Detailed Description

IDetailedModel implementation for a non linear evolution problem.

This is a detailed model for a problem of this type

\begin{align} \partial_t u(x;t,\mu) - {\cal L}(t,\mu)[u(x,t;\mu)] &= 0 \qquad &\text{in }& \Omega \times [0, T], \\\ u(x;0,\mu) &= u_0(x;\mu) \qquad &\text{in }& \Omega \times \{0\} \end{align}

with suitable boundary conditions and (non-linear) operators \({\cal L}(t,\mu) : {\cal W} \to {\cal W}\) parametrized over the Time-parameter domain \([0,T] \times {\cal M}\).

The numerical scheme for the numerical solution of this problem is assumed to be of the

following type:
\begin{align*} u_h(\cdot;t^0,\mu) &= {\cal M}_h[u_0] \\\ \left(\text{Id} - \Delta t^k {\cal L}_{h, I}(t^{k+1},\mu)\right)[u_h(\cdot;t^{k+1},\mu)] &= \left(\text{Id} + \Delta t^k {\cal L}_{h, E}(t^{k},\mu)\right)[u_h(\cdot;t^{k},\mu)] \end{align*}
or with a Newton solver for non-linearities in the implicit case:
\[ u_h(\cdot;t^0,\mu) = {\cal M}_h[u_0], \]
where the equation
\[ F[u_h(\cdot;t^{k+1},\mu), u_h(\cdot;t^k,\mu)] =0 \]
is solved subsequently for the operator \(F:V_h\times V_h \to V_h\) defined by
\[ F[u_h;v_h] := \left(\text{Id} + \Delta t^k {\cal L}_{h, I}(t^{k+1},\mu)\right)[u_h] - \left(\text{Id} + \Delta t^k {\cal L}_{h, E}(t^{k},\mu)\right)[v_h] \]
with a Newton scheme, minimizing defects \(\| \delta_h^{k+1,\nu+1}\| \leq R_{\text{Newton}}\) by solving
\[ D_1 F|_{u_h^{k+1,\nu}}[\delta_h^{k+1,\nu+1};u_h^k] = -F[u_h^{k+1,\nu};u_h^k] \]
with \(u_h^{k+1,\nu+1} := u_h^{k+1,\nu}\), \(u_h^{k,0}:=u_h^k\), \(u_h^{k+1}:=u_h^{k+1, \nu_{\max}(k)}\) and \(\nu_{\max}(k) = \min_{\nu} \| \delta_h^{k+1,\nu} \| \leq R_{\text{Newton}}\)

Furthermore the discretization consists of

a time discretization \(0 = t^0 \leq t^1 = t^0 + \Delta t^0 \leq \ldots \leq t^{K+1} = t^{K} + \Delta t^{K} = T\),
discrete space operators \({\cal L}_{h, I}(t,\mu):{\cal W}_h \to {\cal W}_h\) and \({\cal L}_{h, E}(t,\mu):{\cal W}_h \to {\cal W}_h\) parametrized over the Time-Parameter domain \([0,T] \times {\cal M}\) and
a projection operator \({\cal M}_h : {\cal W} \to {\cal W}_h\)

Definition at line 18 of file DetailedModel.m.

Inheritance diagram for NonlinEvol.DetailedModel:

[legend]

Collaboration diagram for NonlinEvol.DetailedModel:

[legend]

Public Member Functions
	DetailedModel (ModelDescr descr)
	constructor

function sim_data =	detailed_simulation (ModelData model_data)
	executes a detailed simulation for a given parameter

function ModelData model_data =	gen_model_data ()
	method which produces all \(H\)-dependent data, that is required for a detailed simulations and is independent of the parameter \(\mu\).

function p =	plot_sim_data (ModelData model_data, sim_data, plot_params)
	plots the simulation data as returned by detailed_simulation()

Public Member Functions inherited from Greedy.User.FVDetailedModelDefault
	FVDetailedModelDefault (ModelDescr descr)
	Constructor for this class.

function UON =	orthonormalize (ModelData model_data, U)
	orthonormalizes a set of discrete functions \(\{ v_h^l \}_{l=1}^L\)

function U0 =	init_values_algorithm (ModelData model_data)
	computes the Dof vectors for the initial value function \(u_h^0(\cdot;t\mu) = {\cal P}_h[u_0(\mu)]\)

function UV =	inner_product (ModelData model_data, U, V)
	computes the inner project between the Dof vectors of two discrete functions

Public Member Functions inherited from Greedy.User.IDetailedModel
	IDetailedModel (descr)
	constructor

Public Member Functions inherited from IDetailedModel
	IDetailedModel (descr)
	constructor

function isequal =	eq (IDetailedModel other)
	overloaded equality (`==`) operator, by default comparing the descr variables

function IDetailedModel this =	set_mu (mu)
	sets the active parameter vector \(\mu \in {\cal M}\)

function mu =	get_mu ()
	returns the active parameter vector \(\boldsymbol\mu \in {\cal M}\)

virtual function sim_data =	detailed_simulation (model_data)
	executes a detailed simulation for a given parameter

virtual function p =	plot_sim_data (model_data, sim_data, plot_params)
	plots the simulation data as returned by detailed_simulation()

Static Public Member Functions
static function U =	get_dofs_from_sim_data (sim_data)
	extracts the \(H\) dimensional Dof vector from the `sim_data` structure

static function snapshot =	get_dofs_at_time (sim_data, time_index)

Static Public Member Functions inherited from Greedy.User.FVDetailedModelDefault
static function errs =	l2_error_sequence_algorithm (U, Uapprox,ModelData model_data)
	computes the \(L^2(\Omega)\) error for each "snapshot" of a trajectories and returns the sequence of these errors.

static function errs =	linfty_error_sequence_algorithm (U, Uapprox,ModelData model_data)
	computes the \(L^{\infty}(\Omega)\) error for each "snapshot" of a trajectories and returns the sequence of these errors.

static function W =	get_inner_product_matrix (ModelData model_data)
	returns the inner product matrix for efficient computation of inner products on \({\cal W}_h\)

Public Attributes
	t
	current time instance

	tstep
	current time step

	compute_conditions = false
	boolean flag indicating whether conditions of system matrices shall be computed.

Static Public Attributes
static const	model_checks
	This constant is used for a consistency check of the model descr with help of DetailedModelBaseIf.struct_check()

static const	newton_checks
	This constant is used for a consistency check of the model descr with help of DetailedModelBaseIf.struct_check()

Constructor & Destructor Documentation

NonlinEvol.DetailedModel.DetailedModel ( ModelDescr descr )

constructor

This checks and stores the model description structure

Parameters

descr structure

Required fields of descr:

newton_solver — boolean flag indicating whether we are using a Newton scheme for the discretization.

Definition at line 171 of file DetailedModel.m.

Member Function Documentation

function [ sim_data , fl ] = NonlinEvol.DetailedModel.detailed_simulation ( ModelData model_data )

executes a detailed simulation for a given parameter

Parameters

model_data Matlab structure storing (possibly) high dimensional data needed by IDetailedModel.detailed_simulation().

Return values

sim_data	structure holding the \(H\)-dimensional simulation data.
fl	debug output holding the flux evaluations returned by the `L_E_local_ptr` operator.

Required fields of model:

T — final time
nt — number of time-steps \(0 = t_0 \leq t_1 \leq \cdots \leq t^K+1 = T\), i.e. \(K+1\).
init_values_algorithm — function pointer for computing the initvalues-DOF with arguments (model, model_data)
- example: fv_init_values()
L_E_local_ptr — function pointer to the local space-discretization operator evaluation with syntax
INC_local = L_local(U_local_ext, ind_local, grid_local_ext)

where *_local indicates results on a set of elements, *_local_ext includes information including their neighbours, i.e. the grid must be known also for the neighbours, the values of the previous timstep must be known on the neighbours. The subset of elements, which are to be computed are given in ind_local and the values produced are INC_local. A time-step can then be realized by
NU_local = U_local_ext(ind_local) - dt * INC_local
- example: fv_conv_explicit_space()

Optional fields of model:

data_const_in_time — if this optional field is 1, the time evolution is performed with constant operators, i.e. only the initial-time-operators are computed and used throughout the time simulation.

Generated fields of sim_data:

U — matrix of size \(H \times K+1\) holding for each time step the solution dof vector.
Unewton — matrix of size \(H \times K+1+\sum_{k=0}^K n_{\text{Newton}}(k)\) holding for each time step and each Newton iteration the intermediate solution dof vectors.
conds.LI — conds.LI
conds.gradLI — conds.gradLI

Definition at line 19 of file detailed_simulation.m.

function ModelData model_data = NonlinEvol.DetailedModel.gen_model_data ( )

virtual

method which produces all \(H\)-dependent data, that is required for a detailed simulations and is independent of the parameter \(\mu\).

Return values

model_data model data

Generated fields of model_data:

W — inner product weighting matrix
grid — a structure containing geometry information of a mesh used for discretizations

Implements IDetailedModel.

Definition at line 19 of file gen_model_data.m.

static function U = NonlinEvol.DetailedModel.get_dofs_from_sim_data ( sim_data )

staticvirtual

extracts the \(H\) dimensional Dof vector from the sim_data structure

Parameters

sim_data matlab struct with simulation data generated e.g. by detailed_simulation() or rb_reconstruction() methods.

Return values

U	\(H\) dimensional Dof vector

Implements Greedy.User.IDetailedModel.

Definition at line 204 of file DetailedModel.m.

function p = NonlinEvol.DetailedModel.plot_sim_data	(	ModelData	model_data,
			sim_data,
			plot_params
	)

plots the simulation data as returned by detailed_simulation()

function plotting simulation results

Parameters

model_data	Matlab structure storing (possibly) high dimensional data needed by IDetailedModel.detailed_simulation().
sim_data	simulation data structure as returned by detailed_simulation()
plot_params	structure which controls the plot output

Return values

p	Matlab GUI handle to figure window

Required fields of sim_data:

U — DOF vector of solution snapshots

Required fields of model_data:

grid — a structure containing geometry information of a mesh used for discretizations

Optional fields of plot_params:

bind_to_model — Sometimes the plot can adapted according to the parameters. For example if the domain of the simulation changes with the parameter. In this case this boolean flag can be set, and the copy_model_descr_to_plot_params() function is called to copy important parameters to the plot_params (default = false)
plot — function pointer to a function plotting a single snapshot for a specific time instant. (default = descr.plot)

Definition at line 19 of file plot_sim_data.m.

Member Data Documentation

NonlinEvol.DetailedModel.compute_conditions = false

boolean flag indicating whether conditions of system matrices shall be computed.

Default: false

Definition at line 160 of file DetailedModel.m.

NonlinEvol.DetailedModel.model_checks

static

Initial value:

= struct("   \
      'fv_impl_diff_weight',          {{\@isscalar}},   \
      'fv_expl_diff_weight',          {{\@isscalar}},   \
      'fv_impl_conv_weight',          {{\@isscalar}},   \
      'fv_expl_conv_weight',          {{\@isscalar}},   \
      'newton_solver',                {{\@isscalar}},   \
      'implicit_nonlinear',           {{\@isscalar}},   \
      'L_E_local_ptr',                {{\@(x) isequal(class(x), 'function_handler')}},   \
      'L_I_local_ptr',                {{\@(x) isequal(class(x), 'function_handler')}}...
    ")

This constant is used for a consistency check of the model descr with help of DetailedModelBaseIf.struct_check()

Default: struct(" \ 'fv_impl_diff_weight', {{@isscalar}}, \ 'fv_expl_diff_weight', {{@isscalar}}, \ 'fv_impl_conv_weight', {{@isscalar}}, \ 'fv_expl_conv_weight', {{@isscalar}}, \ 'newton_solver', {{@isscalar}}, \ 'implicit_nonlinear', {{@isscalar}}, \ 'L_E_local_ptr', {{@(x) isequal(class(x), 'function_handler')}}, \ 'L_I_local_ptr', {{@(x) isequal(class(x), 'function_handler')}}... ")

Definition at line 82 of file DetailedModel.m.

NonlinEvol.DetailedModel.newton_checks

static

Initial value:

= struct("   \
      'newton_steps',                 {{\@isscalar}},   \
      'implicit_operators_algorithm', {{\@(x) isequal(class(x), 'function_handler')}},   \
      'implicit_gradient_operators_algorithm',   \
                                      {{\@(x) isequal(class(x), 'function_handler')}},   \
      'newton_epsilon',               {{\@isscalar}}...
    ")

This constant is used for a consistency check of the model descr with help of DetailedModelBaseIf.struct_check()

This check is only executed if we use a Newton scheme

Default: struct(" \ 'newton_steps', {{@isscalar}}, \ 'implicit_operators_algorithm', {{@(x) isequal(class(x), 'function_handler')}}, \ 'implicit_gradient_operators_algorithm', \ {{@(x) isequal(class(x), 'function_handler')}}, \ 'newton_epsilon', {{@isscalar}}... ")

Definition at line 110 of file DetailedModel.m.

NonlinEvol.DetailedModel.t

current time instance

Note: This property has the MATLAB attribute Dependent set to true.; Matlab documentation of property attributes.

Definition at line 136 of file DetailedModel.m.

NonlinEvol.DetailedModel.tstep

current time step

Note: This property has the MATLAB attribute Dependent set to true.; Matlab documentation of property attributes.

Definition at line 147 of file DetailedModel.m.

The documentation for this class was generated from the following files:

rbasis/problem_types/oop/+NonlinEvol/@DetailedModel/DetailedModel.m
rbasis/problem_types/oop/+NonlinEvol/@DetailedModel/detailed_simulation.m
rbasis/problem_types/oop/+NonlinEvol/@DetailedModel/gen_model_data.m
rbasis/problem_types/oop/+NonlinEvol/@DetailedModel/plot_sim_data.m

Detailed Description

Public Member Functions

Static Public Member Functions

Public Attributes

Static Public Attributes

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation