Reference documentation for deal.II version 9.1.1

#include <deal.II/sundials/arkode.h>
Classes  
class  AdditionalData 
Public Member Functions  
ARKode (const AdditionalData &data=AdditionalData(), const MPI_Comm mpi_comm=MPI_COMM_WORLD)  
~ARKode ()  
unsigned int  solve_ode (VectorType &solution) 
void  reset (const double t, const double h, const VectorType &y) 
DeclException1 (ExcARKodeError, int,<< "One of the SUNDIALS ARKode internal functions "<< " returned a negative error code: "<< arg1<< ". Please consult SUNDIALS manual.")  
Public Attributes  
std::function< void(VectorType &)>  reinit_vector 
std::function< int(const double t, const VectorType &y, VectorType &explicit_f)>  explicit_function 
std::function< int(const double t, const VectorType &y, VectorType &res)>  implicit_function 
std::function< int(const int convfail, const double t, const double gamma, const VectorType &ypred, const VectorType &fpred, bool &j_is_current)>  setup_jacobian 
std::function< int(const double t, const double gamma, const VectorType &ycur, const VectorType &fcur, const VectorType &rhs, VectorType &dst)>  solve_jacobian_system 
std::function< int(const double t)>  setup_mass 
std::function< int(const VectorType &rhs, VectorType &dst)>  solve_mass_system 
std::function< void(const double t, const VectorType &sol, const unsigned int step_number)>  output_step 
std::function< bool(const double t, VectorType &sol)>  solver_should_restart 
std::function< VectorType &()>  get_local_tolerances 
Private Member Functions  
DeclException1 (ExcFunctionNotProvided, std::string,<< "Please provide an implementation for the function << arg1<< "\")  
void  set_functions_to_trigger_an_assert () 
Private Attributes  
AdditionalData  data 
void *  arkode_mem 
N_Vector  yy 
N_Vector  abs_tolls 
MPI_Comm  communicator 
GrowingVectorMemory< VectorType >  mem 
Interface to SUNDIALS additive RungeKutta methods (ARKode).
The class ARKode is a wrapper to SUNDIALS variablestep, embedded, additive RungeKutta solver which is a general purpose solver for systems of ordinary differential equations characterized by the presence of both fast and slow dynamics.
Fast dynamics are treated implicitly, and slow dynamics are treated explicitly, using nested families of implicit and explicit RungeKutta solvers.
Citing directly from ARKode documentation:
ARKode solves ODE initial value problems (IVPs) in \(R^N\). These problems should be posed in explicit form as
\[ M\dot y = f_E(t, y) + f_I (t, y), \qquad y(t_0) = y_0. \]
Here, \(t\) is the independent variable (e.g. time), and the dependent variables are given by \(y \in R^N\), and we use notation \(\dot y\) to denote \(dy/dt\). \(M\) is a usersupplied nonsingular operator from \(R^N \to R^N\). This operator may depend on \(t\) but not on \(y\).
For standard systems of ordinary differential equations and for problems arising from the spatial semidiscretization of partial differential equations using finite difference or finite volume methods, \(M\) is typically the identity matrix, \(I\). For PDEs using a finiteelement spatial semidiscretization \(M\) is typically a wellconditioned mass matrix.
The two righthand side functions may be described as:
ARKode may be used to solve stiff, nonstiff and multirate problems. Roughly speaking, stiffness is characterized by the presence of at least one rapidly damped mode, whose time constant is small compared to the time scale of the solution itself. In the implicit/explicit (ImEx) splitting above, these stiff components should be included in the righthand side function \(f_I (t, y)\).
For multirate problems, a user should provide both of the functions \(f_E\) and \(f_I\) that define the IVP system.
For nonstiff problems, only \(f_E\) should be provided, and \(f_I\) is assumed to be zero, i.e. the system reduces to the nonsplit IVP:
\[ M\dot y = f_E(t, y), \qquad y(t_0) = y_0. \]
In this scenario, the ARK methods reduce to classical explicit RungeKutta methods (ERK). For these classes of methods, ARKode allows orders of accuracy \(q = \{2, 3, 4, 5, 6, 8\}\), with embeddings of orders \(p = \{1, 2, 3, 4, 5, 7\}\). These default to the HeunEuler212, BogackiShampine423, Zonneveld534, CashKarp645, Verner856 and Fehlberg1378 methods, respectively.
Finally, for stiff (linear or nonlinear) problems the user may provide only \(f_I\), implying that \(f_E = 0\), so that the system reduces to the nonsplit IVP
\[ M\dot y = f_I(t, y), \qquad y(t_0) = y_0. \]
Similarly to ERK methods, in this scenario the ARK methods reduce to classical diagonallyimplicit RungeKutta methods (DIRK). For these classes of methods, ARKode allows orders of accuracy \(q = \{2, 3, 4, 5\}\), with embeddings of orders \(p = \{1, 2, 3, 4\}\). These default to the SDIRK212, ARK423 (implicit), SDIRK534 and ARK845 (implicit) methods, respectively.
For both DIRK and ARK methods, an implicit system of the form
\[ G(z_i) \dealcoloneq M z_i − h_n A^I_{i,i} f_I (t^I_{n,i}, z_i) − a_i = 0 \]
must be solved for each stage \(z_i , i = 1, \ldots, s\), where we have the data
\[ a_i \dealcoloneq M y_{n−1} + h_n \sum_{j=1}^{i−1} [ A^E_{i,j} f_E(t^E_{n,j}, z_j) + A^I_{i,j} f_I (t^I_{n,j}, z_j)] \]
for the ARK methods, or
\[ a_i \dealcoloneq M y_{n−1} + h_n \sum_{j=1}^{i−1} A^I_{i,j} f_I (t^I_{n,j}, z_j) \]
for the DIRK methods. Here \(A^I_{i,j}\) and \(A^E_{i,j}\) are the Butcher's tables for the chosen solver.
If \(f_I(t,y)\) depends nonlinearly on \(y\) then the systems above correspond to a nonlinear system of equations; if \(f_I (t, y)\) depends linearly on \(y\) then this is a linear system of equations. By specifying the flag implicit_function_is_linear
, ARKode takes some shortcuts that allow a faster solution process.
For systems of either type, ARKode allows a choice of solution strategy. The default solver choice is a variant of Newton’s method,
\[ z_i^{m+1} = z_i^m +\delta^{m+1}, \]
where \(m\) is the Newton index, and the Newton update \(\delta^{m+1}\) requires the solution of the linear Newton system
\[ N(z_i^m) \delta^{m+1} = G(z_i^m), \]
where
\[ N \dealcoloneq M  \gamma J, \quad J \dealcoloneq \frac{\partial f_I}{\partial y}, \qquad \gamma\dealcoloneq h_n A^I_{i,i}. \]
As an alternate to Newton’s method, ARKode may solve for each stage \(z_i ,i = 1, \ldots , s\) using an Andersonaccelerated fixed point iteration
\[ z_i^{m+1} = g(z_i^{m}), m=0,1,\ldots. \]
Unlike with Newton’s method, this option does not require the solution of a linear system at each iteration, instead opting for solution of a lowdimensional leastsquares solution to construct the nonlinear update.
Finally, if the user specifies implicit_function_is_linear
, i.e., \(f_I(t, y)\) depends linearly on \(y\), and if the Newtonbased nonlinear solver is chosen, then the system will be solved using only a single Newton iteration. Notice that in order for the Newton solver to be used, at least the solve_jacobian_system() function should be supplied. If this function is not supplied, then only the fixedpoint iteration will be supported, and the implicit_function_is_linear
setting is ignored.
The optimal solver (Newton vs fixedpoint) is highly problemdependent. Since fixedpoint solvers do not require the solution of any linear systems, each iteration may be significantly less costly than their Newton counterparts. However, this can come at the cost of slower convergence (or even divergence) in comparison with Newtonlike methods. These fixedpoint solvers do allow for user specification of the Andersonaccelerated subspace size, \(m_k\). While the required amount of solver memory grows proportionately to \(m_k N\), larger values of \(m_k\) may result in faster convergence.
This improvement may be significant even for "small" values, e.g. \(1 \leq m_k \leq 5\), and convergence may not improve (or even deteriorate) for larger values of \(m_k\). While ARKode uses a Newtonbased iteration as its default solver due to its increased robustness on very stiff problems, it is highly recommended that users also consider the fixedpoint solver for their cases when attempting a new problem.
For either the Newton or fixedpoint solvers, it is wellknown that both the efficiency and robustness of the algorithm intimately depends on the choice of a good initial guess. In ARKode, the initial guess for either nonlinear solution method is a predicted value \(z_i(0)\) that is computed explicitly from the previouslycomputed data (e.g. \(y_{n−2}, y_{n−1}\), and \(z_j\) where \(j < i\)). Additional information on the specific predictor algorithms implemented in ARKode is provided in ARKode documentation.
The user has to provide the implementation of the following std::function:
If the mass matrix is different from the identity, the user should supply
If the use of a Newton method is desired, then the user should also supply
Also the following functions could be rewritten. By default they do nothing, or are not required.
To produce output at fixed steps, overload the function
To provide a simple example, consider the harmonic oscillator problem:
\[ \begin{split} u'' & = k^2 u \\ u (0) & = 0 \\ u'(0) & = k \end{split} \]
We write it in terms of a first order ode:
\[ \begin{matrix} y_0' & = y_1 \\ y_1' & =  k^2 y_0 \end{matrix} \]
That is \(y' = A y\) where
\[ A \dealcoloneq \begin{matrix} 0 & 1 \\ k^2 &0 \end{matrix} \]
and \(y(0)=(0, k)\).
The exact solution is \(y_0(t) = \sin(k t)\), \(y_1(t) = y_0'(t) = k \cos(k *t)\), \(y_1'(t) = k^2 \sin(k t)\).
A minimal implementation, using only explicit RK methods, is given by the following code snippet:
SUNDIALS::ARKode< VectorType >::ARKode  (  const AdditionalData &  data = AdditionalData() , 
const MPI_Comm  mpi_comm = MPI_COMM_WORLD 

) 
SUNDIALS::ARKode< VectorType >::~ARKode  (  ) 
unsigned int SUNDIALS::ARKode< VectorType >::solve_ode  (  VectorType &  solution  ) 
void SUNDIALS::ARKode< VectorType >::reset  (  const double  t, 
const double  h,  
const VectorType &  y  
) 
Clear internal memory and start with clean objects. This function is called when the simulation starts and when the user returns true to a call to solver_should_restart().
By default solver_should_restart() returns false. If the user needs to implement, for example, local adaptivity in space, he or she may assign a different function to solver_should_restart() that performs all mesh changes, transfers the solution to the new mesh, and returns true.
[in]  t  The new starting time 
[in]  h  The new starting time step 
[in,out]  y  The new initial solution 
SUNDIALS::ARKode< VectorType >::DeclException1  (  ExcARKodeError  , 
int  ,  
<< "One of the SUNDIALS ARKode< VectorType > internal functions "<< " returned a negative error code: "<< arg1<< ". Please consult SUNDIALS manual."  
) 
Handle ARKode exceptions.

private 
Throw an exception when a function with the given name is not implemented.

private 
std::function<void(VectorType &)> SUNDIALS::ARKode< VectorType >::reinit_vector 
std::function< int(const double t, const VectorType &y, VectorType &explicit_f)> SUNDIALS::ARKode< VectorType >::explicit_function 
A function object that users may supply and that is intended to compute the explicit part of the IVP right hand side. Sets \(explicit_f = f_E(t, y)\).
At least one of explicit_function() or implicit_function() must be provided. According to which one is provided, explicit, implicit, or mixed RK methods are used.
This function should return:
std::function<int(const double t, const VectorType &y, VectorType &res)> SUNDIALS::ARKode< VectorType >::implicit_function 
A function object that users may supply and that is intended to compute the implicit part of the IVP right hand side. Sets \(implicit_f = f_I(t, y)\).
At least one of explicit_function() or implicit_function() must be provided. According to which one is provided, explicit, implicit, or mixed RK methods are used.
This function should return:
std::function<int(const int convfail, const double t, const double gamma, const VectorType &ypred, const VectorType &fpred, bool & j_is_current)> SUNDIALS::ARKode< VectorType >::setup_jacobian 
A function object that users may supply and that is intended to prepare the linear solver for subsequent calls to solve_jacobian_system().
Make sure that after a call to this function, we know how to compute solutions of systems \(A x = b\), where \(A\) is some approximation to the Newton matrix, \(M − \gamma \partial f_I/\partial y\). This function is optional. If the user does not provide it, then solve_jacobian_system() is assumed to also perform the setup internally.
The setup_jacobian() function may call a usersupplied function to compute needed data related to the Jacobian matrix. Alternatively, it may choose to retrieve and use stored values of this data. In either case, setup_jacobian() may also preprocess that data as needed for solve_jacobian_system(), which may involve calling a generic function (such as for LU factorization).
This data may be intended either for direct use (in a direct linear solver) or for use in a preconditioner (in a preconditioned iterative linear solver). The setup_jacobian() function is not called at every stage solve (or even every time step), but only as frequently as the solver determines that it is appropriate to perform the setup task. In this way, Jacobianrelated data generated by setup_jacobian() is expected to be used over a number of time steps.
If the user uses a matrix based computation of the Jacobian, then this is the right place where an assembly routine shoulde be called to assemble both a matrix and a preconditioner for the Jacobian system. Subsequent calls (possibly more than one) to solve_jacobian_system() can assume that this function has been called at least once.
Notice that no assumption is made by this interface on what the user should do in this function. ARKode only assumes that after a call to setup_jacobian() it is possible to call solve_jacobian_system(), to obtain a solution \(x\) to the system \(J x = b\). If this function is not provided, then it is never called.
Arguments to the function are
[in]  t  the current time 
[in]  gamma  the current factor to use in the jacobian computation 
[in]  ypred  is the predicted \(y\) vector for the current ARKode internal step 
[in]  fpred  is the value of the implicit righthand side at ypred, \(f_I (t_n, ypred)\). 
[in]  convfail  – an input flag used to indicate any problem that occurred during the solution of the nonlinear equation on the current time step for which the linear solver is being used. This flag can be used to help decide whether the Jacobian data kept by a linear solver needs to be updated or not. Its possible values are: 
[out]  j_is_current  a boolean to be filled in by setup_jacobian(). The value should be set to true if the Jacobian data is current after the call, and should be set to false if its Jacobian data is not current. If setup_jacobian() calls for reevaluation of Jacobian data (based on convfail and ARKode state data), then it should set j_is_current to true unconditionally, otherwise an infinite loop can result. 
This function should return:
std::function<int(const double t, const double gamma, const VectorType &ycur, const VectorType &fcur, const VectorType &rhs, VectorType & dst)> SUNDIALS::ARKode< VectorType >::solve_jacobian_system 
A function object that users may supply and that is intended to solve the Jacobian linear system. This function will be called by ARKode (possibly several times) after setup_jacobian() has been called at least once. ARKode tries to do its best to call setup_jacobian() the minimum amount of times. If convergence can be achieved without updating the Jacobian, then ARKode does not call setup_jacobian() again. If, on the contrary, internal ARKode convergence tests fail, then ARKode calls again setup_jacobian() with updated vectors and coefficients so that successive calls to solve_jacobian_systems() lead to better convergence in the Newton process.
If you do not specify a solve_jacobian_system() function, then a fixed point iteration is used instead of a Newton method. Notice that this may not converge, or may converge very slowly.
The jacobian \(J\) should be (an approximation of) the system Jacobian
\[ J = M  \gamma \frac{\partial f_I}{\partial y} \]
evaluated at t
, ycur
. fcur
is \(f_I(t,ycur)\).
A call to this function should store in dst
the result of \(J^{1}\) applied to src
, i.e., J*dst = src
. It is the users responsibility to set up proper solvers and preconditioners inside this function.
Arguments to the function are
[in]  t  the current time 
[in]  gamma  the current factor to use in the jacobian computation 
[in]  ycur  is the current \(y\) vector for the current ARKode internal step 
[in]  fcur  is the current value of the implicit righthand side at ycur, \(f_I (t_n, ypred)\). 
This function should return:
std::function<int(const double t)> SUNDIALS::ARKode< VectorType >::setup_mass 
A function object that users may supply and that is intended to setup the mass matrix. This function is called by ARKode any time a mass matrix update is required. The user should compute the mass matrix (or update all the variables that allow the application of the mass matrix). This function is called by ARKode once, before any call to solve_mass_system().
ARKode supports the case where the mass matrix may depend on time, but not the case where the mass matrix depends on the solution itself.
If the user does not provide a solve_mass_matrix() function, then the identity is used. If the setup_mass() function is not provided, then solve_mass_system() should do all the work by itself.
If the user uses a matrix based computation of the mass matrix, then this is the right place where an assembly routine shoulde be called to assemble both a matrix and a preconditioner. Subsequent calls (possibly more than one) to solve_mass_system() can assume that this function has been called at least once.
Notice that no assumption is made by this interface on what the user should do in this function. ARKode only assumes that after a call to setup_mass() it is possible to call solve_mass_system(), to obtain a solution \(x\) to the system \(M x = b\).
This function should return:
std::function<int(const VectorType &rhs, VectorType &dst)> SUNDIALS::ARKode< VectorType >::solve_mass_system 
A function object that users may supply and that is intended to solve the mass matrix linear system. This function will be called by ARKode (possibly several times) after setup_mass() has been called at least once. ARKode tries to do its best to call setup_mass() the minimum amount of times.
A call to this function should store in dst
the result of \(M^{1}\) applied to src
, i.e., M*dst = src
. It is the users responsibility to set up proper solvers and preconditioners inside this function.
This function should return:
std::function<void(const double t, const VectorType & sol, const unsigned int step_number)> SUNDIALS::ARKode< VectorType >::output_step 
A function object that users may supply and that is intended to postprocess the solution. This function is called by ARKode at fixed time increments (every output_period
seconds), and it is passed a polynomial interpolation of the solution, computed using the current ARK order and the (internally stored) previously computed solution steps.
Notice that it is well possible that internally ARKode computes a time step which is much larger than the output_period
step, and therefore calls this function consecutively several times by simply performing all intermediate interpolations. There is no relationship between how many times this function is called and how many time steps have actually been computed.
std::function<bool(const double t, VectorType &sol)> SUNDIALS::ARKode< VectorType >::solver_should_restart 
A function object that users may supply and that is intended to evaluate whether the solver should be restarted (for example because the number of degrees of freedom has changed).
This function is supposed to perform all operations that are necessary in sol
to make sure that the resulting vectors are consistent, and of the correct final size.
For example, one may decide that a local refinement is necessary at time t. This function should then return true, and change the dimension of sol
to reflect the new dimension. Since ARKode does not know about the new dimension, an internal reset is necessary.
The default implementation simply returns false
, i.e., no restart is performed during the evolution.
std::function<VectorType &()> SUNDIALS::ARKode< VectorType >::get_local_tolerances 

private 

private 

private 

private 

private 

private 