Reference documentation for deal.II version Git 8198d0d 20180224 12:45:34 0600

#include <deal.II/sundials/ida.h>
Classes  
class  AdditionalData 
Public Member Functions  
IDA (const AdditionalData &data=AdditionalData(), const MPI_Comm mpi_comm=MPI_COMM_WORLD)  
~IDA ()  
unsigned int  solve_dae (VectorType &solution, VectorType &solution_dot) 
void  reset (const double &t, const double &h, VectorType &y, VectorType &yp) 
DeclException1 (ExcIDAError, int,<< "One of the SUNDIALS IDA 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, const VectorType &y_dot, VectorType &res)>  residual 
std::function< int(const double t, const VectorType &y, const VectorType &y_dot, const double alpha)>  setup_jacobian 
std::function< int(const VectorType &rhs, VectorType &dst)>  solve_jacobian_system 
std::function< void(const double t, const VectorType &sol, const VectorType &sol_dot, const unsigned int step_number)>  output_step 
std::function< bool(const double t, VectorType &sol, VectorType &sol_dot)>  solver_should_restart 
std::function< IndexSet()>  differential_components 
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 *  ida_mem 
N_Vector  yy 
N_Vector  yp 
N_Vector  abs_tolls 
N_Vector  diff_id 
MPI_Comm  communicator 
GrowingVectorMemory< VectorType >  mem 
Interface to SUNDIALS Implicit DifferentialAlgebraic (IDA) solver.
The class IDA is a wrapper to SUNDIALS Implicit DifferentialAlgebraic solver which is a general purpose solver for systems of DifferentialAlgebraic Equations (DAEs).
The user has to provide the implementation of the following std::functions:
Optionally, also the following functions could be rewritten. By default they do nothing, or are not required. If you call the constructor in a way that requires a notimplemented function, an Assertion will be thrown.
To output steps, connect a function to the signal
Citing from the SUNDIALS documentation:
Consider a system of DifferentialAlgebraic Equations written in the general form
\[ \begin{cases} F(t,y,\dot y) = 0\, , \\ y(t_0) = y_0\, , \\ \dot y (t_0) = \dot y_0\, . \end{cases} \]
where \(y,\dot y\) are vectors in \(\R^n\), \(t\) is often the time (but can also be a parametric quantity), and \(F:\R\times\R^n\times\R^n\rightarrow\R^n\). Such problem is solved using Newton iteration augmented with a line search global strategy. The integration method used in IDA is the variableorder, variablecoefficient BDF (Backward Differentiation Formula), in fixedleadingcoefficient. The method order ranges from 1 to 5, with the BDF of order \(q\) given by the multistep formula
\[ \sum_{i=0}^q \alpha_{n,i}\,y_{ni}=h_n\,\dot y_n\, , \label{eq:bdf} \]
where \(y_n\) and \(\dot y_n\) are the computed approximations of \(y(t_n)\) and \(\dot y(t_n)\), respectively, and the step size is \(h_n=t_nt_{n1}\). The coefficients \(\alpha_{n,i}\) are uniquely determined by the order \(q\), and the history of the step sizes. The application of the BDF method to the DAE system results in a nonlinear algebraic system to be solved at each time step:
\[ G(y_n)\equiv F\left(t_n,y_n,\dfrac{1}{h_n}\sum_{i=0}^q \alpha_{n,i}\,y_{ni}\right)=0\, . \]
The Newton method leads to a linear system of the form
\[ J[y_{n(m+1)}y_{n(m)}]=G(y_{n(m)})\, , \]
where \(y_{n(m)}\) is the \(m\)th approximation to \(y_n\), and \(J\) is the approximation of the system Jacobian
\[ J=\dfrac{\partial G}{\partial y} = \dfrac{\partial F}{\partial y} + \alpha \dfrac{\partial F}{\partial \dot y}\, , \]
and \(\alpha = \alpha_{n,0}/h_n\). It is worth mentioning that the scalar \(\alpha\) changes whenever the step size or method order changes.
To provide a simple example, consider the following 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:
\[ \begin{matrix} y_0' & y_1 & = 0 \\ y_1' & + k^2 y_0 & = 0 \end{matrix} \]
That is \(F(y', y, t) = y' + A y = 0 \) where
\[ \begin{matrix} 0 & 1 \\ k^2 &0 \end{matrix} \]
and \(y(0)=(0, k)\), \(y'(0) = (k, 0)\).
The exact solution is \(y_0(t) = \sin(k t)\), \(y_1(t) = y_0'(t) = k \cos(k t)\), \(y_1'(t) = k^2 \sin(k t)\).
The Jacobian to assemble is the following: \(J = \alpha I + A\).
This is achieved by the following snippet of code:
SUNDIALS::IDA< VectorType >::IDA  (  const AdditionalData &  data = AdditionalData() , 
const MPI_Comm  mpi_comm = MPI_COMM_WORLD 

) 
Constructor. It is possible to fine tune the SUNDIALS IDA solver by passing an AdditionalData() object that sets all of the solver parameters.
IDA is a Differential Algebraic solver. As such, it requires initial conditions also for the first order derivatives. If you do not provide consistent initial conditions, (i.e., conditions for which F(y_dot(0), y(0), 0) = 0), you can ask SUNDIALS to compute initial conditions for you by using the ic_type
parameter at construction time.
You have three options
By default, this class assumes that all components are differential, and that you want to solve a standard ode. In this case, the initial component type is set to use_y_diff
, so that the y_dot
at time t=initial_time
is computed by solving the nonlinear problem \(F(y_dot, y(t0), t0) = 0\) in the variable y_dot
.
Notice that a Newton solver is used for this computation. The Newton solver parameters can be tweaked by acting on ic_alpha
and ic_max_iter
.
If you reset the solver at some point, you may want to select a different computation for the initial conditions after reset. Say, for example, that you have refined a grid, and after transferring the solution to the new grid, the initial conditions are no longer consistent. Then you can choose how these are made consistent, using the same three options that you used for the initial conditions in reset_type
.
The MPI communicator is simply ignored in the serial case.
data  IDA configuration data 
mpi_comm  MPI communicator 
SUNDIALS::IDA< VectorType >::~IDA  (  ) 
unsigned int SUNDIALS::IDA< VectorType >::solve_dae  (  VectorType &  solution, 
VectorType &  solution_dot  
) 
void SUNDIALS::IDA< VectorType >::reset  (  const double &  t, 
const double &  h,  
VectorType &  y,  
VectorType &  yp  
) 
Clear internal memory and start with clean objects. This function is called when the simulation start 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 and the solution dot to the new mesh, and returns true.
During reset(), both y and yp are checked for consistency, and according to what was specified as ic_type (if t==initial_time) or reset_type (if t>initial_time), yp, y, or both are modified to obtain a consistent set of initial data.
[in]  t  The new starting time 
[in]  h  The new (tentative) starting time step 
[in,out]  y  The new (tentative) initial solution 
[in,out]  yp  The new (tentative) initial solution_dot 
SUNDIALS::IDA< VectorType >::DeclException1  (  ExcIDAError  , 
int  ,  
<< "One of the SUNDIALS IDA< VectorType > internal functions "<< " returned a negative error code: "<< arg1<< ". Please consult SUNDIALS manual."  
) 
Handle IDA exceptions.

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

private 
std::function<void(VectorType &)> SUNDIALS::IDA< VectorType >::reinit_vector 
std::function<int(const double t, const VectorType &y, const VectorType &y_dot, VectorType &res)> SUNDIALS::IDA< VectorType >::residual 
Compute residual. Return \(F(t, y, \dot y)\).
This function should return:
std::function<int(const double t, const VectorType &y, const VectorType &y_dot, const double alpha)> SUNDIALS::IDA< VectorType >::setup_jacobian 
Compute Jacobian. This function is called by IDA any time a Jacobian update is required. The user should compute the Jacobian (or update all the variables that allow the application of the Jacobian). This function is called by IDA once, before any call to solve_jacobian_system().
The Jacobian \(J\) should be a (possibly inexact) computation of
\[ J=\dfrac{\partial G}{\partial y} = \dfrac{\partial F}{\partial y} + \alpha \dfrac{\partial F}{\partial \dot y}. \]
If the user uses a matrix based computation of the Jacobian, than this is the right place where an assembly routine should 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. IDA 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 \(J x = b\).
This function should return:
std::function<int(const VectorType &rhs, VectorType &dst)> SUNDIALS::IDA< VectorType >::solve_jacobian_system 
Solve the Jacobian linear system. This function will be called by IDA (possibly several times) after setup_jacobian() has been called at least once. IDA tries to do its best to call setup_jacobian() the minimum amount of times. If convergence can be achieved without updating the Jacobian, then IDA does not call setup_jacobian() again. If, on the contrary, internal IDA convergence tests fail, then IDA 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.
The jacobian \(J\) should be (an approximation of) the system Jacobian
\[ J=\dfrac{\partial G}{\partial y} = \dfrac{\partial F}{\partial y} + \alpha \dfrac{\partial F}{\partial \dot y}. \]
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.
This function should return:
std::function<void (const double t, const VectorType &sol, const VectorType &sol_dot, const unsigned int step_number)> SUNDIALS::IDA< VectorType >::output_step 
Process solution. This function is called by IDA at fixed time steps, every output_period
seconds, and it is passed a polynomial interpolation of the solution and of its time derivative, computed using the current BDF order and the (internally stored) previously computed solution steps.
Notice that it is well possible that internally IDA 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, VectorType &sol_dot)> SUNDIALS::IDA< VectorType >::solver_should_restart 
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
and sol_dot
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 both sol and sol_dot to reflect the new dimension. Since IDA 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<IndexSet()> SUNDIALS::IDA< VectorType >::differential_components 
Return an index set containing the differential components. Implementation of this function is optional. The default is to return a complete index set. If your equation is also algebraic (i.e., it contains algebraic constraints, or Lagrange multipliers), you should overwrite this function in order to return only the differential components of your system.
std::function<VectorType&()> SUNDIALS::IDA< VectorType >::get_local_tolerances 

private 

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private 