ode.hpp

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// Copyright (c) 2010-2022, Lawrence Livermore National Security, LLC. Produced
// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
// LICENSE and NOTICE for details. LLNL-CODE-806117.
//
// This file is part of the MFEM library. For more information and source code
// availability visit https://mfem.org.
//
// MFEM is free software; you can redistribute it and/or modify it under the
// terms of the BSD-3 license. We welcome feedback and contributions, see file
// CONTRIBUTING.md for details.

#ifndef MFEM_ODE
#define MFEM_ODE

#include "../config/config.hpp"
#include "operator.hpp"

namespace mfem
{

/// Abstract class for solving systems of ODEs: dx/dt = f(x,t)
class ODESolver
{
protected:
   /// Pointer to the associated TimeDependentOperator.
   TimeDependentOperator *f;  // f(.,t) : R^n --> R^n
   MemoryType mem_type;

public:
   ODESolver() : f(NULL) { mem_type = Device::GetHostMemoryType(); }

   /// Associate a TimeDependentOperator with the ODE solver.
   /** This method has to be called:
       - Before the first call to Step().
       - When the dimensions of the associated TimeDependentOperator change.
       - When a time stepping sequence has to be restarted.
       - To change the associated TimeDependentOperator. */
   virtual void Init(TimeDependentOperator &f_);

   /** @brief Perform a time step from time @a t [in] to time @a t [out] based
       on the requested step size @a dt [in]. */
   /** @param[in,out] x   Approximate solution.
       @param[in,out] t   Time associated with the approximate solution @a x.
       @param[in,out] dt  Time step size.

       The following rules describe the common behavior of the method:
       - The input @a x [in] is the approximate solution for the input time
         @a t [in].
       - The input @a dt [in] is the desired time step size, defining the desired
         target time: t [target] = @a t [in] + @a dt [in].
       - The output @a x [out] is the approximate solution for the output time
         @a t [out].
       - The output @a dt [out] is the last time step taken by the method which
         may be smaller or larger than the input @a dt [in] value, e.g. because
         of time step control.
       - The method may perform more than one time step internally; in this case
         @a dt [out] is the last internal time step size.
       - The output value of @a t [out] may be smaller or larger than
         t [target], however, it is not smaller than @a t [in] + @a dt [out], if
         at least one internal time step was performed.
       - The value @a x [out] may be obtained by interpolation using internally
         stored data.
       - In some cases, the contents of @a x [in] may not be used, e.g. when
         @a x [out] from a previous Step() call was obtained by interpolation.
       - In consecutive calls to this method, the output @a t [out] of one
         Step() call has to be the same as the input @a t [in] to the next
         Step() call.
       - If the previous rule has to be broken, e.g. to restart a time stepping
         sequence, then the ODE solver must be re-initialized by calling Init()
         between the two Step() calls. */
   virtual void Step(Vector &x, double &t, double &dt) = 0;

   /// Perform time integration from time @a t [in] to time @a tf [in].
   /** @param[in,out] x   Approximate solution.
       @param[in,out] t   Time associated with the approximate solution @a x.
       @param[in,out] dt  Time step size.
       @param[in]     tf  Requested final time.

       The default implementation makes consecutive calls to Step() until
       reaching @a tf.
       The following rules describe the common behavior of the method:
       - The input @a x [in] is the approximate solution for the input time
         @a t [in].
       - The input @a dt [in] is the initial time step size.
       - The output @a dt [out] is the last time step taken by the method which
         may be smaller or larger than the input @a dt [in] value, e.g. because
         of time step control.
       - The output value of @a t [out] is not smaller than @a tf [in]. */
   virtual void Run(Vector &x, double &t, double &dt, double tf)
   {
      while (t < tf) { Step(x, t, dt); }
   }

   /// Function for getting and setting the state vectors
   virtual int   GetMaxStateSize() { return 0; }
   virtual int   GetStateSize() { return 0; }
   virtual const Vector &GetStateVector(int i)
   {
      mfem_error("ODESolver has no state vectors");
      Vector *s = NULL; return *s; // Make some compiler happy
   }
   virtual void  GetStateVector(int i, Vector &state)
   {
      mfem_error("ODESolver has no state vectors");
   }
   virtual void  SetStateVector(int i, Vector &state)
   {
      mfem_error("ODESolver has no state vectors");
   }

   virtual ~ODESolver() { }
};


/// The classical forward Euler method
class ForwardEulerSolver : public ODESolver
{
private:
   Vector dxdt;

public:
   void Init(TimeDependentOperator &f_) override;

   void Step(Vector &x, double &t, double &dt) override;
};


/** A family of explicit second-order RK2 methods. Some choices for the
    parameter 'a' are:
    a = 1/2 - the midpoint method
    a =  1  - Heun's method
    a = 2/3 - default, has minimal truncation error. */
class RK2Solver : public ODESolver
{
private:
   double a;
   Vector dxdt, x1;

public:
   RK2Solver(const double a_ = 2./3.) : a(a_) { }

   void Init(TimeDependentOperator &f_) override;

   void Step(Vector &x, double &t, double &dt) override;
};


/// Third-order, strong stability preserving (SSP) Runge-Kutta method
class RK3SSPSolver : public ODESolver
{
private:
   Vector y, k;

public:
   void Init(TimeDependentOperator &f_) override;

   void Step(Vector &x, double &t, double &dt) override;
};


/// The classical explicit forth-order Runge-Kutta method, RK4
class RK4Solver : public ODESolver
{
private:
   Vector y, k, z;

public:
   void Init(TimeDependentOperator &f_) override;

   void Step(Vector &x, double &t, double &dt) override;
};


/** An explicit Runge-Kutta method corresponding to a general Butcher tableau
    +--------+----------------------+
    | c[0]   | a[0]                 |
    | c[1]   | a[1] a[2]            |
    | ...    |    ...               |
    | c[s-2] | ...   a[s(s-1)/2-1]  |
    +--------+----------------------+
    |        | b[0] b[1] ... b[s-1] |
    +--------+----------------------+ */
class ExplicitRKSolver : public ODESolver
{
private:
   int s;
   const double *a, *b, *c;
   Vector y, *k;

public:
   ExplicitRKSolver(int s_, const double *a_, const double *b_,
                    const double *c_);

   void Init(TimeDependentOperator &f_) override;

   void Step(Vector &x, double &t, double &dt) override;

   virtual ~ExplicitRKSolver();
};


/** An 8-stage, 6th order RK method. From Verner's "efficient" 9-stage 6(5)
    pair. */
class RK6Solver : public ExplicitRKSolver
{
private:
   static const double a[28], b[8], c[7];

public:
   RK6Solver() : ExplicitRKSolver(8, a, b, c) { }
};


/** A 12-stage, 8th order RK method. From Verner's "efficient" 13-stage 8(7)
    pair. */
class RK8Solver : public ExplicitRKSolver
{
private:
   static const double a[66], b[12], c[11];

public:
   RK8Solver() : ExplicitRKSolver(12, a, b, c) { }
};


/** An explicit Adams-Bashforth method. */
class AdamsBashforthSolver : public ODESolver
{
private:
   int s, smax;
   const double *a;
   Vector *k;
   Array<int> idx;
   ODESolver *RKsolver;

public:
   AdamsBashforthSolver(int s_, const double *a_);

   void Init(TimeDependentOperator &f_) override;

   void Step(Vector &x, double &t, double &dt) override;

   int  GetMaxStateSize() override { return smax; };
   int  GetStateSize() override { return s; };
   const Vector &GetStateVector(int i) override;
   void GetStateVector(int i, Vector &state) override;
   void SetStateVector(int i, Vector &state) override;

   ~AdamsBashforthSolver()
   {
      if (RKsolver) { delete RKsolver; }
      delete [] k;
   }
};


/** A 1-stage, 1st order AB method.  */
class AB1Solver : public AdamsBashforthSolver
{
private:
   static const double a[1];

public:
   AB1Solver() : AdamsBashforthSolver(1, a) { }
};

/** A 2-stage, 2st order AB method.  */
class AB2Solver : public AdamsBashforthSolver
{
private:
   static const double a[2];

public:
   AB2Solver() : AdamsBashforthSolver(2, a) { }
};

/** A 3-stage, 3st order AB method.  */
class AB3Solver : public AdamsBashforthSolver
{
private:
   static const double a[3];

public:
   AB3Solver() : AdamsBashforthSolver(3, a) { }
};

/** A 4-stage, 4st order AB method.  */
class AB4Solver : public AdamsBashforthSolver
{
private:
   static const double a[4];

public:
   AB4Solver() : AdamsBashforthSolver(4, a) { }
};

/** A 5-stage, 5st order AB method.  */
class AB5Solver : public AdamsBashforthSolver
{
private:
   static const double a[5];

public:
   AB5Solver() : AdamsBashforthSolver(5, a) { }
};


/** An implicit Adams-Moulton method. */
class AdamsMoultonSolver : public ODESolver
{
private:
   int s, smax;
   const double *a;
   Vector *k;
   Array<int> idx;
   ODESolver *RKsolver;

public:
   AdamsMoultonSolver(int s_, const double *a_);

   void Init(TimeDependentOperator &f_) override;

   void Step(Vector &x, double &t, double &dt) override;

   int  GetMaxStateSize() override { return smax-1; };
   int  GetStateSize() override { return s-1; };
   const Vector &GetStateVector(int i) override;
   void GetStateVector(int i, Vector &state) override;
   void SetStateVector(int i, Vector &state) override;

   ~AdamsMoultonSolver()
   {
      if (RKsolver) { delete RKsolver; }
      delete [] k;
   };
};

/** A 0-stage, 1st order AM method. */
class AM0Solver : public AdamsMoultonSolver
{
private:
   static const double a[1];

public:
   AM0Solver() : AdamsMoultonSolver(0, a) { }
};


/** A 1-stage, 2st order AM method. */
class AM1Solver : public AdamsMoultonSolver
{
private:
   static const double a[2];

public:
   AM1Solver() : AdamsMoultonSolver(1, a) { }
};

/** A 2-stage, 3st order AM method. */
class AM2Solver : public AdamsMoultonSolver
{
private:
   static const double a[3];

public:
   AM2Solver() : AdamsMoultonSolver(2, a) { }
};

/** A 3-stage, 4st order AM method. */
class AM3Solver : public AdamsMoultonSolver
{
private:
   static const double a[4];

public:
   AM3Solver() : AdamsMoultonSolver(3, a) { }
};

/** A 4-stage, 5st order AM method. */
class AM4Solver : public AdamsMoultonSolver
{
private:
   static const double a[5];

public:
   AM4Solver() : AdamsMoultonSolver(4, a) { }
};


/// Backward Euler ODE solver. L-stable.
class BackwardEulerSolver : public ODESolver
{
protected:
   Vector k;

public:
   void Init(TimeDependentOperator &f_) override;

   void Step(Vector &x, double &t, double &dt) override;
};


/// Implicit midpoint method. A-stable, not L-stable.
class ImplicitMidpointSolver : public ODESolver
{
protected:
   Vector k;

public:
   void Init(TimeDependentOperator &f_) override;

   void Step(Vector &x, double &t, double &dt) override;
};


/** Two stage, singly diagonal implicit Runge-Kutta (SDIRK) methods;
    the choices for gamma_opt are:
    0 - 3rd order method, not A-stable
    1 - 3rd order method, A-stable, not L-stable (default)
    2 - 2nd order method, L-stable
    3 - 2nd order method, L-stable (has solves outside [t,t+dt]). */
class SDIRK23Solver : public ODESolver
{
protected:
   double gamma;
   Vector k, y;

public:
   SDIRK23Solver(int gamma_opt = 1);

   void Init(TimeDependentOperator &f_) override;

   void Step(Vector &x, double &t, double &dt) override;
};


/** Three stage, singly diagonal implicit Runge-Kutta (SDIRK) method of
    order 4. A-stable, not L-stable. */
class SDIRK34Solver : public ODESolver
{
protected:
   Vector k, y, z;

public:
   void Init(TimeDependentOperator &f_) override;

   void Step(Vector &x, double &t, double &dt) override;
};


/** Three stage, singly diagonal implicit Runge-Kutta (SDIRK) method of
    order 3. L-stable. */
class SDIRK33Solver : public ODESolver
{
protected:
   Vector k, y;

public:
   void Init(TimeDependentOperator &f_) override;

   void Step(Vector &x, double &t, double &dt) override;
};


/** Two stage, explicit singly diagonal implicit Runge-Kutta (ESDIRK) method
    of order 2. A-stable. */
class TrapezoidalRuleSolver : public ODESolver
{
protected:
   Vector k, y;

public:
   virtual void Init(TimeDependentOperator &f_);

   virtual void Step(Vector &x, double &t, double &dt);
};


/** Three stage, explicit singly diagonal implicit Runge-Kutta (ESDIRK) method
    of order 2. L-stable. */
class ESDIRK32Solver : public ODESolver
{
protected:
   Vector k, y, z;

public:
   virtual void Init(TimeDependentOperator &f_);

   virtual void Step(Vector &x, double &t, double &dt);
};


/** Three stage, explicit singly diagonal implicit Runge-Kutta (ESDIRK) method
    of order 3. A-stable. */
class ESDIRK33Solver : public ODESolver
{
protected:
   Vector k, y, z;

public:
   virtual void Init(TimeDependentOperator &f_);

   virtual void Step(Vector &x, double &t, double &dt);
};


/// Generalized-alpha ODE solver from "A generalized-α method for integrating
/// the filtered Navier-Stokes equations with a stabilized finite element
/// method" by K.E. Jansen, C.H. Whiting and G.M. Hulbert.
class GeneralizedAlphaSolver : public ODESolver
{
protected:
   mutable Vector xdot,k,y;
   double alpha_f, alpha_m, gamma;
   int  nstate;

   void SetRhoInf(double rho_inf);
   void PrintProperties(std::ostream &out = mfem::out);
public:

   GeneralizedAlphaSolver(double rho = 1.0) { SetRhoInf(rho); };

   void Init(TimeDependentOperator &f_) override;

   void Step(Vector &x, double &t, double &dt) override;

   int  GetMaxStateSize() override { return 1; };
   int  GetStateSize() override { return nstate; };
   const Vector &GetStateVector(int i) override;
   void GetStateVector(int i, Vector &state) override;
   void SetStateVector(int i, Vector &state) override;
};


/// The SIASolver class is based on the Symplectic Integration Algorithm
/// described in "A Symplectic Integration Algorithm for Separable Hamiltonian
/// Functions" by J. Candy and W. Rozmus, Journal of Computational Physics,
/// Vol. 92, pages 230-256 (1991).

/** The Symplectic Integration Algorithm (SIA) is designed for systems of first
    order ODEs derived from a Hamiltonian.
       H(q,p,t) = T(p) + V(q,t)
    Which leads to the equations:
       dq/dt = dT/dp
       dp/dt = -dV/dq
    In the integrator the operators P and F are defined to be:
       P = dT/dp
       F = -dV/dq
 */
class SIASolver
{
public:
   SIASolver() : F_(NULL), P_(NULL) {}

   virtual void Init(Operator &P, TimeDependentOperator & F);

   virtual void Step(Vector &q, Vector &p, double &t, double &dt) = 0;

   virtual void Run(Vector &q, Vector &p, double &t, double &dt, double tf)
   {
      while (t < tf) { Step(q, p, t, dt); }
   }

   virtual ~SIASolver() {}

protected:
   TimeDependentOperator * F_; // p_{i+1} = p_{i} + dt F(q_{i})
   Operator              * P_; // q_{i+1} = q_{i} + dt P(p_{i+1})

   mutable Vector dp_;
   mutable Vector dq_;
};

/// First Order Symplectic Integration Algorithm
class SIA1Solver : public SIASolver
{
public:
   SIA1Solver() {}
   void Step(Vector &q, Vector &p, double &t, double &dt) override;
};

/// Second Order Symplectic Integration Algorithm
class SIA2Solver : public SIASolver
{
public:
   SIA2Solver() {}
   void Step(Vector &q, Vector &p, double &t, double &dt) override;
};

/// Variable order Symplectic Integration Algorithm (orders 1-4)
class SIAVSolver : public SIASolver
{
public:
   SIAVSolver(int order);
   void Step(Vector &q, Vector &p, double &t, double &dt) override;

private:
   int order_;

   Array<double> a_;
   Array<double> b_;
};



/// Abstract class for solving systems of ODEs: d2x/dt2 = f(x,dx/dt,t)
class SecondOrderODESolver
{
protected:
   /// Pointer to the associated TimeDependentOperator.
   SecondOrderTimeDependentOperator *f;  // f(.,.,t) : R^n x R^n --> R^n
   MemoryType mem_type;

public:
   SecondOrderODESolver() : f(NULL) { mem_type = MemoryType::HOST; }

   /// Associate a TimeDependentOperator with the ODE solver.
   /** This method has to be called:
       - Before the first call to Step().
       - When the dimensions of the associated TimeDependentOperator change.
       - When a time stepping sequence has to be restarted.
       - To change the associated TimeDependentOperator. */
   virtual void Init(SecondOrderTimeDependentOperator &f);

   /** @brief Perform a time step from time @a t [in] to time @a t [out] based
       on the requested step size @a dt [in]. */
   /** @param[in,out] x    Approximate solution.
       @param[in,out] dxdt Approximate rate.
       @param[in,out] t    Time associated with the
                           approximate solution @a x and rate @ dxdt
       @param[in,out] dt   Time step size.

       The following rules describe the common behavior of the method:
       - The input @a x [in] is the approximate solution for the input time
         @a t [in].
       - The input @a dxdt [in] is the approximate rate for the input time
         @a t [in].
       - The input @a dt [in] is the desired time step size, defining the desired
         target time: t [target] = @a t [in] + @a dt [in].
       - The output @a x [out] is the approximate solution for the output time
         @a t [out].
       - The output @a dxdt [out] is the approximate rate for the output time
         @a t [out].
       - The output @a dt [out] is the last time step taken by the method which
         may be smaller or larger than the input @a dt [in] value, e.g. because
         of time step control.
       - The method may perform more than one time step internally; in this case
         @a dt [out] is the last internal time step size.
       - The output value of @a t [out] may be smaller or larger than
         t [target], however, it is not smaller than @a t [in] + @a dt [out], if
         at least one internal time step was performed.
       - The value @a x [out] may be obtained by interpolation using internally
         stored data.
       - In some cases, the contents of @a x [in] may not be used, e.g. when
         @a x [out] from a previous Step() call was obtained by interpolation.
       - In consecutive calls to this method, the output @a t [out] of one
         Step() call has to be the same as the input @a t [in] to the next
         Step() call.
       - If the previous rule has to be broken, e.g. to restart a time stepping
         sequence, then the ODE solver must be re-initialized by calling Init()
         between the two Step() calls. */
   virtual void Step(Vector &x, Vector &dxdt, double &t, double &dt) = 0;

   /// Perform time integration from time @a t [in] to time @a tf [in].
   /** @param[in,out] x    Approximate solution.
       @param[in,out] dxdt Approximate rate.
       @param[in,out] t    Time associated with the approximate solution @a x.
       @param[in,out] dt   Time step size.
       @param[in]     tf   Requested final time.

       The default implementation makes consecutive calls to Step() until
       reaching @a tf.
       The following rules describe the common behavior of the method:
       - The input @a x [in] is the approximate solution for the input time
         @a t [in].
       - The input @a dxdt [in] is the approximate rate for the input time
         @a t [in].
       - The input @a dt [in] is the initial time step size.
       - The output @a dt [out] is the last time step taken by the method which
         may be smaller or larger than the input @a dt [in] value, e.g. because
         of time step control.
       - The output value of @a t [out] is not smaller than @a tf [in]. */
   virtual void Run(Vector &x, Vector &dxdt, double &t, double &dt, double tf)
   {
      while (t < tf) { Step(x, dxdt, t, dt); }
   }

   /// Function for getting and setting the state vectors
   virtual int   GetMaxStateSize() { return 0; };
   virtual int   GetStateSize() { return 0; }
   virtual const Vector &GetStateVector(int i)
   {
      mfem_error("ODESolver has no state vectors");
      Vector *s = NULL; return *s; // Make some compiler happy
   }
   virtual void  GetStateVector(int i, Vector &state)
   {
      mfem_error("ODESolver has no state vectors");
   }
   virtual void  SetStateVector(int i, Vector &state)
   {
      mfem_error("ODESolver has no state vectors");
   }

   virtual ~SecondOrderODESolver() { }
};

/// The classical newmark method.
/// Newmark, N. M. (1959) A method of computation for structural dynamics.
/// Journal of Engineering Mechanics, ASCE, 85 (EM3) 67-94.
class NewmarkSolver : public SecondOrderODESolver
{
private:
   Vector d2xdt2;

   double beta, gamma;
   bool first;

public:
   NewmarkSolver(double beta_ = 0.25, double gamma_ = 0.5) { beta = beta_; gamma = gamma_; };

   void PrintProperties(std::ostream &out = mfem::out);

   void Init(SecondOrderTimeDependentOperator &f_) override;

   void Step(Vector &x, Vector &dxdt, double &t, double &dt) override;
};

class LinearAccelerationSolver : public NewmarkSolver
{
public:
   LinearAccelerationSolver() : NewmarkSolver(1.0/6.0, 0.5) { };
};

class CentralDifferenceSolver : public NewmarkSolver
{
public:
   CentralDifferenceSolver() : NewmarkSolver(0.0, 0.5) { };
};

class FoxGoodwinSolver : public NewmarkSolver
{
public:
   FoxGoodwinSolver() : NewmarkSolver(1.0/12.0, 0.5) { };
};

/// Generalized-alpha ODE solver
/// A Time Integration Algorithm for Structural Dynamics With Improved
/// Numerical Dissipation: The Generalized-α Method
/// J.Chung and G.M. Hulbert,  J. Appl. Mech 60(2), 371-375, 1993
/// https://doi.org/10.1115/1.2900803
/// rho_inf in [0,1]
class GeneralizedAlpha2Solver : public SecondOrderODESolver
{
protected:
   Vector xa,va,aa,d2xdt2;
   double alpha_f, alpha_m, beta, gamma;
   int nstate;

public:
   GeneralizedAlpha2Solver(double rho_inf = 1.0)
   {
      rho_inf = (rho_inf > 1.0) ? 1.0 : rho_inf;
      rho_inf = (rho_inf < 0.0) ? 0.0 : rho_inf;

      alpha_m = (2.0 - rho_inf)/(1.0 + rho_inf);
      alpha_f = 1.0/(1.0 + rho_inf);
      beta    = 0.25*pow(1.0 + alpha_m - alpha_f,2);
      gamma   = 0.5 + alpha_m - alpha_f;
   };

   void PrintProperties(std::ostream &out = mfem::out);

   void Init(SecondOrderTimeDependentOperator &f_) override;

   void Step(Vector &x, Vector &dxdt, double &t, double &dt) override;

   int  GetMaxStateSize() override { return 1; };
   int  GetStateSize() override { return nstate; };
   const Vector &GetStateVector(int i) override;
   void GetStateVector(int i, Vector &state) override;
   void SetStateVector(int i, Vector &state) override;
};

/// The classical midpoint method.
class AverageAccelerationSolver : public GeneralizedAlpha2Solver
{
public:
   AverageAccelerationSolver()
   {
      alpha_m = 0.5;
      alpha_f = 0.5;
      beta    = 0.25;
      gamma   = 0.5;
   };
};

/// HHT-alpha ODE solver
/// Improved numerical dissipation for time integration algorithms
/// in structural dynamics
/// H.M. Hilber, T.J.R. Hughes and R.L. Taylor 1977
/// https://doi.org/10.1002/eqe.4290050306
/// alpha in [2/3,1] --> Defined differently than in paper.
class HHTAlphaSolver : public GeneralizedAlpha2Solver
{
public:
   HHTAlphaSolver(double alpha = 1.0)
   {
      alpha = (alpha > 1.0) ? 1.0 : alpha;
      alpha = (alpha < 2.0/3.0) ? 2.0/3.0 : alpha;

      alpha_m = 1.0;
      alpha_f = alpha;
      beta    = (2-alpha)*(2-alpha)/4;
      gamma   = 0.5 + alpha_m - alpha_f;
   };

};

/// WBZ-alpha ODE solver
/// An alpha modification of Newmark's method
/// W.L. Wood, M. Bossak and O.C. Zienkiewicz 1980
/// https://doi.org/10.1002/nme.1620151011
/// rho_inf in [0,1]
class WBZAlphaSolver : public GeneralizedAlpha2Solver
{
public:
   WBZAlphaSolver(double rho_inf = 1.0)
   {
      rho_inf = (rho_inf > 1.0) ? 1.0 : rho_inf;
      rho_inf = (rho_inf < 0.0) ? 0.0 : rho_inf;

      alpha_f = 1.0;
      alpha_m = 2.0/(1.0 + rho_inf);
      beta    = 0.25*pow(1.0 + alpha_m - alpha_f,2);
      gamma   = 0.5 + alpha_m - alpha_f;
   };

};

}

#endif