NAME
characteristic - Lagrange-Galerkin method (rheolef-7.2)
DESCRIPTION
The class characteristic implements the Lagrange-Galerkin method. It is the extension of the method of characteristic from the finite difference to the finite element context.
Note that the Lagrange-Galerkin method applies to diffusion-convection problems when the convection term is not dominant. For more serious problems, please refer to the discontinuous Galerkin method in the Rheolef library.
EXAMPLE
Consider the bilinear form lh defined by
/
|
lh(x) = | uh(x+dh(x)) v(x) dx
|
/ Omega
where dh is a deformation vector field. The characteristic is defined by X(x)=x+dh(x) for any x in Omega, and the previous integral writes equivalently:
/
|
lh(x) = | uh(X(x)) v(x) dx
|
/ Omega
For instance, in Lagrange-Galerkin methods, the deformation field dh(x)=-dt*uh(x) where uh is the advection field and dt a time step. The following code implements the computation of lh:
field dh = ...;
field uh = ...;
characteristic X (dh);
test v (Xh);
field lh = integrate (compose(uh, X)*v, qopt);
The Gauss-Lobatto quadrature formula is recommended for the computation of integrals involving the characteristic X of the Lagrange-Galerkin method. The order equal to the polynomial order of Xh (order 1: trapeze, order 2: simpson, etc). Recall that this choice of quadrature formula guaranties inconditionnal stability at any polynomial order. Alternative quadrature formulae or order can be used by using the additional integrate_option(3) argument to the integrate(3) function.
IMPLEMENTATION
This documentation has been generated from file main/lib/characteristic.h
The characteristic class is simply an alias to the characteristic_basic class
typedef characteristic_basic<Float> characteristic;
The characteristic_basic class provides an interface, via the smart_pointer(7) class family, to a data container:
template<class
T, class M = rheo_default_memory_model>
class characteristic_basic : public
smart_pointer<characteristic_rep<T,M> > {
public:
typedef characteristic_rep<T,M> rep;
typedef smart_pointer<rep> base;
// allocator:
characteristic_basic(const field_basic<T,M>& dh);
// accesors:
const field_basic<T,M>& get_displacement() const;
};
AUTHOR
Pierre Saramito <Pierre.Saramito [AT] imag.fr>
COPYRIGHT
Copyright (C) 2000-2018 Pierre Saramito <Pierre.Saramito [AT] imag.fr> GPLv3+: GNU GPL version 3 or later <http://gnu.org/licenses/gpl.html>;. This is free software: you are free to change and redistribute it. There is NO WARRANTY, to the extent permitted by law.