. When used with the MPI-based parallel, PETSc, hypre, CUDA, and Raja NVECTOR modules, or a user-provided vector data structure, only the Krylov solvers are available, although a user may supply their own linear solver for any data structures if desired. For the serial or threaded vector structures, we provide a banded preconditioner module called ARKBANDPRE that may be used with the Krylov solvers, while for the MPI-based parallel vector structure there is a preconditioner module called ARKBBDPRE which provides a band-block-diagonal preconditioner. Additionally, a user may supply more optimal, problem-specific preconditioner routines.
An additional NVECTOR implementation was added for the Tpetra vector from the Trilinos library to facilitate interoperability between SUNDIALS and Trilinos. This implementation is accompanied by additions to user documentation and SUNDIALS examples.
A bug was fixed where a nonlinear solver object could be freed twice in some use cases.
The EXAMPLES_ENABLE_RAJA CMake option has been removed. The option EXAMPLES_ENABLE_CUDA
enables all examples that use CUDA including the RAJA examples with a CUDA back end
(if the RAJA NVECTOR is enabled).
The implementation header file arkode_impl.h is no longer installed. This means users
who are directly manipulating the ARKodeMem structure will need to update their code
to use ARKode’s public API.
Python is no longer required to run make test and make test_install.
Fixed a bug in ARKodeButcherTable_Write when printing a Butcher table
without an embedding.
Added information on how to contribute to SUNDIALS and a contributing agreement.
A bug in ARKode where single precision builds would fail to compile has been fixed.
The ARKode library has been entirely rewritten to support a modular
approach to one-step methods, which should allow rapid research and
development of novel integration methods without affecting existing
solver functionality. To support this, the existing ARK-based methods
have been encapsulated inside the new ARKStep time-stepping
module. Two new time-stepping modules have been added:
ERKStep module provides an optimized implementation for explicit
Runge-Kutta methods with reduced storage and number of calls to the ODE
right-hand side function.MRIStep module implements two-rate explicit-explicit multirate
infinitesimal step methods utilizing different step sizes for slow
and fast processes in an additive splitting.This restructure has resulted in numerous small changes to the user
interface, particularly the suite of “Set” routines for user-provided
solver parameters and and “Get” routines to access solver statistics,
that are now prefixed with the name of time-stepping module (e.g., ARKStep
or ERKStep) instead of ARKode. Aside from affecting the names of these
routines, user-level changes have been kept to a minimum. However, we recommend
that users consult both this documentation and the ARKode example programs for
further details on the updated infrastructure.
As part of the ARKode restructuring an ARKodeButcherTable structure
has been added for storing Butcher tables. Functions for creating new Butcher
tables and checking their analytic order are provided along with other utility
routines. For more details see Butcher Table Data Structure.
Two changes were made in the initial step size algorithm:
ARKode’s dense output infrastructure has been improved to support higher-degree Hermite polynomial interpolants (up to degree 5) over the last successful time step.
ARKode’s previous direct and iterative linear solver interfaces, ARKDLS and
ARKSPILS, have been merged into a single unified linear solver interface, ARKLS,
to support any valid SUNLINSOL module. This includes DIRECT and
ITERATIVE types as well as the new MATRIX_ITERATIVE type. Details
regarding how ARKLS utilizes linear solvers of each type as well as discussion
regarding intended use cases for user-supplied SUNLinSol implementations are
included in the chapter Description of the SUNLinearSolver module. All ARKode examples programs and the
standalone linear solver examples have been updated to use the unified linear
solver interface.
The user interface for the new ARKLS module is very similar to the previous
ARKDLS and ARKSPILS interfaces. Additionally, we note that Fortran users will
need to enlarge their iout array of optional integer outputs, and update the
indices that they query for certain linear-solver-related statistics.
The names of all constructor routines for SUNDIALS-provided SUNLinSol
implementations have been updated to follow the naming convention
SUNLinSol_* where * is the name of the linear solver. The new names are
SUNLinSol_Band, SUNLinSol_Dense, SUNLinSol_KLU,
SUNLinSol_LapackBand, SUNLinSol_LapackDense, SUNLinSol_PCG,
SUNLinSol_SPBCGS, SUNLinSol_SPFGMR, SUNLinSol_SPGMR,
SUNLinSol_SPTFQMR, and SUNLinSol_SuperLUMT. Solver-specific “set”
routine names have been similarly standardized. To minimize challenges in user
migration to the new names, the previous routine names may still be used; these
will be deprecated in future releases, so we recommend that users migrate to the
new names soon. All ARKode example programs and the standalone linear solver
examples have been updated to use the new naming convention.
The SUNBandMatrix constructor has been simplified to remove the
storage upper bandwidth argument.
SUNDIALS integrators have been updated to utilize generic nonlinear solver modules defined through the SUNNONLINSOL API. This API will ease the addition of new nonlinear solver options and allow for external or user-supplied nonlinear solvers. The SUNNONLINSOL API and SUNDIALS provided modules are described in Nonlinear Solver Data Structures and follow the same object oriented design and implementation used by the NVector, SUNMatrix, and SUNLinSol modules. Currently two SUNNONLINSOL implementations are provided, SUNNonlinSol_Newton and SUNNonlinSol_FixedPoint. These replicate the previous integrator specific implementations of a Newton iteration and an accelerated fixed-point iteration, respectively. Example programs using each of these nonlinear solver modules in a standalone manner have been added and all ARKode example programs have been updated to use generic SUNNonlinSol modules.
As with previous versions, ARKode will use the Newton solver (now
provided by SUNNonlinSol_Newton) by default. Use of the
ARKStepSetLinear() routine (previously named
ARKodeSetLinear) will indicate that the problem is
linearly-implicit, using only a single Newton iteration per implicit
stage. Users wishing to switch to the accelerated fixed-point solver
are now required to create a SUNNonlinSol_FixedPoint object and attach
that to ARKode, instead of calling the previous
ARKodeSetFixedPoint routine. See the documentation sections
A skeleton of the user’s main program,
Nonlinear solver interface functions, and
The SUNNonlinearSolver_FixedPoint implementation for further details, or the serial C
example program ark_brusselator_fp.c for an example.
Three fused vector operations and seven vector array operations have been added
to the NVECTOR API. These optional operations are disabled by default and may
be activated by calling vector specific routines after creating an NVector (see
Description of the NVECTOR Modules for more details). The new operations are intended
to increase data reuse in vector operations, reduce parallel communication on
distributed memory systems, and lower the number of kernel launches on systems
with accelerators. The fused operations are N_VLinearCombination,
N_VScaleAddMulti, and N_VDotProdMulti, and the vector array operations
are N_VLinearCombinationVectorArray, N_VScaleVectorArray,
N_VConstVectorArray, N_VWrmsNormVectorArray,
N_VWrmsNormMaskVectorArray, N_VScaleAddMultiVectorArray, and
N_VLinearCombinationVectorArray. If an NVector implementation defines any of
these operations as NULL, then standard NVector operations will
automatically be called as necessary to complete the computation.
Multiple changes to the CUDA NVECTOR were made:
- Changed the
N_VMake_Cudafunction to take a host data pointer and a device data pointer instead of anN_VectorContent_Cudaobject.- Changed
N_VGetLength_Cudato return the global vector length instead of the local vector length.- Added
N_VGetLocalLength_Cudato return the local vector length.- Added
N_VGetMPIComm_Cudato return the MPI communicator used.- Removed the accessor functions in the namespace
suncudavec.- Added the ability to set the
cudaStream_tused for execution of the CUDA NVECTOR kernels. See the functionN_VSetCudaStreams_Cuda.- Added
N_VNewManaged_Cuda,N_VMakeManaged_Cuda, andN_VIsManagedMemory_Cudafunctions to accommodate using managed memory with the CUDA NVECTOR.
Multiple changes to the RAJA NVECTOR were made:
- Changed
N_VGetLength_Rajato return the global vector length instead of the local vector length.- Added
N_VGetLocalLength_Rajato return the local vector length.- Added
N_VGetMPIComm_Rajato return the MPI communicator used.- Removed the accessor functions in the namespace
sunrajavec.
A new NVECTOR implementation for leveraging OpenMP 4.5+ device offloading has been added, NVECTOR_OpenMPDEV. See The NVECTOR_OPENMPDEV Module for more details.
Fixed a bug in the CUDA NVECTOR where the N_VInvTest operation could
write beyond the allocated vector data.
Fixed library installation path for multiarch systems. This fix changes the default
library installation path to CMAKE_INSTALL_PREFIX/CMAKE_INSTALL_LIBDIR
from CMAKE_INSTALL_PREFIX/lib. CMAKE_INSTALL_LIBDIR is automatically
set, but is available as a CMAKE option that can modified.
Fixed a problem with setting sunindextype which would occur with
some compilers (e.g. armclang) that did not define __STDC_VERSION__.
Added hybrid MPI/CUDA and MPI/RAJA vectors to allow use of more than one MPI rank when using a GPU system. The vectors assume one GPU device per MPI rank.
Changed the name of the RAJA NVECTOR library to
libsundials_nveccudaraja.lib from
libsundials_nvecraja.lib to better reflect that we only support CUDA
as a backend for RAJA currently.
Several changes were made to the build system:
SUNDIALS_INDEX_TYPE CMake option and
added the SUNDIALS_INDEX_SIZE CMake option to select the sunindextype
integer size.CMAKE_<language>_COMPILER can compile MPI programs before
trying to locate and use an MPI installation.MPI_C_COMPILER, MPI_CXX_COMPILER, MPI_Fortran_COMPILER,
and MPIEXEC_EXECUTABLE.LAPACK_ENABLE
is ON) the build system will infer the scheme from the Fortran
compiler. If a Fortran compiler is not available or the inferred or default
scheme needs to be overridden, the advanced options
SUNDIALS_F77_FUNC_CASE and SUNDIALS_F77_FUNC_UNDERSCORES can
be used to manually set the name-mangling scheme and bypass trying to infer
the scheme.src and example directories to make the CMake configuration file
structure more modular.Updated the minimum required version of CMake to 2.8.12 and enabled using rpath by default to locate shared libraries on OSX.
Fixed Windows specific problem where sunindextype was not correctly
defined when using 64-bit integers for the SUNDIALS index type. On Windows
sunindextype is now defined as the MSVC basic type __int64.
Added sparse SUNMatrix “Reallocate” routine to allow specification of the nonzero storage.
Updated the KLU SUNLinearSolver module to set constants for the two reinitialization types, and fixed a bug in the full reinitialization approach where the sparse SUNMatrix pointer would go out of scope on some architectures.
Updated the “ScaleAdd” and “ScaleAddI” implementations in the sparse SUNMatrix module to more optimally handle the case where the target matrix contained sufficient storage for the sum, but had the wrong sparsity pattern. The sum now occurs in-place, by performing the sum backwards in the existing storage. However, it is still more efficient if the user-supplied Jacobian routine allocates storage for the sum \(I+\gamma J\) or \(M+\gamma J\) manually (with zero entries if needed).
Changed LICENSE install path to instdir/include/sundials.
Fixed a potential memory leak in the SPGMR and SPFGMR linear solvers: if “Initialize” was called multiple times then the solver memory was reallocated (without being freed).
Fixed a minor bug in the ARKReInit routine, where a flag was incorrectly set to indicate that the problem had been resized (instead of just re-initialized).
Fixed C++11 compiler errors/warnings about incompatible use of string literals.
Updated KLU SUNLinearSolver module to use a typedef for the
precision-specific solve function to be used (to avoid compiler
warnings).
Added missing typecasts for some (void*) pointers (again, to avoid
compiler warnings).
Bugfix in sunmatrix_sparse.c where we had used int instead of
sunindextype in one location.
Added missing #include <stdio.h> in NVECTOR and SUNMATRIX header files.
Added missing prototype for ARKSpilsGetNumMTSetups.
Fixed an indexing bug in the CUDA NVECTOR implementation of
N_VWrmsNormMask and revised the RAJA NVECTOR implementation of
N_VWrmsNormMask to work with mask arrays using values other than
zero or one. Replaced double with realtype in the RAJA vector
test functions.
Fixed compilation issue with GCC 7.3.0 and Fortran programs that do not require a SUNMatrix or SUNLinearSolver module (e.g. iterative linear solvers, explicit methods, fixed point solver, etc.).
Added NVECTOR print functions that write vector data to a specified
file (e.g. N_VPrintFile_Serial).
Added make test and make test_install options to the build
system for testing SUNDIALS after building with make and
installing with make install respectively.
All interfaces to matrix structures and linear solvers have been reworked, and all example programs have been updated. The goal of the redesign of these interfaces was to provide more encapsulation and ease in interfacing custom linear solvers and interoperability with linear solver libraries.
Specific changes include:
Two additional NVECTOR implementations were added – one for CUDA and one for RAJA vectors. These vectors are supplied to provide very basic support for running on GPU architectures. Users are advised that these vectors both move all data to the GPU device upon construction, and speedup will only be realized if the user also conducts the right-hand-side function evaluation on the device. In addition, these vectors assume the problem fits on one GPU. Further information about RAJA, users are referred to the web site, https://software.llnl.gov/RAJA/. These additions are accompanied by additions to various interface functions and to user documentation.
All indices for data structures were updated to a new sunindextype
that can be configured to be a 32- or 64-bit integer data index type.
sunindextype is defined to be int32_t or int64_t when
portable types are supported, otherwise it is defined as int or
long int. The Fortran interfaces continue to use long int for
indices, except for their sparse matrix interface that now uses the
new sunindextype. This new flexible capability for index types
includes interfaces to PETSc, hypre, SuperLU_MT, and KLU with either
32-bit or 64-bit capabilities depending how the user configures
SUNDIALS.
To avoid potential namespace conflicts, the macros defining
booleantype values TRUE and FALSE have been changed to
SUNTRUE and SUNFALSE respectively.
Temporary vectors were removed from preconditioner setup and solve routines for all packages. It is assumed that all necessary data for user-provided preconditioner operations will be allocated and stored in user-provided data structures.
The file include/sundials_fconfig.h was added. This file contains
SUNDIALS type information for use in Fortran programs.
Added functions SUNDIALSGetVersion and SUNDIALSGetVersionNumber to get SUNDIALS release version information at runtime.
The build system was expanded to support many of the xSDK-compliant keys. The xSDK is a movement in scientific software to provide a foundation for the rapid and efficient production of high-quality, sustainable extreme-scale scientific applications. More information can be found at, https://xsdk.info.
In addition, numerous changes were made to the build system.
These include the addition of separate BLAS_ENABLE and BLAS_LIBRARIES
CMake variables, additional error checking during CMake configuration,
minor bug fixes, and renaming CMake options to enable/disable examples
for greater clarity and an added option to enable/disable Fortran 77 examples.
These changes included changing ENABLE_EXAMPLES to ENABLE_EXAMPLES_C,
changing CXX_ENABLE to EXAMPLES_ENABLE_CXX, changing F90_ENABLE to
EXAMPLES_ENABLE_F90, and adding an EXAMPLES_ENABLE_F77 option.
Corrections and additions were made to the examples, to installation-related files, and to the user documentation.
We have included numerous bugfixes and enhancements since the v1.0.2 release.
The bugfixes include:
linit function. This ensures that these solver
counters are initialized upon linear solver instantiation as well as
at the beginning of the problem solution.The feature changes/enhancements include:
N_VGetVectorID,
that returns the NVECTOR module name.ARKSpilsGetNumMtimesEvals() function was added
– this had been included in the previous documentation but had not
been implemented.This user guide is a combination of general usage instructions and specific example programs. 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.
The structure of this document is as follows:
All SUNDIALS packages are released open source, under the BSD 3-Clause license. The only requirements of the license are preservation of copyright and a standard disclaimer of liability. The full text of the license and an additional notice are provided below and may also be found in the LICENSE and NOTICE files provided with all SUNDIALS packages.
PLEASE NOTE If you are using SUNDIALS with any third party libraries linked in (e.g., LAPACK, KLU, SuperLU_MT, PETSc, or hypre), be sure to review the respective license of the package as that license may have more restrictive terms than the SUNDIALS license. For example, if someone builds SUNDIALS with a statically linked KLU, the build is subject to terms of the more-restrictive LGPL license (which is what KLU is released with) and not the SUNDIALS BSD license anymore.
Copyright (c) 2002-2019, Lawrence Livermore National Security and Southern Methodist University.
All rights reserved.
Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ‘’AS IS’’ AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
This work was produced under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.
This work was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights.
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LLNL-CODE-667205 (ARKODE)
UCRL-CODE-155951 (CVODE)
UCRL-CODE-155950 (CVODES)
UCRL-CODE-155952 (IDA)
UCRL-CODE-237203 (IDAS)
LLNL-CODE-665877 (KINSOL)