string_simulation.cpp

# include <cstdlib>
# include <iostream>
# include <iomanip>
# include <cmath>
# include <ctime>
# include <fstream>

using namespace std;

int main ( );
double f ( double x );
double g ( double x );
void timestamp ( );

//****************************************************************************80

int main ( )

//****************************************************************************80
//
//  Purpose:
//
//    MAIN is the main program for STRING_SIMULATION.
//
//  Discussion:
//
//    This program solves the 1D wave equation of the form:
//
//      Utt = c^2 Uxx
//
//    over the spatial interval [X1,X2] and time interval [T1,T2],
//    with initial conditions:
//
//      U(T1,X)  = U_T1(X),
//      Ut(T1,X) = UT_T1(X),
//
//    and boundary conditions of Dirichlet type:
//
//      U(T,X1) = U_X1(T),
//      U(T,X2) = U_X2(T).
//
//    The value C represents the propagation speed of waves.
//
//    The program uses the finite difference method, and marches
//    forward in time, solving for all the values of U at the next
//    time step by using the values known at the previous two time steps.
//
//    Central differences may be used to approximate both the time
//    and space derivatives in the original differential equation.
//
//    Thus, assuming we have available the approximated values of U
//    at the current and previous times, we may write a discretized
//    version of the wave equation as follows:
//
//      Uxx(T,X) = ( U(T,   X+dX) - 2 U(T,X) + U(T,   X-dX) ) / dX^2
//      Utt(T,X) = ( U(T+dt,X   ) - 2 U(T,X) + U(T-dt,X   ) ) / dT^2
//
//    If we multiply the first term by C^2 and solve for the single
//    unknown value U(T+dt,X), we have:
//
//      U(T+dT,X) =        (     C^2 * dT^2 / dX^2 ) * U(T,   X+dX)
//                  +  2 * ( 1 - C^2 * dT^2 / dX^2 ) * U(T,   X   )
//                  +      (     C^2 * dT^2 / dX^2 ) * U(T,   X-dX)
//                  -                                  U(T-dT,X   )
//
//    (Equation to advance from time T to time T+dT, except for FIRST step)
//
//    However, on the very first step, we only have the values of U
//    for the initial time, but not for the previous time step.
//    In that case, we use the initial condition information for dUdT
//    which can be approximated by a central difference that involves
//    U(T+dT,X) and U(T-dT,X):
//
//      dU/dT(T,X) = ( U(T+dT,X) - U(T-dT,X) ) / ( 2 * dT )
//
//    and so we can estimate U(T-dT,X) as
//
//      U(T-dT,X) = U(T+dT,X) - 2 * dT * dU/dT(T,X)
//
//    If we replace the "missing" value of U(T-dT,X) by the known values
//    on the right hand side, we now have U(T+dT,X) on both sides of the
//    equation, so we have to rearrange to get the formula we use
//    for just the first time step:
//
//      U(T+dT,X) =   1/2 * (     C^2 * dT^2 / dX^2 ) * U(T,   X+dX)
//                  +       ( 1 - C^2 * dT^2 / dX^2 ) * U(T,   X   )
//                  + 1/2 * (     C^2 * dT^2 / dX^2 ) * U(T,   X-dX)
//                  +  dT *                         dU/dT(T,   X   )
//
//    (Equation to advance from time T to time T+dT for FIRST step.)
//
//    It should be clear now that the quantity ALPHA = C * DT / DX will affect
//    the stability of the calculation.  If it is greater than 1, then
//    the middle coefficient 1-C^2 DT^2 / DX^2 is negative, and the
//    sum of the magnitudes of the three coefficients becomes unbounded.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    25 December 2012
//
//  Author:
//
//    John Burkardt
//
//  Local Parameters:
//
//    Local, double ALPHA, the CFL stability parameter.
//
//    Local, double C, the wave speed.
//
//    Local, double DT, the time step.
//
//    Local, double DX, the spatial step.
//
//    Local, int M, the number of time steps.
//
//    Local, int N, the number of spatial intervals.
//
//    Local, double T1, T2, the initial and final times.
//
//    Local, double U[M+1][N+1], the computed solution.
//
//    Local, double X1, X2, the left and right spatial endpoints.
//
{
# define m 30
# define n 40

  double alpha;
  double c = 0.25;
  ofstream command_unit;
  ofstream data_unit;
  double dt;
  double dx;
  int i;
  int j;
  double t;
  double t1 = 0.0;
  double t2 = 3.0;
  double u[m+1][n+1];
  double x;
  double x1 = 0.0;
  double x2 = 1.0;

  timestamp ( );
  cout << "\n";
  cout << "STRING_SIMULATION:\n";
  cout << "  C++ version\n";
  cout << "  Simulate the behavior of a vibrating string.\n";

  dx = ( x2 - x1 ) / ( double ) n;
  dt = ( t2 - t1 ) / ( double ) m;
  alpha = pow ( c * dt / dx, 2 );
  cout << "  ALPHA = ( C * dT / dX )^2 = " << alpha << "\n";
//
//  Warn the user if ALPHA will cause an unstable computation.
//
  if ( 1.0 < alpha )
  {
    cout << "\n";
    cout << "  Warning!\n";
    cout << "  ALPHA is greater than 1.\n";
    cout << "  The computation is unstable.\n";
  }
//
//  Time step 0: 
//  Use the initial condition for U.
//
  u[0][0] = 0.0;
  for ( j = 1; j < n; j++ )
  {
    x = j * dx;
    u[0][j] = f ( x );
  }
  u[0][n] = 0.0;
//
//  Time step 1:
//  Use the initial condition for dUdT.
//
  u[1][0] = 0.0;
  for ( j = 1; j < n; j++ )
  {
    x = j * dx;
    u[1][j] = 
        ( alpha / 2.0 ) * u[0][j-1]
      + ( 1.0 - alpha ) * u[0][j] 
      + ( alpha / 2.0 ) * u[0][j+1]
      + dt * g ( x );
  }
  u[1][n] = 0.0;
//
//  Time steps 2 through M:
//
  for ( i = 2; i <= m; i++ )
  {
    u[i][0] = 0.0;
    for ( j = 1; j < n; j++ )
    {
      u[i][j] = 
                        alpha   * u[i-1][j-1]
        + 2.0 * ( 1.0 - alpha ) * u[i-1][j] 
        +               alpha   * u[i-1][j+1]
        -                         u[i-2][j];
    }
    u[i][n] = 0.0;
  }
//
//  Write data file.
//
  data_unit.open ( "string_data.txt" );

  for ( i = 0; i <= m; i++ )
  {
    t = i * dt;
    for ( j = 0; j <= n; j++ )
    {
      x = j * dx;
      data_unit << "  " << x
                << "  " << t
                << "  " << u[i][j] << "\n";
    }
    data_unit << "\n";
  }
  data_unit.close ( );

  cout << "\n";
  cout << "  Plot data written to the file \"string_data.txt\".\n";
//
//  Write gnuplot command file.
//
  command_unit.open ( "string_commands.txt" );

  command_unit << "set term png\n";
  command_unit << "set output \"string.png\"\n";
  command_unit << "set grid\n";
  command_unit << "set style data lines\n";
  command_unit << "unset key\n";
  command_unit << "set xlabel '<---X--->'\n";
  command_unit << "set ylabel '<---Time--->'\n";
  command_unit << "splot \"string_data.txt\" using 1:2:3 with lines\n";
  command_unit << "quit\n";

  command_unit.close ( );

  cout << "  Gnuplot command data written to the file \"string_commands.txt\".\n";
//
//  Terminate.
//
  cout << "\n";
  cout << "STRING_SIMULATION:\n";
  cout << "  Normal end of execution.\n";
  cout << "\n";
  timestamp ( );

  return 0;
# undef m
# undef n
}
//****************************************************************************80

double f ( double x )

//****************************************************************************80
//
//  Purpose:
//
//    F supplies the initial condition.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    01 December 2012
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, the location.
//
//    Output, double F, the value of the solution at time 0 and location X.
//
{
  const double pi = 3.14159265;
  double value;

  if ( 0.25 <= x && x <= 0.50 )
  {
    value = ( x - 0.25 ) * ( 0.50 - x );
  }
  else
  {
    value = 0.0;
  }

  return value;
}
//****************************************************************************80

double g ( double x )

//****************************************************************************80
//
//  Purpose:
//
//    G supplies the initial derivative.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    01 December 2012
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, the location.
//
//    Output, double G, the value of the time derivative of the solution 
//    at time 0 and location X.
//
{
  double value;

  value = 0.0;

  return value;
}
//****************************************************************************80

void timestamp ( )

//****************************************************************************80
//
//  Purpose:
//
//    TIMESTAMP prints the current YMDHMS date as a time stamp.
//
//  Example:
//
//    31 May 2001 09:45:54 AM
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    01 December 2012
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    None
//
{
# define TIME_SIZE 40

  static char time_buffer[TIME_SIZE];
  const struct std::tm *tm_ptr;
  size_t len;
  std::time_t now;

  now = std::time ( NULL );
  tm_ptr = std::localtime ( &now );

  len = std::strftime ( time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm_ptr );

  std::cout << time_buffer << "\n";

  return;
# undef TIME_SIZE
}