5.1. Introduction

IDA is part of a software family called SUNDIALS: SUite of Nonlinear and DIfferential/ALgebraic equation Solvers [70]. This suite consists of CVODE, ARKODE, KINSOL, and IDA, and variants of these with sensitivity analysis capabilities, CVODES and IDAS.

IDA is a general purpose solver for the initial value problem (IVP) for systems of differential-algebraic equations (DAEs). The name IDA stands for Implicit Differential-Algebraic solver. IDA is based on DASPK [23, 24], but is written in ANSI-standard C rather than Fortran77. Its most notable features are that, (1) in the solution of the underlying nonlinear system at each time step, it offers a choice of Newton/direct methods and a choice of Inexact Newton/Krylov (iterative) methods; and (2) it is written in a data-independent manner in that it acts on generic vectors and matrices without any assumptions on the underlying organization of the data. Thus IDA shares significant modules previously written within CASC at LLNL to support the ordinary differential equation (ODE) solvers CVODE [38, 73] and PVODE [29, 30], and also the nonlinear system solver KINSOL [74].

At present, IDA 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) [106], FGMRES (Flexible Generalized Minimum RESidual) [105], Bi-CGStab (Bi-Conjugate Gradient Stabilized) [131], TFQMR (Transpose-Free Quasi-Minimal Residual) [56], and PCG (Preconditioned Conjugate Gradient) [65] linear iterative methods. As Krylov methods, these require little matrix storage for solving the Newton equations as compared to direct methods. However, the algorithms allow for a user-supplied preconditioner, and, for most problems, preconditioning is essential for an efficient solution.

For very large DAE 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-CGFStab 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.

5.1.1. Changes to SUNDIALS in release 7.1.0

Major Features

Created shared user interface functions for ARKODE to allow more uniform control over time-stepping algorithms, improved extensibility, and simplified code maintenance. The corresponding stepper-specific user-callable functions are now deprecated and will be removed in a future major release.

Added CMake infrastructure that enables externally maintained addons/plugins to be optionally built with SUNDIALS. See Contributing for details.

New Features and Enhancements

Added support for Kokkos Kernels v4.

Added the following Runge-Kutta Butcher tables

  • ARKODE_FORWARD_EULER_1_1

  • ARKODE_RALSTON_EULER_2_1_2

  • ARKODE_EXPLICIT_MIDPOINT_EULER_2_1_2

  • ARKODE_BACKWARD_EULER_1_1

  • ARKODE_IMPLICIT_MIDPOINT_1_2

  • ARKODE_IMPLICIT_TRAPEZOIDAL_2_2

Added the following MRI coupling tables

  • ARKODE_MRI_GARK_FORWARD_EULER

  • ARKODE_MRI_GARK_RALSTON2

  • ARKODE_MRI_GARK_RALSTON3

  • ARKODE_MRI_GARK_BACKWARD_EULER

  • ARKODE_MRI_GARK_IMPLICIT_MIDPOINT

  • ARKODE_IMEX_MRI_GARK_EULER

  • ARKODE_IMEX_MRI_GARK_TRAPEZOIDAL

  • ARKODE_IMEX_MRI_GARK_MIDPOINT

Added ARKodeButcherTable_ERKIDToName() and ARKodeButcherTable_DIRKIDToName() to convert a Butcher table ID to a string representation.

Added the function ARKodeSetAutonomous() in ARKODE to indicate that the implicit right-hand side function does not explicitly depend on time. When using the trivial predictor, an autonomous problem may reuse implicit function evaluations across stage solves to reduce the total number of function evaluations.

Users may now disable interpolated output in ARKODE by passing ARK_INTERP_NONE to ARKodeSetInterpolantType(). When interpolation is disabled, rootfinding is not supported, implicit methods must use the trivial predictor (the default option), and interpolation at stop times cannot be used (interpolating at stop times is disabled by default). With interpolation disabled, calling ARKodeEvolve() in ARK_NORMAL mode will return at or past the requested output time (setting a stop time may still be used to halt the integrator at a specific time). Disabling interpolation will reduce the memory footprint of an integrator by two or more state vectors (depending on the interpolant type and degree) which can be beneficial when interpolation is not needed e.g., when integrating to a final time without output in between or using an explicit fast time scale integrator with an MRI method.

Added “Resize” capability to ARKODE’s SPRKStep time-stepping module.

Enabled the Fortran interfaces to build with 32-bit sunindextype.

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).

Renamed the DPCPP value for the SUNDIALS_GINKGO_BACKENDS CMake option to SYCL to match Ginkgo’s updated naming convention.

Changed the CMake version compatibility mode for SUNDIALS to AnyNewerVersion instead of SameMajorVersion. This fixes the issue seen here.

Fixed a CMake bug that caused an MPI linking error for our C++ examples in some instances. Fixes GitHub Issue #464.

Fixed the runtime library installation path for windows systems. This fix changes the default library installation path from CMAKE_INSTALL_PREFIX/CMAKE_INSTALL_LIBDIR to CMAKE_INSTALL_PREFIX/CMAKE_INSTALL_BINDIR.

Fixed conflicting .lib files between shared and static libs when using MSVC on Windows

Fixed invalid SUNDIALS_EXPORT generated macro when building both shared and static libs.

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 in the HIP execution policies where WARP_SIZE would not be set with ROCm 6.0.0 or newer.

Fixed a bug that caused error messages to be cut off in some cases. Fixes GitHub Issue #461.

Fixed a memory leak when an error handler was added to a SUNContext. Fixes GitHub Issue #466.

Fixed a bug where MRIStepEvolve() would not handle a recoverable error produced from evolving the inner stepper.

Added missing SetRootDirection and SetNoInactiveRootWarn functions to ARKODE’s SPRKStep time-stepping module.

Fixed a bug in ARKodeSPRKTable_Create() where the coefficient arrays were not allocated.

Fix bug on LLP64 platforms (like Windows 64-bit) where KLU_INDEXTYPE could be 32 bits wide even if SUNDIALS_INT64_T is defined.

Check if size of SuiteSparse_long is 8 if the size of sunindextype is 8 when using KLU.

Fixed several build errors with the Fortran interfaces on Windows systems.

Deprecation Notices

Numerous ARKODE stepper-specific functions are now deprecated in favor of ARKODE-wide functions.

Deprecated the ARKStepSetOptimalParams function. Since this function does not have an ARKODE-wide equivalent, instructions have been added to the user guide for how to retain the current functionality using other user-callable functions.

The unsupported implementations of N_VGetArrayPointer and N_VSetArrayPointer for the hypre and PETSc vectors are now deprecated. Users should access the underlying wrapped external library vector objects instead with N_VGetVector_ParHyp and N_VGetVector_Petsc, respectively.

For changes in prior versions of SUNDIALS see §15.

5.1.2. Reading this User Guide

The structure of this document is as follows:

  • In Chapter §5.2, we give short descriptions of the numerical methods implemented by IDA for the solution of initial value problems for systems of DAEs, along with short descriptions of preconditioning (§5.2.3) and rootfinding (§5.2.4).

  • The following chapter describes the software organization of the IDA solver (§5.3).

  • Chapter §5.4 is the main usage document for IDA for C and C++ applications. It includes a complete description of the user interface for the integration of DAE initial value problems. This is followed by documentation for using IDA with Fortran applications and on GPU accelerated systems.

  • Chapter §8 gives a brief overview of the generic N_Vector module shared among the various components of SUNDIALS, as well as details on the N_Vector implementations provided with SUNDIALS.

  • Chapter §9 gives a brief overview of the generic SUNMatrix module shared among the various components of SUNDIALS, and details on the SUNMatrix implementations provided with SUNDIALS.

  • Chapter §10 gives a brief overview of the generic SUNLinearSolver module shared among the various components of SUNDIALS. This chapter contains details on the SUNLinearSolver implementations provided with SUNDIALS. The chapter also contains details on the SUNLinearSolver implementations provided with SUNDIALS that interface with external linear solver libraries.

  • Chapter §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 IDA, within the structure of SUNDIALS (Appendix §1.1), as well as a list of all the constants used for input to and output from IDA functions (Appendix §5.5).

5.1.3. 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.