Schema_RK_Rationnel.h

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#ifndef Schema_RK_Rationnel_included
#define Schema_RK_Rationnel_included
 
#include <TRUSTSchema_RK.h>
 
/// \cond DO_NOT_DOCUMENT
class Schema_RK_Rationnel
{  };
/// \endcond
 
/*! @brief : class RRK2 This class represents a second-order rational Runge-Kutta scheme:
 *
 *      U(n+1) = U(n) + (2g1 (g1,b) - b(g1,g1))/(b,b)
 *      where: g1 = dt f(U(n))
 *             g2 = dt f(U(n) + c2 g1)
 *             b = b1 g1 + b2 g2
 *      with b1=2, b2=-1, c2=1/2
 *
 * @sa : TRUSTSchema_RK, 03/07/2017 ABn. Some notes on RRK2:, The reference paper for the Rational Runge-Kutta 2 implementation appears to be:, Wambeck - Rational Runge-Kutta methods for solving systems of ordinary differential equations:, https://link.springer.com/article/10.1007/BF02252381, The "rational" refers to the rational fraction used in the expansion., The rest matches the implementation closely. Notably, we find the interesting definition:, a.b / d = a(b,d)+b(d,a)-(a,b) / ||d||^2, for vectors a, b and d., In brief (from the conclusion):, Although rational methods require more computational work than linear ones, they can have some other properties, such as a stable behaviour with explicitness, which make them preferable., 05/07/2017 ABn. Additional notes on RRK2:, Application of RRK2 in CFD:, https://link.springer.com/content/pdf/10.1007%2F3-540-13917-6_112.pdf, rho*u stays at time n for an intermediate evaluation of f., Said differently, from the time scheme perspective, f is only a function of Y1.
 */
 
class RRK2: public TRUSTSchema_RK<Ordre_RK::RATIO_DEUX>
{
  Declare_instanciable(RRK2);
public :
  int faire_un_pas_de_temps_eqn_base(Equation_base&) override;
};
 
#endif /* Schema_RK_Rationnel_included */