IBVPDiffusion.cpp File Reference

Solving the heat diffusion equation. More...

#include <IBVP_bundle.h>

Go to the source code of this file.

Classes
class	CppNoddy::Example::Diff_equation
class	CppNoddy::Example::Diff_both_BC

Namespaces
namespace	CppNoddy
	A collection of OO numerical routines aimed at simple (typical) applied problems in continuum mechanics.
namespace	CppNoddy::Example

Enumerations
enum	{ f , fd }

Functions
int	main ()

Detailed Description

Solving the heat diffusion equation.

subject to and with initial condition . The solution is computed over a range of and the maximum deviation away from the analytical solution is found. The test fails if this deviation is larger than a set tolerance . The analytical solution is the Fourier series:

The series is truncated at the -th term, in such a way that the first neglected term is of magnitude .

Definition in file IBVPDiffusion.cpp.

Enumeration Type Documentation

anonymous enum

anonymous enum

Enumerator
f
fd

Definition at line 19 of file IBVPDiffusion.cpp.

{ f, fd };

Function Documentation

main()

int main

(

)

Definition at line 83 of file IBVPDiffusion.cpp.

{
  cout << "\n";
  cout << "=== IBVP: An unsteady diffusion eqn =================\n";
  cout << "\n";
 
  // Diffusion equation
  Example::Diff_equation problem;
  // boundary conditions
  Example::Diff_both_BC BC_both;
 
  // domain definition
  double left = 0.0;
  double right = 1.0;
  // number of (spatial) nodal points
  unsigned ny = 201;
  // time step
  double dt = 0.005;
  // number of time steps
  unsigned max_steps = ( unsigned ) ( 3.0 / dt );
  // test tolerance
  double tol = 1.e-4;
 
  // construct our IBVP
  PDE_IBVP<double> heat( &problem, Utility::uniform_node_vector( left, right, ny ), &BC_both, &BC_both );
 
  for ( unsigned i = 0; i < ny; ++i )
  {
    double y = heat.solution().coord( i );
    heat.solution()( i, f ) = y * ( 1 - y );
    heat.solution()( i, fd ) = 1 - 2 * y;
  }
 
  // maximum difference between numerical and series solution at centre point
  double max_diff( 0.0 );
  // time step
  for ( unsigned i = 1; i < max_steps; ++i )
  {
    // take a time step
    try
    {
      heat.step2( dt );
    }
    catch (const std::runtime_error &error )
    {
      cout << " \033[1;31;48m  * FAILED THROUGH EXCEPTION BEING RAISED \033[0m\n";
      return 1;
    }
 
    // evaluate analytical Fourier series solution at y = 0.5
    unsigned mid = ( ny - 1 ) / 2;
    double y = left + mid * ( right - left ) / ( ny - 1 );
    double u( 0.0 );
    int en( -1 );
    double correction( 0.0 );
    do
    {
      en += 2;
      correction = 8 / ( std::pow( en * M_PI, 3 ) )
                   * std::exp( -std::pow( en * M_PI, 2 ) * heat.coord() ) * std::sin( en * M_PI * y );
      u += correction;
    }
    while ( std::abs( correction ) > tol / 10. );
    // examine the difference between the numerical and series solutions
    max_diff = std::max( max_diff, std::abs( u - heat.solution()( mid, f ) ) );
  }
 
  bool failed = true;
  if ( abs( max_diff ) < tol )
  {
    failed = false;
  }
 
  if ( failed )
  {
    cout << "\033[1;31;48m  * FAILED \033[0m\n";
    return 1;
  }
  else
  {
    cout << "\033[1;32;48m  * PASSED \033[0m\n";
    return 0;
  }
 
}

References CppNoddy::PDE_IBVP< _Type >::coord(), f, fd, CppNoddy::PDE_IBVP< _Type >::solution(), CppNoddy::PDE_IBVP< _Type >::step2(), u, and CppNoddy::Utility::uniform_node_vector().