Solving the linear equation. More...

#include <IBVP_double_bundle.h>
#include "../Utils_Fill.h"

Classes
class	CppNoddy::Example::diffusion_double

class	CppNoddy::Example::dd_BC

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

namespace	CppNoddy::Example

Enumerations
enum	{ U , Ud }

Functions
double	CppNoddy::Example::source (const double &x, const double &y, const double &t)

int	main ()

Detailed Description

Solving the linear equation.

$- u_x - u_t + u_{yy} = ( ( 1 - y^2 )( x + t ) - 2 ) e^{-xt}\,,$

subject to and $f(x, y = \pm 1,t ) = 0$ . The solution is computed over a range of for $(x,y) \in [0,10]\times [-1,1]$ and the maximum deviation away from the exact solution $u = (1-y^2) e^{-xt}$ is found. The test fails if this deviation is larger than a set tolerance $\tau$ . The example is solved using the PDE_double_IBVP for problems that are parabolic in 2 coordinates with a BVP in .

Definition in file IBVPLinear.cpp.

Enumeration Type Documentation

◆ anonymous enum

anonymous enum

Enumerator
U
Ud

Definition at line 18 of file IBVPLinear.cpp.

18{ U, Ud };

U

@ U

Definition: IBVPLinear.cpp:18

Ud

@ Ud

Definition: IBVPLinear.cpp:18

Function Documentation

◆ main()

int main ( )

Definition at line 104 of file IBVPLinear.cpp.

{
  cout << "\n";
  cout << "=== double_IBVP: linear diffusive problem ===========\n";
  cout << "\n";
 
  // instantiate the problem
  Example::diffusion_double problem;
  Example::dd_BC BC_bottom;
  Example::dd_BC BC_top;
 
  // domain definition
  double top = 1.0;
  double bottom = -1.0;
  double left = 0.0;
  double right = 10.0;
  // number of (spatial) nodal points
  unsigned ny = 301;
  unsigned nx = 301;
  // time and time step
  double t_end = 1.0;
  double dt = 0.01;
 
  // construct our IBVP
  PDE_double_IBVP<double> diffusion( &problem,
                                     Utility::uniform_node_vector( left, right, nx ),
                                     Utility::uniform_node_vector( bottom, top, ny ),
                                     &BC_bottom, &BC_top );
 
  // initial conditions
  for ( unsigned i = 0; i < nx; ++i )
  {
    for ( unsigned j = 0; j < ny; ++j )
    {
      double y = diffusion.solution().coord( i, j ).second;
      diffusion.solution()( i, j, U ) = ( 1.0 - y * y );
      diffusion.solution()( i, j, Ud ) = - 2 * y ;
    }
  }
 
  double max_error( 0.0 );
  do
  {
    diffusion.update_previous_solution();
    diffusion.step2( dt );
    for ( unsigned i = 0; i < nx; ++i )
    {
      for ( unsigned j = 0; j < ny; ++j )
      {
        double x = diffusion.solution().coord( i, j ).first;
        double y = diffusion.solution().coord( i, j ).second;
        double exact_u = ( 1 - y * y ) * exp( - x * diffusion.t() );
        max_error = max( max_error, abs( exact_u - diffusion.solution()( i, j, U ) ) );
      }
    }
  }
  while ( diffusion.t() < t_end );
 
  const double tol( 1.e-4 );
  // check the BL transpiration vs the known solution
  if ( max_error > tol )
  {
    cout << "\033[1;31;48m  * FAILED \033[0m\n";
    cout << " Difference = " << max_error << "\n";
    return 1;
  }
  else
  {
    cout << "\033[1;32;48m  * PASSED \033[0m\n";
    return 0;
  }
 
}

References CppNoddy::PDE_double_IBVP< _Type >::solution(), CppNoddy::PDE_double_IBVP< _Type >::step2(), CppNoddy::PDE_double_IBVP< _Type >::t(), U, Ud, CppNoddy::Utility::uniform_node_vector(), and CppNoddy::PDE_double_IBVP< _Type >::update_previous_solution().

Classes

Namespaces

Enumerations

Functions

Detailed Description

Enumeration Type Documentation

◆ anonymous enum

Function Documentation

◆ main()