2.1. Introduction

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 four time-stepping modules, ARKStep, ERKStep, SPRKStep, and MRIStep.

ARKStep supports ODE systems posed in split, linearly-implicit form,

(2.1)\[M(t)\, \dot{y} = f^E(t,y) + f^I(t,y), \qquad y(t_0) = y_0,\]

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:

  • \(f^E(t,y)\) contains the “nonstiff” time scale components to be integrated explicitly, and

  • \(f^I(t,y)\) contains the “stiff” time scale components to be integrated implicitly.

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 (2.1), 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 2-9, adaptive implicit methods of orders 2-5, and adaptive ImEx methods of orders 2-5.

ERKStep focuses specifically on problems posed in explicit form,

(2.2)\[\dot{y} = f(t,y), \qquad y(t_0) = y_0.\]

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 2-9.

SPRKStep focuses on Hamiltonian systems posed in the form,

\[H(t, p, q) = T(t, p) + V(t, q)\]
(2.3)\[\dot{p} = f_1(t,q) = \frac{\partial V(t,q)}{\partial q}, \quad \dot{q} = f_2(t,p) = \frac{\partial T(t,p)}{\partial p},\]

allowing for conservation of quadratic invariants.

MRIStep focuses specifically on problems posed in additive form,

(2.4)\[\dot{y} = f^E(t,y) + f^I(t,y) + f^F(t,y), \qquad y(t_0) = y_0.\]

where here the right-hand side function is additively split into three components:

  • \(f^E(t,y)\) contains the “slow-nonstiff” components of the system (this will be integrated using an explicit method and a large time step \(h^S\)),

  • \(f^I(t,y)\) contains the “slow-stiff” components of the system (this will be integrated using an implicit method and a large time step \(h^S\)), and

  • \(f^F(t,y)\) contains the “fast” components of the system (this will be integrated using a possibly different method than the slow time scale and a small time step \(h^F \ll h^S\)).

For such problems, MRIStep provides fixed-step slow step multirate infinitesimal step (MIS), multirate infinitesimal GARK (MRI-GARK), and implicit-explicit MRI-GARK (IMEX-MRI-GARK) methods, allowing for evolution of the problem (2.4) using multirate methods having orders of accuracy 2-4.

For ARKStep or MRIStep 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 SUNNonlinearSolver module, supplied either by the user or from the underlying SUNDIALS infrastructure. For nonlinear solver algorithms that internally require a linear solver, ARKODE may use a variety of SUNLinearSolver modules provided with SUNDIALS, or again may utilize a user-supplied module.

2.1.1. Changes to SUNDIALS in release 6.1.0

Bug Fixes

Fixed a bug in v7.1.0 with the SYCL N_Vector N_VSpace function.

For changes in prior versions of SUNDIALS see §15.

2.1.2. Reading this User Guide

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:

2.1.3. 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.