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.