# User-supplied functions¶

The user-supplied functions for ARKStep 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),
• a function that handles adaptive time step error control (optional),
• a function that handles explicit time step stability (optional),
• 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 Newton-based nonlinear iteration is chosen (optional),
• one or two functions that define the preconditioner for use in any of the Krylov iterative algorithms, if a Newton-based nonlinear iteration and iterative linear solver are chosen (optional), and
• if the problem involves a non-identity mass matrix $$M\ne I$$:
• 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 in 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() (optional).

## 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:

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 $$f_E(t,y) + f_I(t,y)$$.
• user_data – the user_data pointer that was passed to ARKStepSetUserData().

Return value: An ARKRhsFn should return 0 if successful, a positive value if a recoverable error occurred (in which case ARKStep 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 the ARKStep module. 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, ARKStep will attempt to recover (possibly repeating the nonlinear iteration, or reducing the step size) 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 ARKStep returns ARK_FIRST_RHSFUNC_ERR). Another is when a recoverable error is reported by ARKRhsFn after the integrator completes a successful stage, in which case ARKStep returns ARK_UNREC_RHSFUNC_ERR).

## 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()), 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 ARKStep and its sub-modules.

Arguments:
• error_code – the error code.
• module – the name of the ARKStep 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().

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.

## 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 $$\|v\|_{WRMS} = \left(\frac{1}{n} \sum_{i=1}^n \left(ewt_i\; v_i\right)^2 \right)^{1/2}$$. These weights will be used in place of those defined in the section Error norms.

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().

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 ARKStep.

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.

## Residual weight function¶

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 $$\|v\|_{WRMS} = \left(\frac{1}{n} \sum_{i=1}^n \left(rwt_i\; v_i\right)^2 \right)^{1/2}$$. These weights will be used in place of those defined in the section Error norms.

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.

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 as the maximum value such that the error estimates remain 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 $$h$$ satisfying 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().

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

## Explicit stability function¶

A user may supply a function to predict the maximum stable step size for the explicit portion of the ImEx system, $$f_E(t,y)$$. 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 $$f_E(t,y)$$ contains moderately stiff terms. In this scenario, a user may provide a function of type ARKExpStabFn to provide this stability information to ARKStep. 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 portions of the ImEx 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().

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 ARKStep will assume that there is no explicit stability restriction on the time step size.

## Rootfinding function¶

If a rootfinding problem is to be solved during the 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 the roots of the nrtfn components $$g_i(t,y)$$ are sought.

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().

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 ARKStep returns ARK_RTFUNC_FAIL).

Notes: Allocation of memory for gout is handled within ARKStep.

## Jacobian construction (matrix-based linear solvers)¶

If a matrix-based linear solver module is used (i.e., a non-NULL SUNMatrix object was supplied to ARKStepSetLinearSolver() in section A skeleton of the user’s main program), the user may provide a function of type ARKLsJacFn to provide the Jacobian approximation.

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 $$J = \frac{\partial f_I}{\partial y}$$ (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().
• 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 ARKStep 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() returns ARK_LSETUP_FAIL and ARKLS sets last_flag to ARKLS_JACFUNC_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 the section Matrix Data Structures for details).

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.

If the user’s ARKLsJacFn function uses difference quotient approximations, then 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* functions listed in Optional output functions. The unit roundoff can be accessed as UNIT_ROUNDOFF, which is defined in the header file sundials_types.h.

dense:

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 $$(t,y)$$. The accessor macros SM_ELEMENT_D and SM_COLUMN_D allow the user to read and write dense matrix elements without making explicit references to the underlying representation of the SUNMATRIX_DENSE type. SM_ELEMENT_D(J, i, j) references the (i,j)-th element of the dense matrix J (for i, j between 0 and N-1). This macro is meant for small problems for which efficiency of access is not a major concern. Thus, in terms of the indices $$m$$ and $$n$$ ranging from 1 to N, the Jacobian element $$J_{m,n}$$ can be set using the statement SM_ELEMENT_D(J, m-1, n-1) = $$J_{m,n}$$. Alternatively, SM_COLUMN_D(J, j) returns a pointer to the first element of the j-th column of J (for j ranging from 0 to N-1), and the elements of the j-th column can then be accessed using ordinary array indexing. Consequently, $$J_{m,n}$$ can be loaded using the statements col_n = SM_COLUMN_D(J, n-1); col_n[m-1] = $$J_{m,n}$$. For large problems, it is more efficient to use SM_COLUMN_D than to use SM_ELEMENT_D. Note that both of these macros number rows and columns starting from 0. The SUNMATRIX_DENSE type and accessor macros are documented in section The SUNMATRIX_DENSE Module.

band:

A user-supplied banded Jacobian function must load the band matrix Jac with the elements of the Jacobian $$J(t,y)$$ at the point $$(t,y)$$. The accessor macros SM_ELEMENT_B, SM_COLUMN_B, and SM_COLUMN_ELEMENT_B allow the user to read and write band matrix elements without making specific references to the underlying representation of the SUNMATRIX_BAND type. SM_ELEMENT_B(J, i, j) references the (i,j)-th element of the band matrix J, counting from 0. This macro is meant for use in small problems for which efficiency of access is not a major concern. Thus, in terms of the indices $$m$$ and $$n$$ ranging from 1 to N with $$(m, n)$$ within the band defined by mupper and mlower, the Jacobian element $$J_{m,n}$$ can be loaded using the statement SM_ELEMENT_B(J, m-1, n-1) $$= J_{m,n}$$. The elements within the band are those with -mupper $$\le m-n \le$$ mlower. Alternatively, SM_COLUMN_B(J, j) returns a pointer to the diagonal element of the j-th column of J, and if we assign this address to realtype *col_j, then the i-th element of the j-th column is given by SM_COLUMN_ELEMENT_B(col_j, i, j), counting from 0. Thus, for $$(m,n)$$ within the band, $$J_{m,n}$$ can be loaded by setting col_n = SM_COLUMN_B(J, n-1); SM_COLUMN_ELEMENT_B(col_n, m-1, n-1) $$= J_{m,n}$$ . The elements of the j-th column can also be accessed via ordinary array indexing, but this approach requires knowledge of the underlying storage for a band matrix of type SUNMATRIX_BAND. The array col_n can be indexed from -mupper to mlower. For large problems, it is more efficient to use SM_COLUMN_B and SM_COLUMN_ELEMENT_B than to use the SM_ELEMENT_B macro. As in the dense case, these macros all number rows and columns starting from 0. The SUNMATRIX_BAND type and accessor macros are documented in section The SUNMATRIX_BAND Module.

sparse:

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. The amount of allocated space in a SUNMATRIX_SPARSE object may be accessed using the macro SM_NNZ_S or the routine SUNSparseMatrix_NNZ(). The SUNMATRIX_SPARSE type is further documented in the section The SUNMATRIX_SPARSE Module.

## Jacobian-vector product (matrix-free linear solvers)¶

When using a matrix-free linear solver modules for the implicit stage solves (i.e., a NULL-valued SUNMATRIX argument was supplied to ARKStepSetLinearSolver() in the section A skeleton of the user’s main program), 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 $$Jv = \left(\frac{\partial f_I}{\partial y}\right)v$$ (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().
• 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* functions listed in Optional output functions. The unit roundoff can be accessed as UNIT_ROUNDOFF, which is defined in the header file sundials_types.h.

## Jacobian-vector product setup (matrix-free linear solvers)¶

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().

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* functions listed in Optional output functions. The unit roundoff can be accessed as UNIT_ROUNDOFF, which is defined in the header file sundials_types.h.

## Preconditioner solve (iterative linear solvers)¶

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 $$A=M-\gamma J$$, where $$M$$ is the mass matrix (typically $$M=I$$ unless working in a finite-element setting) and $$J = \frac{\partial f_I}{\partial y}$$ If preconditioning is done on both sides, the product of the two preconditioner matrices should approximate $$A$$.

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 $$A=M-\gamma J$$.
• 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. $$\left(\sum_{i=1}^n \left(Res_i * ewt_i\right)^2 \right)^{1/2} < \delta$$, 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).

## Preconditioner setup (iterative linear solvers)¶

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 $$A=M-\gamma J$$.
• 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 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 $$A = M - \gamma J$$.

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 ODE right-hand side.

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* functions listed in Optional output functions. The unit roundoff can be accessed as UNIT_ROUNDOFF, which is defined in the header file sundials_types.h.

## Mass matrix construction (matrix-based linear solvers)¶

If a matrix-based mass-matrix linear solver is used (i.e., a non-NULL SUNMATRIX was supplied to ARKStepSetMassLinearSolver() in the section A skeleton of the user’s main program), 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 $$M$$ (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 the section Matrix Data Structures for details).

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

dense:

A user-supplied dense mass matrix function must load the N by N dense matrix M with an approximation to the mass matrix $$M$$. As discussed above in section Jacobian construction (matrix-based linear solvers), the accessor macros SM_ELEMENT_D and SM_COLUMN_D allow the user to read and write dense matrix elements without making explicit references to the underlying representation of the SUNMATRIX_DENSE type. Similarly, the SUNMATRIX_DENSE type and accessor macros SM_ELEMENT_D and SM_COLUMN_D are documented in the section The SUNMATRIX_DENSE Module.

band:

A user-supplied banded mass matrix function must load the band matrix M with the elements of the mass matrix $$M$$. As discussed above in section Jacobian construction (matrix-based linear solvers), the accessor macros SM_ELEMENT_B, SM_COLUMN_B, and SM_COLUMN_ELEMENT_B allow the user to read and write band matrix elements without making specific references to the underlying representation of the SUNMATRIX_BAND type. Similarly, the SUNMATRIX_BAND type and the accessor macros SM_ELEMENT_B, SM_COLUMN_B, and SM_COLUMN_ELEMENT_B are documented in the section The SUNMATRIX_BAND Module.

sparse:

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 $$M$$. 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. The type of M is SUNMATRIX_SPARSE, and the amount of allocated space in a SUNMATRIX_SPARSE object may be accessed using the macro SM_NNZ_S or the routine SUNSparseMatrix_NNZ(). The SUNMATRIX_SPARSE type is further documented in the section The SUNMATRIX_SPARSE Module.

## Mass matrix-vector product (matrix-free linear solvers)¶

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 the section A skeleton of the user’s main program), the user must provide a function of type ARKLsMassTimesVecFn in the following form, to compute matrix-vector products $$Mv$$.

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

This function computes the product $$M*v$$ (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.

## Mass matrix-vector product setup (matrix-free linear solvers)¶

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.

## Mass matrix preconditioner solve (iterative linear solvers)¶

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 $$M$$. If preconditioning is done on both sides, the product of the two preconditioner matrices should approximate $$M$$.

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. $$\left(\sum_{i=1}^n \left(Res_i * ewt_i\right)^2 \right)^{1/2} < \delta$$, 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).

## Mass matrix preconditioner setup (iterative linear solvers)¶

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.

## 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 ARKStep integrator may be “resized” between integration steps, through calls to the ARKStepResize() 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() so that all internal ARKStep 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:
• y – the vector to resize.
• ytemplate – a vector of the desired size.
• user_data – a pointer to user data, the same as the resize_data parameter that was passed to ARKStepResize().

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 ARKStep will instead destroy the vector y and clone a new vector y off of ytemplate.