BVPKarmanJacobian.cpp

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/// \file BVPKarmanJacobian.cpp
/// \ingroup Tests
/// \ingroup BVP
/// Solving the Karman rotating-disk equations for the flow above an
/// infinite rotating disk by applying a user-provided Jacobian
/// \f[ U''(y) = U^2(y) + V(y)U'(y) - W^2(y) \f]
/// \f[ W''(y) = 2U(y)W(y) + V(y)W'(y)  \f]
/// \f[ 2U(y) + V'(y) = 0 \f]
/// with boundary conditions \f$ U(0)=V(0)=0 \f$, \f$ W(0)=1 \f$
/// and \f$ U(\infty ) \to 0 \f$, \f$ W(\infty ) \to 0 \f$.
/// The class constructs and solves the
/// global matrix problem using 2nd-order finite differences.
 
#include <BVP_bundle.h>
 
#include "../Utils_Fill.h"
 
// enumerate the variables in the ODE system
enum { U, Ud, V, W, Wd };
namespace CppNoddy
{
  namespace Example
  {
    /// Define the Karman equations
    class Karman_equations : public Equation_1matrix<double>
    {
    public:
 
      /// The Karman system is a 5th order real system of ODEs
      Karman_equations() : Equation_1matrix<double>( 5 ) {}
 
      /// Define the Karman system
      void residual_fn( const DenseVector<double> &z, DenseVector<double> &f ) const
      {
        // The 5th order system for ( U, U', V, W, W' )
        f[ U ] = z[ Ud ];
        f[ Ud ] = z[ U ] * z[ U ] + z[ V ] * z[ Ud ] - z[ W ] * z[ W ];
        f[ V ] = -2 * z[ U ];
        f[ W ] = z[ Wd ];
        f[ Wd ] = 2 * z[ U ] * z[ W ] + z[ V ] * z[ Wd ];
      }
 
      /// Provide the exact Jacobian of above RHS rather than using finite-differences
      void jacobian( const DenseVector<double> &z, DenseMatrix<double> &jac ) const
      {
        jac( 0, Ud ) = 1.0;
        jac( 1, U ) = 2 * z[ U ];
        jac( 1, Ud ) = z[ V ];
        jac( 1, V ) = z[ Ud ];
        jac( 1, W ) = -2 * z[ W ];
        jac( 2, U ) = -2.0;
        jac( 3, Wd ) = 1.0;
        jac( 4, U ) = 2 * z[ W ];
        jac( 4, V ) = z[ Wd ];
        jac( 4, W ) = 2 * z[ U ];
        jac( 4, Wd ) = z[ V ];
      }
 
      void matrix0( const DenseVector<double>&x, DenseMatrix<double> &m ) const
      {
        Utils_Fill::fill_identity(m);
      }
    };
 
    class Karman_left_BC : public Residual<double>
    {
    public:
      // 3 BCs for 5 unknowns
      Karman_left_BC() : Residual<double> ( 3, 5 ) {}
 
      void residual_fn( const DenseVector<double> &z, DenseVector<double> &B ) const
      {
        B[ 0 ] = z[ U ];
        B[ 1 ] = z[ V ];
        B[ 2 ] = z[ W ] - 1.0;
      }
    };
 
    class Karman_right_BC : public Residual<double>
    {
    public:
      // 2 BCs for 5 unknowns
      Karman_right_BC() : Residual<double> ( 2, 5 ) {}
 
      void residual_fn( const DenseVector<double> &z, DenseVector<double> &B ) const
      {
        B[ 0 ] = z[ U ];
        B[ 1 ] = z[ W ];
      }
    };
 
 
  } // end Example namespace
} // end CppNoddy namespace
 
using namespace CppNoddy;
using namespace std;
 
int main()
{
  cout << "\n";
  cout << "=== BVP: Karman equations with analytic Jacobian ====\n";
  cout << "\n";
 
  Example::Karman_equations problem;
  Example::Karman_left_BC BC_left;
  Example::Karman_right_BC BC_right;
 
  // domain for the ODE problem
  double left = 0.0;
  double right = 20.0;
  // number of nodal points
  int N = 801;
  // mesh
  DenseVector<double> nodes( Utility::power_node_vector( left, right, N, 1.2 ) );
 
  // pass it to the ode
  ODE_BVP<double> ode( &problem, nodes, &BC_left, &BC_right );
 
  // set the solution to the initial guess
  for ( int i = 0; i < N; ++i )
  {
    double y = ode.solution().coord( i );
    ode.solution()( i, U ) = 0.0;
    ode.solution()( i, Ud ) = 0.0;
    ode.solution()( i, V ) = 0.0;
    ode.solution()( i, W ) = exp( -y );
    ode.solution()( i, Wd ) = -exp( -y );
  }
 
  try
  {
    ode.solve2();
  }
  catch ( const std::runtime_error &error )
  {
    cout << " \033[1;31;48m  * FAILED THROUGH EXCEPTION BEING RAISED \033[0m\n";
    return 1;
  }
 
  const double tol( 1.e-4 );
  // check the BL transpiration vs the known solution
  if ( abs( ode.solution()( N - 1, V ) + 0.88447 ) > tol )
  {
    cout << "\033[1;31;48m  * FAILED \033[0m\n";
    cout << " Difference = " << abs( ode.solution()( N - 1, V ) + 0.88447 ) << "\n";
    return 1;
  }
  else
  {
    cout << "\033[1;32;48m  * PASSED \033[0m\n";
    return 0;
  }
 
}