.TH "characteristic" 2rheolef "Version 7.2" "rheolef" \" -*- nroff -*-
.ad l
.nh
.SH NAME
characteristic \- Lagrange-Galerkin method (rheolef-7\&.2)
.PP
  
.SH "DESCRIPTION"
.PP
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\&.
.PP
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\&.
.SH "EXAMPLE"
.PP
Consider the bilinear form \fClh\fP defined by 
.PP
.nf
        /
        |
lh(x) = | uh(x+dh(x)) v(x) dx
        |
        / Omega

.fi
.PP
 where \fCdh\fP is a deformation vector field\&. The characteristic is defined by \fCX(x)=x+dh(x)\fP for any x in Omega, and the previous integral writes equivalently: 
.PP
.nf
        /
        |
lh(x) = | uh(X(x)) v(x) dx
        |
        / Omega

.fi
.PP
 For instance, in Lagrange-Galerkin methods, the deformation field \fCdh(x)=-dt*uh(x)\fP where \fCuh\fP is the advection field and \fCdt\fP a time step\&. The following code implements the computation of \fClh\fP: 
.PP
.nf
    field dh = \&.\&.\&.;
    field uh = \&.\&.\&.;
    characteristic X (dh);
    test v (Xh);
    field lh = integrate (compose(uh, X)*v, qopt);

.fi
.PP
 The Gauss-Lobatto quadrature formula is recommended for the computation of integrals involving the characteristic \fCX\fP of the Lagrange-Galerkin method\&. The order equal to the polynomial order of \fCXh\fP (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 \fBintegrate_option(3)\fP argument to the \fBintegrate(3)\fP function\&.
.SH "IMPLEMENTATION"
.PP
This documentation has been generated from file main/lib/characteristic\&.h
.PP
The \fCcharacteristic\fP class is simply an alias to the \fC\fBcharacteristic_basic\fP\fP class
.PP
.PP
.nf
typedef characteristic_basic<Float> characteristic;
.fi
.PP
 
.PP
\fB\fP
.RS 4

.RE
.PP
The \fC\fBcharacteristic_basic\fP\fP class provides an interface, via the \fBsmart_pointer(7)\fP class family, to a data container:
.PP
.PP
.nf
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;
.fi
.PP
 
.PP
.nf
};

.fi
.PP
 
.SH AUTHOR
Pierre  Saramito  <Pierre.Saramito@imag.fr>
.SH 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.