A skeleton of the user’s main program

The following is a skeleton of the user’s main program (or calling program) for the integration of an ODE IVP using the MRIStep module. Most of the steps are independent of the NVECTOR, SUNMATRIX, SUNLINSOL and SUNNONLINSOL implementations used. For the steps that are not, refer to §9, §10, §11, and §12 for the specific name of the function to be called or macro to be referenced.

  1. Initialize parallel or multi-threaded environment, if appropriate.

    For example, call MPI_Init to initialize MPI if used, or set num_threads, the number of threads to use within the threaded vector functions, if used.

  2. Create the SUNDIALS context object

    Call SUNContext_Create() to allocate the SUNContext object.

  3. Set problem dimensions, etc.

    This generally includes the problem size, N, and may include the local vector length Nlocal.


    The variables N and Nlocal should be of type sunindextype.

  4. Set vector of initial values

    To set the vector y0 of initial values, use the appropriate functions defined by the particular NVECTOR implementation.

    For native SUNDIALS vector implementations (except the CUDA and RAJA based ones), use a call of the form

    y0 = N_VMake_***(..., ydata);

    if the realtype array ydata containing the initial values of \(y\) already exists. Otherwise, create a new vector by making a call of the form

    y0 = N_VNew_***(...);

    and then set its elements by accessing the underlying data where it is located with a call of the form

    ydata = N_VGetArrayPointer_***(y0);

    For details on each of SUNDIALS’ provided vector implementations, see the corresponding sections in §9 for details.

  5. Create an inner stepper object to solve the fast (inner) IVP

    • If using ARKStep as the fast (inner) integrator, create the ARKStep object with ARKStepCreate() and configure the integrator as desired for evolving the fast time scale. See sections § and § for details on configuring ARKStep.

      Once the ARKStep object is setup, create an MRIStepInnerStepper object with ARKStepCreateMRIStepInnerStepper().

    • If supplying a user-defined fast (inner) integrator, create the MRIStepInnerStepper object as described in section §


    When using ARKStep as a fast (inner) integrator it is the user’s responsibility to create, configure, and attach the integrator to the MRIStep module. User-specified options regarding how this fast integration should be performed (e.g., adaptive vs. fixed time step, explicit/implicit/ImEx partitioning, algebraic solvers, etc.) will be respected during evolution of the fast time scale during MRIStep integration.

    Due to the algorithms supported in MRIStep, the ARKStep module used for the fast time scale must be configured with an identity mass matrix.

    If a user_data pointer needs to be passed to user functions called by the fast (inner) integrator then it should be attached here by calling ARKStepSetUserData(). This user_data pointer will only be passed to user-supplied functions that are attached to the fast (inner) integrator. To supply a user_data pointer to user-supplied functions called by the slow (outer) integrator the desired pointer should be attached by calling MRIStepSetUserData() after creating the MRIStep memory below. The user_data pointers attached to the inner and outer integrators may be the same or different depending on what is required by the user code.

    Specifying a rootfinding problem for the fast integration is not supported. Rootfinding problems should be created and initialized with the slow integrator. See the steps below and MRIStepRootInit() for more details.

  6. Create an MRIStep object for the slow (outer) integration

    Create the MRIStep object by calling MRIStepCreate(). One of the inputs to MRIStepCreate() is the MRIStepInnerStepper object for solving the fast (inner) IVP created in the previous step.

  7. Set the slow step size

    Call MRIStepSetFixedStep() to specify the slow time step size.

  8. Create and configure implicit solvers (as appropriate)

    Specifically, if MRIStep is configured with an implicit slow right-hand side function in the prior step, then the following steps are recommended:

    1. Specify integration tolerances

      Call MRIStepSStolerances() or MRIStepSVtolerances() to specify either a scalar relative tolerance and scalar absolute tolerance, or a scalar relative tolerance and a vector of absolute tolerances, respectively. Alternatively, call MRIStepWFtolerances() to specify a function which sets directly the weights used in evaluating WRMS vector norms. See § for details.

    2. Create nonlinear solver object

      If a non-default nonlinear solver object is desired for implicit MRI stage solves (see §, then that nonlinear solver object must be created by using the appropriate functions defined by the particular SUNNONLINSOL implementation (e.g., NLS = SUNNonlinSol_***(...); where *** is the name of the nonlinear solver (see §12 for details).

      For the SUNDIALS-supplied SUNNONLINSOL implementations, the nonlinear solver object may be created using a call of the form

      SUNNonlinearSolver NLS = SUNNonlinSol_*(...);

      where * can be replaced with “Newton”, “FixedPoint”, or other options, as discussed in the sections § and §12.

      Note: by default, MRIStep will use the Newton nonlinear solver (see section §12.7), so a custom nonlinear solver object is only needed when using a different solver, or for the user to exercise additional controls over the Newton solver.

    3. Attach nonlinear solver module

      If a nonlinear solver object was created above, then it must be attached to MRIStep using the call (for details see §

      ier = MRIStepSetNonlinearSolver(...);
    4. Set nonlinear solver optional inputs

      Call the appropriate set functions for the selected nonlinear solver module to change optional inputs specific to that nonlinear solver. These must be called after attaching the nonlinear solver to MRIStep, otherwise the optional inputs will be overridden by MRIStep defaults. See §12 for more information on optional inputs.

    5. Create matrix object

      If a nonlinear solver requiring a linear solver will be used (e.g., a Newton iteration) and if that linear solver will be matrix-based, then a template Jacobian matrix must be created by using the appropriate functions defined by the particular SUNMATRIX implementation.

      For the SUNDIALS-supplied SUNMATRIX implementations, the matrix object may be created using a call of the form

      SUNMatrix A = SUNBandMatrix(...);

      or similar for other matrix modules (see §10 for further information).

    6. Create linear solver object

      If a nonlinear solver requiring a linear solver will be used (e.g., a Newton iteration), then the desired linear solver object(s) must be created by using the appropriate functions defined by the particular SUNLINSOL implementation.

      For any of the SUNDIALS-supplied SUNLINSOL implementations, the linear solver object may be created using a call of the form

      SUNLinearSolver LS = SUNLinSol_*(...);

      where * can be replaced with “Dense”, “SPGMR”, or other options, as discussed in §11.

    7. Set linear solver optional inputs

      Call *Set* functions from the selected linear solver module to change optional inputs specific to that linear solver. See the documentation for each SUNLINSOL module in §11 for details.

    8. Attach linear solver module

      If a linear solver was created above for implicit MRI stage solves, initialize the ARKLS linear solver interface by attaching the linear solver object (and Jacobian matrix object, if applicable) with the call (for details see §

      ier = MRIStepSetLinearSolver(...);
  9. Set optional inputs

    Call MRIStepSet* functions to change any optional inputs that control the behavior of MRIStep from their default values. See § for details.

  10. Specify rootfinding problem

    Optionally, call MRIStepRootInit() to initialize a rootfinding problem to be solved during the integration of the ODE system. See § for general details, and § for relevant optional input calls.

  11. Advance solution in time

    For each point at which output is desired, call

    ier = MRIStepEvolve(arkode_mem, tout, yout, &tret, itask);

    Here, itask specifies the return mode. The vector yout (which can be the same as the vector y0 above) will contain \(y(t_\text{out})\). See § for details.

  12. Get optional outputs

    Call MRIStepGet* and/or ARKStepGet* functions to obtain optional output from the slow or fast integrators respectively. See § and § for details.

  13. Deallocate memory for solution vector

    Upon completion of the integration, deallocate memory for the vector y (or yout) by calling the NVECTOR destructor function:

  14. Free solver memory

    • If ARKStep was used as the fast (inner) IVP integrator, call MRIStepInnerStepper_Free() and ARKStepFree() to free the memory allocated for the fast (inner) integrator.

    • If a user-defined fast (inner) integrator was supplied, free the integrator content and call MRIStepInnerStepper_Free() to free the MRIStepInnerStepper object.

    • Call MRIStepFree() to free the memory allocated for the slow integration object.

  15. Free linear solver and matrix memory (as appropriate)

    Call SUNLinSolFree() and (possibly) SUNMatDestroy() to free any memory allocated for any linear solver and/or matrix objects created above for either the fast or slow integrators.

  16. Free nonlinear solver memory (as appropriate)

    If a user-supplied SUNNonlinearSolver was provided to MRIStep, then call SUNNonlinSolFree() to free any memory allocated for the nonlinear solver object created above.

  17. Free the SUNContext object Call SUNContext_Free() to free the memory allocated for the SUNContext object.

  1. Finalize MPI, if used

    Call MPI_Finalize to terminate MPI.