SUNDIALS Documentation

KINSOL is a package designed to solve nonlinear algebraic systems of the form:

F(u) = 0

It supports several nonlinear solver methods:

• Newton-Krylov
• Picard
• Fixed-point

Both Picard and fixed-point methods can be enhanced using Anderson acceleration.

KINSOL is built with a modular design that leverages shared vector, matrix, and linear solver APIs used across the SUNDIALS suite. This allows it to run in various computing environments, including serial, shared memory, distributed memory, and accelerator-based (e.g., GPU) systems.

CVODE is a package designed to solve stiff and nonstiff ordinary differential equation (ODE) systems (initial value problems) provided in the explicit form:

dy/dt = f(t,y), y(t0) = y0.

It offers two primary variable-order, variable-coefficient multistep methods:

1. Adams-Moulton methods: Optimized for non-stiff problems.
2. BDF (Backward Differentiation Formula) methods: Optimized for stiff problems (using fixed-leading-coefficient form).

CVODE is written in ANSI standard C and is part of the SUNDIALS suite. Its modular design allows it to run in various environments, including serial, shared memory, distributed memory, and accelerator-based (e.g., GPU) systems by leveraging shared APIs for vectors, matrices, linear solvers, and nonlinear solvers.

CVODES is a package designed to solve stiff and nonstiff ordinary differential equation (ODE) systems (initial value problems) provided in the explicit form:

dy/dt = f(t,y,p), y(t0) = y0(p)

Key features include:

• Sensitivity Analysis: Supports both forward and adjoint modes.
• Solver Methods:
  • Adams-Moulton methods: For non-stiff problems.
  • BDF (Backward Differentiation Formula) methods: In fixed-leading-coefficient form for stiff problems.
• Modular Design: Leverages shared SUNDIALS APIs for vectors, matrices, linear solvers, and nonlinear solvers, allowing it to run on serial, shared memory, distributed memory, and accelerator-based (e.g., GPU) systems.

IDA is a package designed to solve differential-algebraic equation (DAE) systems of the form:

F(t, y, y') = 0, y(t0) = y0, y'(t0) = y0'

It utilizes variable-order, variable-coefficient BDF (Backward Differentiation Formula) methods in fixed-leading-coefficient form.

As part of the SUNDIALS suite, IDA features a modular design that allows it to run in various computing environments, including:

• Serial
• Shared memory
• Distributed memory
• Accelerator-based systems (e.g., GPU)

This modularity is achieved by leveraging shared vector, matrix, linear solver, and nonlinear solver APIs used across other SUNDIALS packages like ARKode, CVODE, and KINSOL.

IDAS is a package designed for solving differential-algebraic equation (DAE) systems of the form:

F(t, y, y', p) = 0, y(t0) = y0(p), y'(t0) = y0'(p)

It includes sensitivity analysis capabilities in both forward and adjoint modes. The solver utilizes variable-order, variable-coefficient BDF (Backward Differentiation Formula) methods in fixed-leading-coefficient form.

IDAS is part of the SUNDIALS suite and is written in ANSI standard C. It is designed to be modular, leveraging shared APIs for vectors, matrices, linear solvers, and nonlinear solvers, allowing it to run in serial, shared memory, distributed memory, and accelerator-based (e.g., GPU) environments.

IDA is used for the integration of Differential-Algebraic Equations (DAEs). When using IDA, users must ensure compatibility between different SUNDIALS objects. Specifically, not all SUNLinearSolver and SUNMatrix objects are compatible with all N_Vector implementations.

Key Compatibility Notes:

• NVECTOR_PARALLEL is not compatible with dense, banded, or sparse SUNMatrix types or their corresponding SUNLinearSolver objects.
• The IDABBDPRE preconditioner can only be used with NVECTOR_PARALLEL.
• When using SuperLU_MT, it is not recommended to use a threaded vector object unless it is the NVECTOR_OPENMP module and SuperLU_MT is also compiled with OpenMP.

ARKODE is a package within the SUNDIALS suite designed for solving stiff, nonstiff, and multirate ordinary differential equation (ODE) systems (initial value problems). It is specifically designed for systems given in the linearly implicit form:

M y' = f1(t,y) + f2(t,y), y(t0) = y0.

ARKODE supports several integration methods:

• Explicit and implicit Runge-Kutta methods for standard ODEs.
• Implicit-explicit (IMEX) additive Runge-Kutta methods for systems where different parts of the vector field have different stiffness levels.
• Multirate infinitesimal (MRI) methods for multirate problems.

It is written in ANSI standard C and is highly portable, supporting serial, shared memory, distributed memory, and accelerator-based (e.g., GPU) computing environments due to its modular design.

CVODES is used to solve Initial Value Problems (IVPs) and provides linear multistep methods with sensitivity analysis. When building applications with CVODES, you must ensure compatibility between your chosen N_Vector implementation and your SUNLinearSolver or SUNMatrix modules.

Compatibility Notes:

• NVECTOR_PARALLEL is not compatible with dense, banded, or sparse SUNMatrix types or their corresponding SUNLinearSolver modules.
• The CVBANDPRE preconditioning module is only compatible with NVECTOR_SERIAL, NVECTOR_OPENMP, and NVECTOR_PTHREADS.
• The CVBBDPRE preconditioner module can only be used with NVECTOR_PARALLEL.
• If using SuperLU_MT, it is recommended to use the NVECTOR_OPENMP module, provided SuperLU_MT is also compiled with OpenMP.

The amount of history required for CVodeResizeHistory depends on the integration method and the current order $q$. Let $k = \min{q+1, q_{\text{max}}}$.

Note on History Ordering: Time values in t_hist must be ordered from most recent to oldest ($t_n > t_{n-1} > \dots$ for forward integration; $t_n < t_{n-1} < \dots$ for backward integration).