.\"
.de Id
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.de Sp
.if n .sp
.if t .sp 0.4
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.TH characteristic 2rheolef "rheolef-6.7" "rheolef-6.7" "rheolef-6.7"
.\" label: /*Class:characteristic
.SH NAME
\fBcharacteristic\fP - the Lagrange-Galerkin method implemented
.\" skip: @cindex  Lagrange-Galerkin method
.\" skip: @cindex  method of characteristic
.\" skip: @cindex  quadrature formulae
.\" skip: @findex  riesz
.\" skip: @clindex space
.\" skip: @clindex geo
.PP
.SH SYNOPSYS

 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.
.SH EXAMPLE

 Consider the bilinear form \fBlh\fP defined by
.\" begin_example
.Sp
.nf
            /
            |
    lh(x) = | uh(x+dh(x)) v(x) dx
            |
            / Omega
.Sp
.fi
.\" end_example
 where \fBdh\fP is a deformation vector field.
 The characteristic is defined by \fBX(x)=x+dh(x)\fP
 and the previous integral writes equivalently:
.\" begin_example
.Sp
.nf
            /
            |
    lh(x) = | uh(X(x)) v(x) dx
            |
            / Omega
.Sp
.fi
.\" end_example
.PP
 For instance, in Lagrange-Galerkin methods,
 the deformation field \fBdh(x)=-dt*uh(x)\fP
 where \fBuh\fP is the advection field and \fBdt\fP a time step.
 The following code implements the computation of lh:
.\" begin_example
.Sp
.nf
  field dh = ...;
  field uh = ...;
  characteristic X (dh);
  test v (Xh);
  field lh = integrate (compose(uh, X)*v, qopt);
.Sp
.fi
.\" end_example
 The Gauss-Lobatto quadrature formule is recommended for
 Lagrange-Galerkin methods.
 The order equal to the polynomial order of Xh
 (order 1: trapeze, order 2: simpson, etc).
 Recall that this choice of quadrature formulae guaranties inconditional
 stability at any polynomial order.
 Alternative quadrature formulae or order can be used by using the
 additional quadrature option argument to the \fBintegrate\fP function see integrate(4).
.\" END
.SH IMPLEMENTATION
.\" begin_example
.Sp
.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;

  const characteristic_on_quadrature<T,M>&
  get_pre_computed (
    const space_basic<T,M>&       Xh,
    const field_basic<T,M>&       dh,
    const quadrature_option_type& qopt) const;

};
typedef characteristic_basic<Float> characteristic;
.Sp
.fi
.\" end_example

.\" LENGTH = 1
.SH SEE ALSO
integrate(4)