noble (2) characteristic.2rheolef.gz

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@imag.fr>

COPYRIGHT

       Copyright   (C)  2000-2018  Pierre  Saramito  <Pierre.Saramito@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.