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characteristic

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characteristic: discrete mesh advection by a field: gh(x)=fh(x

NAME

characteristic - discrete mesh advection by a field: gh(x)=fh(x- dt*uh(x))
SYNOPSYS
The class characteristic is a class implementing the Lagrange-Galerkin method. It is the extension of the method of characteristic in the finite element context. This is an expreimental implementation: please use the "geomap" one for practical usage.
EXAMPLE
The following code compute the Riesz representant, denoted by "mgh" of gh(x)=fh(x-dt*uh(x)). geo omega_h; field uh = ...; field fh = ...; characteristic X (omega_h, -dt*uh); field mgh = riesz_representer(Vh, compose(fh, X)); The Riesz representer is the "mgh" vector of values: mgh(i) = integrate fh(x-dt*uh(x)) phi_i(x) dx where phi_i is the i-th basis function in Vh and the integral is evaluated by using a quadrature formulae. By default the quadrature formule is Gauss-Lobatto with the order equal to the polynomial order of Vh. This quadrature formulae guaranties inconditional stability at any polynomial order (order 1: trapeze, order 2: simpson). Extension will accept in the future alternative quadrature formulae.

IMPLEMENTATION

class characteristic { public: characteristic(const geo& bg_omega, const field& ah); const geo& get_background_geo() const; const field& get_advection() const; protected: geo _bg_omega; field _ah; }; class field_o_characteristic { public: field_o_characteristic(const field& fh, const characteristic& X); friend field_o_characteristic compose( const field& fh, const characteristic& X); Float operator() (const point& x) const; point vector_evaluate (const point& x) const; tensor tensor_evaluate (const point& x) const; const field& get_field() const; const geo& get_background_geo() const; const field& get_advection() const; protected: field _fh; characteristic _X; point advect (const point& x0) const; }; CHARACTERISTIC(9)
 
 
 

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