DERIVED TYPE ode_problem: definition of ODE/PDE problems More...

Data Types
interface	clear

interface	ode_problem
	Type ode_problem: definition of ODE/PDE problems. More...

interface	print

interface	valid

Functions/Subroutines
subroutine	ode_problem_clear (pb)
	destructor for ode_problem More...

type(ode_problem) function	ode_problem_create (type, dim, dof, X, M, S, AB, N, Na)
	Constructor for the type ode_problem More...

logical function	ode_problem_valid (pb)
	check ode_problem More...

character(len=15) function, public	name_ode_problem (type)
	ode_problem name More...

subroutine	ode_problem_print (pb)
	print ode_problem More...

Detailed Description

DERIVED TYPE ode_problem: definition of ODE/PDE problems

Choral constants for ODE problems: ODE_PB_xxx, see the list.

Tutorial examples are listed in ode_solver_mod detailed description.

General target problem: ODE_PB_SL_NL =
SemiLinear PDE coupled with a Non-Linear ODE system,
- for \( 1 \le i \le N_a \): \(~~~ d Y_i/dt =a_i(x,t,Y)Y_i + b_i(x,t,Y)Y_i = F_i \)
- for \( N_a < i < N \): \(~~~ d Y_i/dt = b_i(x,t,Y)Y_i \)
  ( \( a_i(x,t,Y) = 0 \) here),
- for \( i=N \): \(~~~ M \partial V/\partial t = -S V + M b_N(x,t,Y) \)
where:
- \( 0 \le N_a \le N \)
- \( M, ~S \) = positive symmetric linear operators (mass and stiffness matrices e. g.),
- \( Y(x,t) \in R^N \)
- \( V(x,t) = Y_N(x,t)\in R \)
- \( x \in \Omega \) and \( 0 \le t \le T \)
Nota bene: if \( N_a = N \), the semilinear equation is:
- \(~~~ M \partial V/\partial t = -S V + M (a_N(x,t,Y)V + b_N(x,t,Y)) \)
ODE_PB_Lin = Linear PDE
- \( N = 1 \), \( a(x,t) = b(x,t) = 0 \)
- \( M \partial V/\partial t = -S V \)
ODE_PB_SL = semiLinear PDE
- \( N = 1 \)
- \( M \partial V/\partial t = -S V + M F(x,t,V) \)
ODE_PB_NL = Non Linear ODE
- \( S=0 \), \( M=Id \)
- for \( 1 \le i \le N_a \): \(~~~ d Y_i/dt = a_i(x,t,Y)Y_i + b_i(x,t,Y)Y_i \)
- for \( N_a < i \le N \): \(~~~ d Y_i/dt = b_i(x,t,Y)Y_i \)

An ode_problem is constructed with

problem = ode_problem(type, atgs)

the argumentss define the problem relativelly to its type, see ode_problem_create.

Among these parameters:

the two functions \( a(x,t)\), \( b(x,t)\) are passed as a single procedural arguments of type ode_reaction
\( (x,t) \in \R^3 \times \R \rightarrow [a(x,t), ~b(x,t)]\in \R^{N_a}\times \R^N \)
\( M \) and \( S \) are passed as procedural arguments of type RnToRn
\( u \in \R^{\rm dof} \rightarrow Mu \in \R^{\rm dof} \)
The domain \( \Omega \) is considered as embedded in \(\R^3\).
- the computation nodes \( x_i \in \Omega\) are passed as a single array \( X \in \R^3 \times \R^{\rm dof} \)
- in case of a 0D problem (a true ODE system) set dim = 0 (no nodes here).

Reaction term decomposition:
We have set

\( 0 \le N_a \le N \)
the two functions
\(~~~ a:~(x,t)\in\Omega\times\R\rightarrow a(x,t)\in\R^{N_a}\)
\(~~~ b:~(x,t)\in\Omega\times\R\rightarrow b(x,t)\in\R^{N}\)

to define the global reaction term \(~~~ F:~(x,t)\in\Omega\times\R\rightarrow F(x,t)\in\R^{N}\)

for \( 1 \le i \le N_a \): \(~~~ F_i(x,t,Y) = a_i(x,t,Y) Y_i + b_i(x, t, Y) \)
for \( N_a < i \le N \): \(~~ F_i(x,t,Y) = b_i(x, t, Y) \)

This linearisation of the reaction term \( F \) is used by several ODE methods such as implicit-explicit Euler or exponential integrators in ode_NL_ms_mod and ode_SL_NL_ms_mod.

Author: Charles Pierre

Function/Subroutine Documentation

◆ name_ode_problem()

character(len=15) function, public ode_problem_mod::name_ode_problem ( integer, intent(in) type )

ode_problem name

Definition at line 319 of file ode_problem_mod.f90.

◆ ode_problem_clear()

subroutine ode_problem_mod::ode_problem_clear ( type(ode_problem), intent(inout) pb )

private

destructor for ode_problem

Definition at line 187 of file ode_problem_mod.f90.

◆ ode_problem_create()

type(ode_problem) function ode_problem_mod::ode_problem_create	(	integer, intent(in)	type,
		integer, optional	dim,
		integer, optional	dof,
		real(rp), dimension(:,:), optional	X,
		procedure(rntorn), optional	M,
		procedure(rntorn), optional	S,
		procedure(ode_reaction), optional	AB,
		integer, optional	N,
		integer, optional	Na
	)

private

Constructor for the type ode_problem

OUTPUT:
- pb = ode_problem

INPUT:
- type = type, ode problem type
optional arguments:
- dof : number of nodes
- X : node coordinates
- M, S : linear operators
- AB : non linear operators
- N, Na : problem dimensions
- dim : set dim = 0 for 0D problems

Details are given in in ode_problem_mod detailed description

Definition at line 224 of file ode_problem_mod.f90.

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