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characteristic

Lagrange-Galerkin method

# 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 argument to the integrate 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 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;
};