ode_stiff.cpp¶

A Stiff Ode: Example and Test¶

Define \(x : \B{R} \rightarrow \B{R}^2\) by

\begin{eqnarray} x_0 (0) & = & 1 \\ x_1 (0) & = & 0 \\ x_0^\prime (t) & = & - a_0 x_0 (t) \\ x_1^\prime (t) & = & + a_0 x_0 (t) - a_1 x_1 (t) \end{eqnarray}

If \(a_0 \gg a_1 > 0\), this is a stiff Ode and the analytic solution is

\begin{eqnarray} x_0 (t) & = & \exp( - a_0 t ) \\ x_1 (t) & = & a_0 [ \exp( - a_1 t ) - \exp( - a_0 t ) ] / ( a_0 - a_1 ) \end{eqnarray}

The example tests Rosen34 using the relations above:

# include <cppad/cppad.hpp>

// To print the comparison, change the 0 to 1 on the next line.
# define CPPAD_ODE_STIFF_PRINT 0

namespace {
   // --------------------------------------------------------------
   class Fun {
   private:
      CPPAD_TESTVECTOR(double) a;
   public:
      // constructor
      Fun(const CPPAD_TESTVECTOR(double)& a_) : a(a_)
      { }
      // compute f(t, x)
      void Ode(
         const double                    &t,
         const CPPAD_TESTVECTOR(double) &x,
         CPPAD_TESTVECTOR(double)       &f)
      {  f[0]  = - a[0] * x[0];
         f[1]  = + a[0] * x[0] - a[1] * x[1];
      }
      // compute partial of f(t, x) w.r.t. t
      void Ode_ind(
         const double                    &t,
         const CPPAD_TESTVECTOR(double) &x,
         CPPAD_TESTVECTOR(double)       &f_t)
      {  f_t[0] = 0.;
         f_t[1] = 0.;
      }
      // compute partial of f(t, x) w.r.t. x
      void Ode_dep(
         const double                    &t,
         const CPPAD_TESTVECTOR(double) &x,
         CPPAD_TESTVECTOR(double)       &f_x)
      {  f_x[0] = -a[0];
         f_x[1] = 0.;
         f_x[2] = +a[0];
         f_x[3] = -a[1];
      }
   };
   // --------------------------------------------------------------
   class RungeMethod {
   private:
      Fun F;
   public:
      // constructor
      RungeMethod(const CPPAD_TESTVECTOR(double) &a_) : F(a_)
      { }
      void step(
         double                     ta ,
         double                     tb ,
         CPPAD_TESTVECTOR(double) &xa ,
         CPPAD_TESTVECTOR(double) &xb ,
         CPPAD_TESTVECTOR(double) &eb )
      {  xb = CppAD::Runge45(F, 1, ta, tb, xa, eb);
      }
      size_t order(void)
      {  return 5; }
   };
   class RosenMethod {
   private:
      Fun F;
   public:
      // constructor
      RosenMethod(const CPPAD_TESTVECTOR(double) &a_) : F(a_)
      { }
      void step(
         double                     ta ,
         double                     tb ,
         CPPAD_TESTVECTOR(double) &xa ,
         CPPAD_TESTVECTOR(double) &xb ,
         CPPAD_TESTVECTOR(double) &eb )
      {  xb = CppAD::Rosen34(F, 1, ta, tb, xa, eb);
      }
      size_t order(void)
      {  return 4; }
   };
}

bool OdeStiff(void)
{  bool ok = true;     // initial return value

   CPPAD_TESTVECTOR(double) a(2);
   a[0] = 1e3;
   a[1] = 1.;
   RosenMethod rosen(a);
   RungeMethod runge(a);
   Fun          gear(a);

   CPPAD_TESTVECTOR(double) xi(2);
   xi[0] = 1.;
   xi[1] = 0.;

   CPPAD_TESTVECTOR(double) eabs(2);
   eabs[0] = 1e-6;
   eabs[1] = 1e-6;

   CPPAD_TESTVECTOR(double) ef(2);
   CPPAD_TESTVECTOR(double) xf(2);
   CPPAD_TESTVECTOR(double) maxabs(2);
   size_t                nstep;

   size_t k;
   for(k = 0; k < 3; k++)
   {
      size_t M    = 5;
      double ti   = 0.;
      double tf   = 1.;
      double smin = 1e-7;
      double sini = 1e-7;
      double smax = 1.;
      double scur = .5;
      double erel = 0.;

      if( k == 0 )
      {  xf = CppAD::OdeErrControl(rosen, ti, tf,
         xi, smin, smax, scur, eabs, erel, ef, maxabs, nstep);
      }
      else if( k == 1 )
      {  xf = CppAD::OdeErrControl(runge, ti, tf,
         xi, smin, smax, scur, eabs, erel, ef, maxabs, nstep);
      }
      else if( k == 2 )
      {  xf = CppAD::OdeGearControl(gear, M, ti, tf,
         xi, smin, smax, sini, eabs, erel, ef, maxabs, nstep);
      }
      double x0 = exp(-a[0]*tf);
      ok &= CppAD::NearEqual(x0, xf[0], 0., eabs[0]);
      ok &= CppAD::NearEqual(0., ef[0], 0., eabs[0]);

      double x1 = a[0] *
         (exp(-a[1]*tf) - exp(-a[0]*tf))/(a[0] - a[1]);
      ok &= CppAD::NearEqual(x1, xf[1], 0., eabs[1]);
      ok &= CppAD::NearEqual(0., ef[1], 0., eabs[0]);
# if CPPAD_ODE_STIFF_PRINT
      const char* method[]={ "Rosen34", "Runge45", "Gear5" };
      std::cout << std::endl;
      std::cout << "method     = " << method[k] << std::endl;
      std::cout << "nstep      = " << nstep  << std::endl;
      std::cout << "x0         = " << x0 << std::endl;
      std::cout << "xf[0]      = " << xf[0] << std::endl;
      std::cout << "x0 - xf[0] = " << x0 - xf[0] << std::endl;
      std::cout << "ef[0]      = " << ef[0] << std::endl;
      std::cout << "x1         = " << x1 << std::endl;
      std::cout << "xf[1]      = " << xf[1] << std::endl;
      std::cout << "x1 - xf[1] = " << x1 - xf[1] << std::endl;
      std::cout << "ef[1]      = " << ef[1] << std::endl;
# endif
   }

   return ok;
}