SIMPLE SOLUTIONS

# CHARACTERISTIC(2RHEOLEF) - man page online | system calls

The Lagrange-Galerkin method implemented.

Chapter
rheolef-6.6
```characteristic(2rheolef)                   rheolef-6.6                   characteristic(2rheolef)

NAME
characteristic - the Lagrange-Galerkin method implemented

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.

EXAMPLE
Consider the bilinear form lh defined by

/
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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)
and the previous integral writes equivalently:

/
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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 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

IMPLEMENTATION
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;