3.4.5. User-supplied functions

The user-supplied functions for ARKODE consist of:

• at least one function defining the ODE (required),

• a function that handles error and warning messages (optional),

• a function that provides the error weight vector (optional),

• a function that provides the residual weight vector (optional, ARKStep only),

• a function that handles adaptive time step error control (optional, ARKStep/ERKStep only),

• a function that handles explicit time step stability (optional, ARKStep/ERKStep only),

• a function that updates the implicit stage prediction (optional, ARKStep/MRIStep only),

• a function that defines the root-finding problem(s) to solve (optional),

• one or two functions that provide Jacobian-related information for the linear solver, if a component is treated implicitly and a Newton-based nonlinear iteration is chosen (optional, ARKStep/MRIStep only),

• one or two functions that define the preconditioner for use in any of the Krylov iterative algorithms, if linear systems of equations are to be solved using an iterative method (optional, ARKStep/MRIStep only),

• if the problem involves a non-identity mass matrix $$M\ne I$$ with ARKStep:

• one or two functions that provide mass-matrix-related information for the linear and mass matrix solvers (required),

• one or two functions that define the mass matrix preconditioner for use if an iterative mass matrix solver is chosen (optional), and

• a function that handles vector resizing operations, if the underlying vector structure supports resizing (as opposed to deletion/recreation), and if the user plans to call ARKStepResize(), ERKStepResize(), or MRIStepResize() (optional).

• MRIStep only: functions to be called before and after each inner integration to perform any communication or memory transfers of forcing data supplied by the outer integrator to the inner integrator, or state data supplied by the inner integrator to the outer integrator.

3.4.5.1. ODE right-hand side

The user must supply at least one function of type ARKRhsFn to specify the explicit and/or implicit portions of the ODE system to ARKStep, the ODE system function to ERKStep, or the “slow” right-hand side of the ODE system to MRIStep:

typedef int (*ARKRhsFn)(realtype t, N_Vector y, N_Vector ydot, void *user_data)

These functions compute the ODE right-hand side for a given value of the independent variable $$t$$ and state vector $$y$$.

Arguments:
• t – the current value of the independent variable.

• y – the current value of the dependent variable vector.

• ydot – the output vector that forms [a portion of] the ODE RHS $$f(t,y)$$.

• user_data – the user_data pointer that was passed to ARKStepSetUserData(), ERKStepSetUserData(), or MRIStepSetUserData().

Return value:

An ARKRhsFn should return 0 if successful, a positive value if a recoverable error occurred (in which case ARKODE will attempt to correct), or a negative value if it failed unrecoverably (in which case the integration is halted and ARK_RHSFUNC_FAIL is returned).

Notes:

Allocation of memory for ydot is handled within ARKODE.

The vector ydot may be uninitialized on input; it is the user’s responsibility to fill this entire vector with meaningful values.

A recoverable failure error return from the ARKRhsFn is typically used to flag a value of the dependent variable $$y$$ that is “illegal” in some way (e.g., negative where only a non-negative value is physically meaningful). If such a return is made, ARKODE will attempt to recover (possibly repeating the nonlinear iteration, or reducing the step size in ARKStep or ERKStep) in order to avoid this recoverable error return. There are some situations in which recovery is not possible even if the right-hand side function returns a recoverable error flag. One is when this occurs at the very first call to the ARKRhsFn (in which case ARKODE returns ARK_FIRST_RHSFUNC_ERR). Another is when a recoverable error is reported by ARKRhsFn after the ARKStep integrator completes a successful stage, in which case ARKStep returns ARK_UNREC_RHSFUNC_ERR). Similarly, since MRIStep does not currently support adaptive time stepping at the slow time scale, it may halt on a recoverable error flag that would normally have resulted in a stepsize reduction.

3.4.5.2. Error message handler function

As an alternative to the default behavior of directing error and warning messages to the file pointed to by errfp (see ARKStepSetErrFile(), ERKStepSetErrFile(), and MRIStepSetErrFile()), the user may provide a function of type ARKErrHandlerFn to process any such messages.

typedef void (*ARKErrHandlerFn)(int error_code, const char *module, const char *function, char *msg, void *user_data)

This function processes error and warning messages from ARKODE and its sub-modules.

Arguments:
• error_code – the error code.

• module – the name of the ARKODE module reporting the error.

• function – the name of the function in which the error occurred.

• msg – the error message.

• user_data – a pointer to user data, the same as the eh_data parameter that was passed to ARKStepSetErrHandlerFn(), ERKStepSetErrHandlerFn(), or MRIStepSetErrHandlerFn().

Return value:

An ARKErrHandlerFn function has no return value.

Notes:

error_code is negative for errors and positive (ARK_WARNING) for warnings. If a function that returns a pointer to memory encounters an error, it sets error_code to 0.

3.4.5.3. Error weight function

As an alternative to providing the relative and absolute tolerances, the user may provide a function of type ARKEwtFn to compute a vector ewt containing the weights in the WRMS norm $$\|v\|_{WRMS} = \left(\dfrac{1}{n} \displaystyle \sum_{i=1}^n \left(ewt_i\; v_i\right)^2 \right)^{1/2}$$. These weights will be used in place of those defined in §3.2.6.

typedef int (*ARKEwtFn)(N_Vector y, N_Vector ewt, void *user_data)

This function computes the WRMS error weights for the vector $$y$$.

Arguments:
• y – the dependent variable vector at which the weight vector is to be computed.

• ewt – the output vector containing the error weights.

• user_data – a pointer to user data, the same as the user_data parameter that was passed to ARKStepSetUserData(), ERKStepSetUserData(), or MRIStepSetUserData().

Return value:

An ARKEwtFn function must return 0 if it successfully set the error weights, and -1 otherwise.

Notes:

Allocation of memory for ewt is handled within ARKODE.

The error weight vector must have all components positive. It is the user’s responsibility to perform this test and return -1 if it is not satisfied.

3.4.5.4. Residual weight function (ARKStep only)

As an alternative to providing the scalar or vector absolute residual tolerances (when the IVP units differ from the solution units), the user may provide a function of type ARKRwtFn to compute a vector rwt containing the weights in the WRMS norm $$\|v\|_{WRMS} = \left(\dfrac{1}{n} \displaystyle \sum_{i=1}^n \left(rwt_i\; v_i\right)^2 \right)^{1/2}$$. These weights will be used in place of those defined in §3.2.6.

typedef int (*ARKRwtFn)(N_Vector y, N_Vector rwt, void *user_data)

This function computes the WRMS residual weights for the vector $$y$$.

Arguments:
• y – the dependent variable vector at which the weight vector is to be computed.

• rwt – the output vector containing the residual weights.

• user_data – a pointer to user data, the same as the user_data parameter that was passed to ARKStepSetUserData().

Return value:

An ARKRwtFn function must return 0 if it successfully set the residual weights, and -1 otherwise.

Notes:

Allocation of memory for rwt is handled within ARKStep.

The residual weight vector must have all components positive. It is the user’s responsibility to perform this test and return -1 if it is not satisfied.

3.4.5.5. Time step adaptivity function (ARKStep and ERKStep only)

As an alternative to using one of the built-in time step adaptivity methods for controlling solution error, the user may provide a function of type ARKAdaptFn to compute a target step size $$h$$ for the next integration step. These steps should be chosen such that the error estimate for the next time step remains below 1.

typedef int (*ARKAdaptFn)(N_Vector y, realtype t, realtype h1, realtype h2, realtype h3, realtype e1, realtype e2, realtype e3, int q, int p, realtype *hnew, void *user_data)

This function implements a time step adaptivity algorithm that chooses $$h$$ to satisfy the error tolerances.

Arguments:
• y – the current value of the dependent variable vector.

• t – the current value of the independent variable.

• h1 – the current step size, $$t_n - t_{n-1}$$.

• h2 – the previous step size, $$t_{n-1} - t_{n-2}$$.

• h3 – the step size $$t_{n-2}-t_{n-3}$$.

• e1 – the error estimate from the current step, $$n$$.

• e2 – the error estimate from the previous step, $$n-1$$.

• e3 – the error estimate from the step $$n-2$$.

• q – the global order of accuracy for the method.

• p – the global order of accuracy for the embedded method.

• hnew – the output value of the next step size.

• user_data – a pointer to user data, the same as the h_data parameter that was passed to ARKStepSetAdaptivityFn() or ERKStepSetAdaptivityFn().

Return value:

An ARKAdaptFn function should return 0 if it successfully set the next step size, and a non-zero value otherwise.

3.4.5.6. Explicit stability function (ARKStep and ERKStep only)

A user may supply a function to predict the maximum stable step size for the explicit portion of the problem, $$f^E(t,y)$$ in ARKStep or the full $$f(t,y)$$ in ERKStep. While the accuracy-based time step adaptivity algorithms may be sufficient for retaining a stable solution to the ODE system, these may be inefficient if the explicit right-hand side function contains moderately stiff terms. In this scenario, a user may provide a function of type ARKExpStabFn to provide this stability information to ARKODE. This function must set the scalar step size satisfying the stability restriction for the upcoming time step. This value will subsequently be bounded by the user-supplied values for the minimum and maximum allowed time step, and the accuracy-based time step.

typedef int (*ARKExpStabFn)(N_Vector y, realtype t, realtype *hstab, void *user_data)

This function predicts the maximum stable step size for the explicit portion of the ODE system.

Arguments:
• y – the current value of the dependent variable vector.

• t – the current value of the independent variable.

• hstab – the output value with the absolute value of the maximum stable step size.

• user_data – a pointer to user data, the same as the estab_data parameter that was passed to ARKStepSetStabilityFn() or ERKStepSetStabilityFn().

Return value:

An ARKExpStabFn function should return 0 if it successfully set the upcoming stable step size, and a non-zero value otherwise.

Notes:

If this function is not supplied, or if it returns hstab $$\le 0.0$$, then ARKODE will assume that there is no explicit stability restriction on the time step size.

3.4.5.7. Implicit stage prediction function (ARKStep and MRIStep only)

A user may supply a function to update the prediction for each implicit stage solution. If supplied, this routine will be called after any existing ARKStep or MRIStep predictor algorithm completes, so that the predictor may be modified by the user as desired. In this scenario, a user may provide a function of type ARKStagePredictFn to provide this implicit predictor to ARKODE. This function takes as input the already-predicted implicit stage solution and the corresponding “time” for that prediction; it then updates the prediction vector as desired. If the user-supplied routine will construct a full prediction (and thus the ARKODE prediction is irrelevant), it is recommended that the user not call ARKStepSetPredictorMethod() or MRIStepSetPredictorMethod(), thereby leaving the default trivial predictor in place.

typedef int (*ARKStagePredictFn)(realtype t, N_Vector zpred, void *user_data)

This function updates the prediction for the implicit stage solution.

Arguments:
• t – the current value of the independent variable containing the “time” corresponding to the predicted solution.

• zpred – the ARKStep-predicted stage solution on input, and the user-modified predicted stage solution on output.

• user_data – a pointer to user data, the same as the user_data parameter that was passed to ARKStepSetUserData() or MRIStepSetUserData().

Return value:

An ARKStagePredictFn function should return 0 if it successfully set the upcoming stable step size, and a non-zero value otherwise.

Notes:

This may be useful if there are bound constraints on the solution, and these should be enforced prior to beginning the nonlinear or linear implicit solver algorithm.

This routine is incompatible with the “minimum correction predictor” – option 5 to the routine ARKStepSetPredictorMethod(). If both are selected, then ARKStep will override its built-in implicit predictor routine to instead use option 0 (trivial predictor).

3.4.5.8. Rootfinding function

If a rootfinding problem is to be solved during integration of the ODE system, the user must supply a function of type ARKRootFn.

typedef int (*ARKRootFn)(realtype t, N_Vector y, realtype *gout, void *user_data)

This function implements a vector-valued function $$g(t,y)$$ such that roots are sought for the components $$g_i(t,y)$$, $$i=0,\ldots,$$ nrtfn-1.

Arguments:
• t – the current value of the independent variable.

• y – the current value of the dependent variable vector.

• gout – the output array, of length nrtfn, with components $$g_i(t,y)$$.

• user_data – a pointer to user data, the same as the user_data parameter that was passed to ARKStepSetUserData(), ERKStepSetUserData(), or MRIStepSetUserData().

Return value:

An ARKRootFn function should return 0 if successful or a non-zero value if an error occurred (in which case the integration is halted and ARKODE returns ARK_RTFUNC_FAIL).

Notes:

Allocation of memory for gout is handled within ARKODE.

3.4.5.9. Jacobian construction (matrix-based linear solvers, ARKStep and MRIStep only)

If a matrix-based linear solver module is used (i.e., a non-NULL SUNMatrix object was supplied to ARKStepSetLinearSolver() or MRIStepSetLinearSolver(), the user may provide a function of type ARKLsJacFn to provide the Jacobian approximation or ARKLsLinSysFn to provide an approximation of the linear system $$\mathcal{A}(t,y) = M(t) - \gamma J(t,y)$$.

typedef int (*ARKLsJacFn)(realtype t, N_Vector y, N_Vector fy, SUNMatrix Jac, void *user_data, N_Vector tmp1, N_Vector tmp2, N_Vector tmp3)

This function computes the Jacobian matrix $$J(t,y) = \dfrac{\partial f^I}{\partial y}(t,y)$$ (or an approximation to it).

Arguments:
• t – the current value of the independent variable.

• y – the current value of the dependent variable vector, namely the predicted value of $$y(t)$$.

• fy – the current value of the vector $$f^I(t,y)$$.

• Jac – the output Jacobian matrix.

• user_data – a pointer to user data, the same as the user_data parameter that was passed to ARKStepSetUserData() or MRIStepSetUserData().

• tmp1, tmp2, tmp3 – pointers to memory allocated to variables of type N_Vector which can be used by an ARKLsJacFn as temporary storage or work space.

Return value:

An ARKLsJacFn function should return 0 if successful, a positive value if a recoverable error occurred (in which case ARKODE will attempt to correct, while ARKLS sets last_flag to ARKLS_JACFUNC_RECVR), or a negative value if it failed unrecoverably (in which case the integration is halted, ARKStepEvolve() or MRIStepEvolve() returns ARK_LSETUP_FAIL and ARKLS sets last_flag to ARKLS_JACFUNC_UNRECVR).

Notes:

Information regarding the specific SUNMatrix structure (e.g.~number of rows, upper/lower bandwidth, sparsity type) may be obtained through using the implementation-specific SUNMatrix interface functions (see §10 for details).

When using a linear solver of type SUNLINEARSOLVER_DIRECT, prior to calling the user-supplied Jacobian function, the Jacobian matrix $$J(t,y)$$ is zeroed out, so only nonzero elements need to be loaded into Jac.

With the default Newton nonlinear solver, each call to the user’s ARKLsJacFn() function is preceded by a call to the implicit ARKRhsFn() user function with the same $$(t,y)$$ arguments. Thus, the Jacobian function can use any auxiliary data that is computed and saved during the evaluation of $$f^I(t,y)$$. In the case of a user-supplied or external nonlinear solver, this is also true if the nonlinear system function is evaluated prior to calling the linear solver setup function (see §12.1.4 for more information).

If the user’s ARKLsJacFn function uses difference quotient approximations, then it may need to access quantities not in the argument list, including the current step size, the error weights, etc. To obtain these, the user will need to add a pointer to the ark_mem structure to their user_data, and then use the ARKStepGet* or MRIStepGet* functions listed in §3.4.2.2.10 or §3.4.4.2.9. The unit roundoff can be accessed as UNIT_ROUNDOFF, which is defined in the header file sundials_types.h.

dense $$J(t,y)$$: A user-supplied dense Jacobian function must load the N by N dense matrix Jac with an approximation to the Jacobian matrix $$J(t,y)$$ at the point $$(t,y)$$. Utility routines and accessor macros for the SUNMATRIX_DENSE module are documented in §10.9.

banded $$J(t,y)$$: A user-supplied banded Jacobian function must load the band matrix Jac with the elements of the Jacobian $$J(t,y)$$ at the point $$(t,y)$$. Utility routines and accessor macros for the SUNMATRIX_BAND module are documented in §10.12.

sparse $$J(t,y)$$: A user-supplied sparse Jacobian function must load the compressed-sparse-column (CSC) or compressed-sparse-row (CSR) matrix Jac with an approximation to the Jacobian matrix $$J(t,y)$$ at the point $$(t,y)$$. Storage for Jac already exists on entry to this function, although the user should ensure that sufficient space is allocated in Jac to hold the nonzero values to be set; if the existing space is insufficient the user may reallocate the data and index arrays as needed. Utility routines and accessor macros for the SUNMATRIX_SPARSE type are documented in §10.14.

typedef int (*ARKLsLinSysFn)(realtype t, N_Vector y, N_Vector fy, SUNMatrix A, SUNMatrix M, booleantype jok, booleantype *jcur, realtype gamma, void *user_data, N_Vector tmp1, N_Vector tmp2, N_Vector tmp3)

This function computes the linear system matrix $$\mathcal{A}(t,y) = M(t) - \gamma J(t,y)$$ (or an approximation to it).

Arguments:
• t – the current value of the independent variable.

• y – the current value of the dependent variable vector, namely the predicted value of $$y(t)$$.

• fy – the current value of the vector $$f^I(t,y)$$.

• A – the output linear system matrix.

• M – the current mass matrix (this input is NULL if $$M = I$$).

• jok – is an input flag indicating whether the Jacobian-related data needs to be updated. The jok argument provides for the reuse of Jacobian data. When jok = SUNFALSE, the Jacobian-related data should be recomputed from scratch. When jok = SUNTRUE the Jacobian data, if saved from the previous call to this function, can be reused (with the current value of gamma). A call with jok = SUNTRUE can only occur after a call with jok = SUNFALSE.

• jcur – is a pointer to a flag which should be set to SUNTRUE if Jacobian data was recomputed, or set to SUNFALSE if Jacobian data was not recomputed, but saved data was still reused.

• gamma – the scalar $$\gamma$$ appearing in the Newton system matrix $$\mathcal{A}=M(t)-\gamma J(t,y)$$.

• user_data – a pointer to user data, the same as the user_data parameter that was passed to ARKStepSetUserData() or MRIStepSetUserData().

• tmp1, tmp2, tmp3 – pointers to memory allocated to variables of type N_Vector which can be used by an ARKLsLinSysFn as temporary storage or work space.

Return value:

An ARKLsLinSysFn function should return 0 if successful, a positive value if a recoverable error occurred (in which case ARKODE will attempt to correct, while ARKLS sets last_flag to ARKLS_JACFUNC_RECVR), or a negative value if it failed unrecoverably (in which case the integration is halted, ARKStepEvolve() or MRIStepEvolve() returns ARK_LSETUP_FAIL and ARKLS sets last_flag to ARKLS_JACFUNC_UNRECVR).

3.4.5.10. Jacobian-vector product (matrix-free linear solvers, ARKStep and MRIStep only)

When using a matrix-free linear solver module for the implicit stage solves (i.e., a NULL-valued SUNMATRIX argument was supplied to ARKStepSetLinearSolver() or MRIStepSetLinearSolver(), the user may provide a function of type ARKLsJacTimesVecFn in the following form, to compute matrix-vector products $$Jv$$. If such a function is not supplied, the default is a difference quotient approximation to these products.

typedef int (*ARKLsJacTimesVecFn)(N_Vector v, N_Vector Jv, realtype t, N_Vector y, N_Vector fy, void *user_data, N_Vector tmp)

This function computes the product $$Jv$$ where $$J(t,y) \approx \dfrac{\partial f^I}{\partial y}(t,y)$$ (or an approximation to it).

Arguments:
• v – the vector to multiply.

• Jv – the output vector computed.

• t – the current value of the independent variable.

• y – the current value of the dependent variable vector.

• fy – the current value of the vector $$f^I(t,y)$$.

• user_data – a pointer to user data, the same as the user_data parameter that was passed to ARKStepSetUserData() or MRIStepSetUserData().

• tmp – pointer to memory allocated to a variable of type N_Vector which can be used as temporary storage or work space.

Return value:

The value to be returned by the Jacobian-vector product function should be 0 if successful. Any other return value will result in an unrecoverable error of the generic Krylov solver, in which case the integration is halted.

Notes:

If the user’s ARKLsJacTimesVecFn function uses difference quotient approximations, it may need to access quantities not in the argument list. These include the current step size, the error weights, etc. To obtain these, the user will need to add a pointer to the ark_mem structure to their user_data, and then use the ARKStepGet* or MRIStepGet* functions listed in §3.4.2.2.10 or §3.4.4.2.9. The unit roundoff can be accessed as UNIT_ROUNDOFF, which is defined in the header file sundials_types.h.

3.4.5.11. Jacobian-vector product setup (matrix-free linear solvers, ARKStep and MRIStep only)

If the user’s Jacobian-times-vector routine requires that any Jacobian-related data be preprocessed or evaluated, then this needs to be done in a user-supplied function of type ARKLsJacTimesSetupFn, defined as follows:

typedef int (*ARKLsJacTimesSetupFn)(realtype t, N_Vector y, N_Vector fy, void *user_data)

This function preprocesses and/or evaluates any Jacobian-related data needed by the Jacobian-times-vector routine.

Arguments:
• t – the current value of the independent variable.

• y – the current value of the dependent variable vector.

• fy – the current value of the vector $$f^I(t,y)$$.

• user_data – a pointer to user data, the same as the user_data parameter that was passed to ARKStepSetUserData() or MRIStepSetUserData().

Return value:

The value to be returned by the Jacobian-vector setup function should be 0 if successful, positive for a recoverable error (in which case the step will be retried), or negative for an unrecoverable error (in which case the integration is halted).

Notes:

Each call to the Jacobian-vector setup function is preceded by a call to the implicit ARKRhsFn user function with the same $$(t,y)$$ arguments. Thus, the setup function can use any auxiliary data that is computed and saved during the evaluation of the implicit ODE right-hand side.

If the user’s ARKLsJacTimesSetupFn function uses difference quotient approximations, it may need to access quantities not in the argument list. These include the current step size, the error weights, etc. To obtain these, the user will need to add a pointer to the ark_mem structure to their user_data, and then use the ARKStepGet* or MRIStepGet* functions listed in §3.4.2.2.10 or §3.4.4.2.9. The unit roundoff can be accessed as UNIT_ROUNDOFF, which is defined in the header file sundials_types.h.

3.4.5.12. Preconditioner solve (iterative linear solvers, ARKStep and MRIStep only)

If a user-supplied preconditioner is to be used with a SUNLinSol solver module, then the user must provide a function of type ARKLsPrecSolveFn to solve the linear system $$Pz=r$$, where $$P$$ corresponds to either a left or right preconditioning matrix. Here $$P$$ should approximate (at least crudely) the Newton matrix $$\mathcal{A}(t,y)=M(t)-\gamma J(t,y)$$, where $$M(t)$$ is the mass matrix and $$J(t,y) = \dfrac{\partial f^I}{\partial y}(t,y)$$ If preconditioning is done on both sides, the product of the two preconditioner matrices should approximate $$\mathcal{A}$$.

typedef int (*ARKLsPrecSolveFn)(realtype t, N_Vector y, N_Vector fy, N_Vector r, N_Vector z, realtype gamma, realtype delta, int lr, void *user_data)

This function solves the preconditioner system $$Pz=r$$.

Arguments:
• t – the current value of the independent variable.

• y – the current value of the dependent variable vector.

• fy – the current value of the vector $$f^I(t,y)$$.

• r – the right-hand side vector of the linear system.

• z – the computed output solution vector.

• gamma – the scalar $$\gamma$$ appearing in the Newton matrix given by $$\mathcal{A}=M(t)-\gamma J(t,y)$$.

• delta – an input tolerance to be used if an iterative method is employed in the solution. In that case, the residual vector $$Res = r-Pz$$ of the system should be made to be less than delta in the weighted $$l_2$$ norm, i.e. $$\left(\displaystyle \sum_{i=1}^n \left(Res_i * ewt_i\right)^2 \right)^{1/2} < \delta$$, where $$\delta =$$ delta. To obtain the N_Vector ewt, call ARKStepGetErrWeights() or MRIStepGetErrWeights().

• lr – an input flag indicating whether the preconditioner solve is to use the left preconditioner (lr = 1) or the right preconditioner (lr = 2).

• user_data – a pointer to user data, the same as the user_data parameter that was passed to ARKStepSetUserData() or MRIStepSetUserData().

Return value:

The value to be returned by the preconditioner solve function is a flag indicating whether it was successful. This value should be 0 if successful, positive for a recoverable error (in which case the step will be retried), or negative for an unrecoverable error (in which case the integration is halted).

3.4.5.13. Preconditioner setup (iterative linear solvers, ARKStep and MRIStep only)

If the user’s preconditioner routine requires that any data be preprocessed or evaluated, then these actions need to occur within a user-supplied function of type ARKLsPrecSetupFn.

typedef int (*ARKLsPrecSetupFn)(realtype t, N_Vector y, N_Vector fy, booleantype jok, booleantype *jcurPtr, realtype gamma, void *user_data)

This function preprocesses and/or evaluates Jacobian-related data needed by the preconditioner.

Arguments:
• t – the current value of the independent variable.

• y – the current value of the dependent variable vector.

• fy – the current value of the vector $$f^I(t,y)$$.

• jok – is an input flag indicating whether the Jacobian-related data needs to be updated. The jok argument provides for the reuse of Jacobian data in the preconditioner solve function. When jok = SUNFALSE, the Jacobian-related data should be recomputed from scratch. When jok = SUNTRUE the Jacobian data, if saved from the previous call to this function, can be reused (with the current value of gamma). A call with jok = SUNTRUE can only occur after a call with jok = SUNFALSE.

• jcurPtr – is a pointer to a flag which should be set to SUNTRUE if Jacobian data was recomputed, or set to SUNFALSE if Jacobian data was not recomputed, but saved data was still reused.

• gamma – the scalar $$\gamma$$ appearing in the Newton matrix given by $$\mathcal{A}=M(t)-\gamma J(t,y)$$.

• user_data – a pointer to user data, the same as the user_data parameter that was passed to ARKStepSetUserData() or MRIStepSetUserData().

Return value:

The value to be returned by the preconditioner setup function is a flag indicating whether it was successful. This value should be 0 if successful, positive for a recoverable error (in which case the step will be retried), or negative for an unrecoverable error (in which case the integration is halted).

Notes:

The operations performed by this function might include forming a crude approximate Jacobian, and performing an LU factorization of the resulting approximation to $$\mathcal{A} = M(t) - \gamma J(t,y)$$.

With the default nonlinear solver (the native SUNDIALS Newton method), each call to the preconditioner setup function is preceded by a call to the implicit ARKRhsFn user function with the same $$(t,y)$$ arguments. Thus, the preconditioner setup function can use any auxiliary data that is computed and saved during the evaluation of the implicit ODE right-hand side. In the case of a user-supplied or external nonlinear solver, this is also true if the nonlinear system function is evaluated prior to calling the linear solver setup function (see §12.1.4 for more information).

This function is not called in advance of every call to the preconditioner solve function, but rather is called only as often as needed to achieve convergence in the Newton iteration.

If the user’s ARKLsPrecSetupFn function uses difference quotient approximations, it may need to access quantities not in the call list. These include the current step size, the error weights, etc. To obtain these, the user will need to add a pointer to the ark_mem structure to their user_data, and then use the ARKStepGet* or MRIStepGet* functions listed in §3.4.2.2.10 or §3.4.4.2.9. The unit roundoff can be accessed as UNIT_ROUNDOFF, which is defined in the header file sundials_types.h.

3.4.5.14. Mass matrix construction (matrix-based linear solvers, ARKStep only)

If a matrix-based mass-matrix linear solver is used (i.e., a non-NULL SUNMATRIX was supplied to ARKStepSetMassLinearSolver(), the user must provide a function of type ARKLsMassFn to provide the mass matrix approximation.

typedef int (*ARKLsMassFn)(realtype t, SUNMatrix M, void *user_data, N_Vector tmp1, N_Vector tmp2, N_Vector tmp3)

This function computes the mass matrix $$M(t)$$ (or an approximation to it).

Arguments:
• t – the current value of the independent variable.

• M – the output mass matrix.

• user_data – a pointer to user data, the same as the user_data parameter that was passed to ARKStepSetUserData().

• tmp1, tmp2, tmp3 – pointers to memory allocated to variables of type N_Vector which can be used by an ARKLsMassFn as temporary storage or work space.

Return value:

An ARKLsMassFn function should return 0 if successful, or a negative value if it failed unrecoverably (in which case the integration is halted, ARKStepEvolve() returns ARK_MASSSETUP_FAIL and ARKLS sets last_flag to ARKLS_MASSFUNC_UNRECVR).

Notes:

Information regarding the structure of the specific SUNMatrix structure (e.g.~number of rows, upper/lower bandwidth, sparsity type) may be obtained through using the implementation-specific SUNMatrix interface functions (see §10 for details).

Prior to calling the user-supplied mass matrix function, the mass matrix $$M(t)$$ is zeroed out, so only nonzero elements need to be loaded into M.

dense $$M(t)$$: A user-supplied dense mass matrix function must load the N by N dense matrix M with an approximation to the mass matrix $$M(t)$$. Utility routines and accessor macros for the SUNMATRIX_DENSE module are documented in §10.9.

banded $$M(t)$$: A user-supplied banded mass matrix function must load the band matrix M with the elements of the mass matrix $$M(t)$$. Utility routines and accessor macros for the SUNMATRIX_BAND module are documented in §10.12.

sparse $$M(t)$$: A user-supplied sparse mass matrix function must load the compressed-sparse-column (CSR) or compressed-sparse-row (CSR) matrix M with an approximation to the mass matrix $$M(t)$$. Storage for M already exists on entry to this function, although the user should ensure that sufficient space is allocated in M to hold the nonzero values to be set; if the existing space is insufficient the user may reallocate the data and row index arrays as needed. Utility routines and accessor macros for the SUNMATRIX_SPARSE type are documented in §10.14.

3.4.5.15. Mass matrix-vector product (matrix-free linear solvers, ARKStep only)

If a matrix-free linear solver is to be used for mass-matrix linear systems (i.e., a NULL-valued SUNMATRIX argument was supplied to ARKStepSetMassLinearSolver() in §3.4.2.1), the user must provide a function of type ARKLsMassTimesVecFn in the following form, to compute matrix-vector products $$M(t)\, v$$.

typedef int (*ARKLsMassTimesVecFn)(N_Vector v, N_Vector Mv, realtype t, void *mtimes_data)

This function computes the product $$M(t)\, v$$ (or an approximation to it).

Arguments:
• v – the vector to multiply.

• Mv – the output vector computed.

• t – the current value of the independent variable.

• mtimes_data – a pointer to user data, the same as the mtimes_data parameter that was passed to ARKStepSetMassTimes().

Return value:

The value to be returned by the mass-matrix-vector product function should be 0 if successful. Any other return value will result in an unrecoverable error of the generic Krylov solver, in which case the integration is halted.

3.4.5.16. Mass matrix-vector product setup (matrix-free linear solvers, ARKStep only)

If the user’s mass-matrix-times-vector routine requires that any mass matrix-related data be preprocessed or evaluated, then this needs to be done in a user-supplied function of type ARKLsMassTimesSetupFn, defined as follows:

typedef int (*ARKLsMassTimesSetupFn)(realtype t, void *mtimes_data)

This function preprocesses and/or evaluates any mass-matrix-related data needed by the mass-matrix-times-vector routine.

Arguments:
• t – the current value of the independent variable.

• mtimes_data – a pointer to user data, the same as the mtimes_data parameter that was passed to ARKStepSetMassTimes().

Return value:

The value to be returned by the mass-matrix-vector setup function should be 0 if successful. Any other return value will result in an unrecoverable error of the ARKLS mass matrix solver interface, in which case the integration is halted.

3.4.5.17. Mass matrix preconditioner solve (iterative linear solvers, ARKStep only)

If a user-supplied preconditioner is to be used with a SUNLINEAR solver module for mass matrix linear systems, then the user must provide a function of type ARKLsMassPrecSolveFn to solve the linear system $$Pz=r$$, where $$P$$ may be either a left or right preconditioning matrix. Here $$P$$ should approximate (at least crudely) the mass matrix $$M(t)$$. If preconditioning is done on both sides, the product of the two preconditioner matrices should approximate $$M(t)$$.

typedef int (*ARKLsMassPrecSolveFn)(realtype t, N_Vector r, N_Vector z, realtype delta, int lr, void *user_data)

This function solves the preconditioner system $$Pz=r$$.

Arguments:
• t – the current value of the independent variable.

• r – the right-hand side vector of the linear system.

• z – the computed output solution vector.

• delta – an input tolerance to be used if an iterative method is employed in the solution. In that case, the residual vector $$Res = r-Pz$$ of the system should be made to be less than delta in the weighted $$l_2$$ norm, i.e. $$\left(\displaystyle \sum_{i=1}^n \left(Res_i * ewt_i\right)^2 \right)^{1/2} < \delta$$, where $$\delta =$$ delta. To obtain the N_Vector ewt, call ARKStepGetErrWeights().

• lr – an input flag indicating whether the preconditioner solve is to use the left preconditioner (lr = 1) or the right preconditioner (lr = 2).

• user_data – a pointer to user data, the same as the user_data parameter that was passed to ARKStepSetUserData().

Return value:

The value to be returned by the preconditioner solve function is a flag indicating whether it was successful. This value should be 0 if successful, positive for a recoverable error (in which case the step will be retried), or negative for an unrecoverable error (in which case the integration is halted).

3.4.5.18. Mass matrix preconditioner setup (iterative linear solvers, ARKStep only)

If the user’s mass matrix preconditioner above requires that any problem data be preprocessed or evaluated, then these actions need to occur within a user-supplied function of type ARKLsMassPrecSetupFn.

typedef int (*ARKLsMassPrecSetupFn)(realtype t, void *user_data)

This function preprocesses and/or evaluates mass-matrix-related data needed by the preconditioner.

Arguments:
• t – the current value of the independent variable.

• user_data – a pointer to user data, the same as the user_data parameter that was passed to ARKStepSetUserData().

Return value:

The value to be returned by the mass matrix preconditioner setup function is a flag indicating whether it was successful. This value should be 0 if successful, positive for a recoverable error (in which case the step will be retried), or negative for an unrecoverable error (in which case the integration is halted).

Notes:

The operations performed by this function might include forming a mass matrix and performing an incomplete factorization of the result. Although such operations would typically be performed only once at the beginning of a simulation, these may be required if the mass matrix can change as a function of time.

If both this function and a ARKLsMassTimesSetupFn are supplied, all calls to this function will be preceded by a call to the ARKLsMassTimesSetupFn, so any setup performed there may be reused.

3.4.5.19. Vector resize function

For simulations involving changes to the number of equations and unknowns in the ODE system (e.g. when using spatial adaptivity in a PDE simulation), the ARKODE integrator may be “resized” between integration steps, through calls to the ARKStepResize(), ERKStepResize(), or MRIStepResize() function. Typically, when performing adaptive simulations the solution is stored in a customized user-supplied data structure, to enable adaptivity without repeated allocation/deallocation of memory. In these scenarios, it is recommended that the user supply a customized vector kernel to interface between SUNDIALS and their problem-specific data structure. If this vector kernel includes a function of type ARKVecResizeFn to resize a given vector implementation, then this function may be supplied to ARKStepResize(), ERKStepResize(), or MRIStepResize(), so that all internal ARKODE vectors may be resized, instead of deleting and re-creating them at each call. This resize function should have the following form:

typedef int (*ARKVecResizeFn)(N_Vector y, N_Vector ytemplate, void *user_data)

This function resizes the vector y to match the dimensions of the supplied vector, ytemplate.

Arguments:
Return value:

An ARKVecResizeFn function should return 0 if it successfully resizes the vector y, and a non-zero value otherwise.

Notes:

If this function is not supplied, then ARKODE will instead destroy the vector y and clone a new vector y off of ytemplate.

3.4.5.20. Pre inner integrator communication function (MRIStep only)

The user may supply a function of type MRIStepPreInnerFn that will be called before each inner integration to perform any communication or memory transfers of forcing data supplied by the outer integrator to the inner integrator for the inner integration.

typedef int (*MRIStepPreInnerFn)(realtype t, N_Vector *f, int num_vecs, void *user_data)
Arguments:
• t – the current value of the independent variable.

• f – an N_Vector array of outer forcing vectors.

• num_vecs – the number of vectors in the N_Vector array.

• user_data – the user_data pointer that was passed to MRIStepSetUserData().

Return value:

An MRIStepPreInnerFn function should return 0 if successful, a positive value if a recoverable error occurred, or a negative value if an unrecoverable error occurred. As the MRIStep module only supports fixed step sizes at this time any non-zero return value will halt the integration.

Notes:

In a heterogeneous computing environment if any data copies between the host and device vector data are necessary, this is where that should occur.

3.4.5.21. Post inner integrator communication function (MRIStep only)

The user may supply a function of type MRIStepPostInnerFn that will be called after each inner integration to perform any communication or memory transfers of state data supplied by the inner integrator to the outer integrator for the outer integration.

typedef int (*MRIStepPostInnerFn)(realtype t, N_Vector y, void *user_data)
Arguments:
• t – the current value of the independent variable.

• y – the current value of the dependent variable vector.

• user_data – the user_data pointer that was passed to MRIStepSetUserData().

Return value:

An MRIStepPostInnerFn() function should return 0 if successful, a positive value if a recoverable error occurred, or a negative value if an unrecoverable error occurred. As the MRIStep module only supports fixed step sizes at this time any non-zero return value will halt the integration.

Notes:

In a heterogeneous computing environment if any data copies between the host and device vector data are necessary, this is where that should occur.