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

using namespace std;

# include "fd1d_wave.hpp"

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

double fd1d_wave_alpha ( int x_num, double x1, double x2, int t_num, double t1, 
  double t2, double c )

//****************************************************************************80
//
//  Purpose:
//
//    FD1D_WAVE_ALPHA computes ALPHA for the 1D wave equation.
//
//  Discussion:
//
//    The explicit timestepping procedure uses the quantity ALPHA, which
//    is determined by this function.
//
//    If the spatial region bounds are X1 <= X <= X2, containing X_NUM equally
//    spaced nodes, including the endpoints, and the time domain similarly
//    extends from T1 <= T <= T2 containing T_NUM equally spaced time values,
//    then
//
//      ALPHA = C * DT / DX
//            = C * ( ( T2 - T1 ) / ( T_NUM - 1 ) )
//                / ( ( X2 - X1 ) / ( X_NUM - 1 ) ).
//
//    For a stable computation, it must be the case that ALPHA < 1.
//
//    If ALPHA 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.  In such a case, the user must reduce the time step 
//    size appropriately.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    24 January 2012
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    George Lindfield, John Penny,
//    Numerical Methods Using MATLAB,
//    Second Edition,
//    Prentice Hall, 1999,
//    ISBN: 0-13-012641-1,
//    LC: QA297.P45.
//
//  Parameters:
//
//    Input, int X_NUM, the number of nodes in the X direction.
//
//    Input, double X1, X2, the first and last X coordinates.
//
//    Input, int T_NUM, the number of time steps, including the 
//    initial condition.
//
//    Input, double T1, T2, the first and last T coordinates.
//
//    Input, double C, a parameter which gives the speed of waves.
//
//    Output, double FD1D_WAVE_ALPHA, the stability coefficient.
//
{
  double alpha;
  double t_delta;
  double x_delta;

  t_delta = ( t2 - t1 ) / ( double ) ( t_num - 1 );
  x_delta = ( x2 - x1 ) / ( double ) ( x_num - 1 );
  alpha = c * t_delta / x_delta;

  cout << "\n";
  cout << "  Stability condition ALPHA = C * DT / DX = " << alpha << "\n";

  if ( 1.0 < fabs ( alpha ) )
  {
    cerr << "\n";
    cerr << "FD1D_WAVE_ALPHA - Warning!\n";
    cerr << "  The stability condition |ALPHA| <= 1 fails.\n";
    cerr << "  Computed results are liable to be inaccurate.\n";
  }

  return alpha;
}
//****************************************************************************80

double *fd1d_wave_start ( int x_num, double x_vec[], double t, double t_delta, 
  double alpha, double u_x1 ( double t ), double u_x2 ( double t ), 
  double *ut_t1 ( int x_num, double x_vec[] ), double u1[] )

//****************************************************************************80
//
//  Purpose:
//
//    FD1D_WAVE_START takes the first step for the wave equation.
//
//  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 MIT license.
//
//  Modified:
//
//    24 January 2012
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    George Lindfield, John Penny,
//    Numerical Methods Using MATLAB,
//    Second Edition,
//    Prentice Hall, 1999,
//    ISBN: 0-13-012641-1,
//    LC: QA297.P45.
//
//  Parameters:
//
//    Input, int X_NUM, the number of nodes in the X direction.
//
//    Input, double X_VEC[X_NUM], the coordinates of the nodes.
//
//    Input, double T, the time after the first step has been taken.
//    In other words, T = T1 + T_DELTA.
//
//    Input, double T_DELTA, the time step.
//
//    Input, double ALPHA, the stability coefficient, computed 
//    by FD1D_WAVE_ALPHA.
//
//    Input, double U_X1 ( double T ), U_X2 ( double T ), functions for the left and 
//    right boundary conditions.
//
//    Input, double *UT_T1 ( int X_NUM, double X_VEC[] ), the function that 
//    evaluates dUdT at the initial time.
//
//    Input, real U1[X_NUM], the initial condition.
//
//    Output, real FD1D_WAVE_START[X_NUM], the solution at the first time step.
//
{
  int j;
  double *u2;
  double *ut;

  ut = ut_t1 ( x_num, x_vec );

  u2 = new double[x_num];

  u2[0] = u_x1 ( t );

  for ( j = 1; j < x_num - 1; j++ )
  {
    u2[j] =           alpha * alpha   * u1[j+1] / 2.0
            + ( 1.0 - alpha * alpha ) * u1[j] 
            +         alpha * alpha   * u1[j-1] / 2.0
            +         t_delta         * ut[j];
  }

  u2[x_num-1] = u_x2 ( t );

  delete [] ut;

  return u2;
}
//****************************************************************************80

double *fd1d_wave_step ( int x_num, double t, double alpha, 
  double u_x1 ( double t ), double u_x2 ( double t ), double u1[], double u2[] )

//****************************************************************************80
//
//  Purpose:
//
//    FD1D_WAVE_STEP computes a step of the 1D wave equation.
//
//  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 MIT license.
//
//  Modified:
//
//    24 January 2012
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    George Lindfield, John Penny,
//    Numerical Methods Using MATLAB,
//    Second Edition,
//    Prentice Hall, 1999,
//    ISBN: 0-13-012641-1,
//    LC: QA297.P45.
//
//  Parameters:
//
//    Input, int X_NUM, the number of nodes in the X direction.
//
//    Input, double T, the new time, that is, the current 
//    time + T_DELTA.
//
//    Input, double ALPHA, the stability coefficient, computed 
//    by FD1D_WAVE_ALPHA.
//
//    Input, double U_X1 ( double T ), U_X2 ( double T ), functions for the left and 
//    right boundary conditions.
//
//    Input, double U1[X_NUM], the solution at the old time.
//
//    Input, double U2[X_NUM], the solution at the current time.
//
//    Output, double FD1D_WAVE_STEP[X_NUM], the solution at the new time.
//
{
  int j;
  double *u3;

  u3 = new double[x_num];

  u3[0] = u_x1 ( t );

  for ( j = 1; j < x_num - 1; j++ )
  {
    u3[j] =                 alpha * alpha   * u2[j+1] 
            + 2.0 * ( 1.0 - alpha * alpha ) * u2[j]
            +               alpha * alpha   * u2[j-1] 
            -                                 u1[j];
  }

  u3[x_num-1] = u_x2 ( t );

  return u3;
}
//****************************************************************************80

double *piecewise_linear ( int nd, double xd[], double yd[], int nv, double xv[] )

//****************************************************************************80
//
//  Purpose:
//
//    PIECEWISE_LINEAR evaluates a piecewise linear spline.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    24 January 2012
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int ND, the number of data points.
//
//    Input, double XD[ND], YD[ND], the data values.
//
//    Input, int NV, the number of evaluation points.
//
//    Input, double XV[NV], the evaluation arguments.
//
//    Output, double PIECEWISE_LINEAR[NV], the values.
//
{
  int id;
  int iv;
  double *yv;

  yv = new double[nv];

  for ( iv = 0; iv < nv; iv++ )
  {
    if ( xv[iv] < xd[0] )
    {
      yv[iv] = yd[0];
    }
    else if ( xd[nd-1] < xv[iv] )
    {
      yv[iv] = yd[nd-1];
    }
    else
    {
      for ( id = 1; id < nd; id++ )
      {
        if ( xv[iv] < xd[id] )
        {
          yv[iv] = ( ( xd[id] - xv[iv]            ) * yd[id-1] 
                   + (          xv[iv] - xd[id-1] ) * yd[id] ) 
                   / ( xd[id]          - xd[id-1] );
          break;
        }
      }
    }
  }
  return yv;
}
//****************************************************************************80

void r8mat_write ( string output_filename, int m, int n, double table[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_WRITE writes an R8MAT file.
//
//  Discussion:
//
//    An R8MAT is an array of R8's.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    29 June 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string OUTPUT_FILENAME, the output filename.
//
//    Input, int M, the spatial dimension.
//
//    Input, int N, the number of points.
//
//    Input, double TABLE[M*N], the data.
//
{
  int i;
  int j;
  ofstream output;
//
//  Open the file.
//
  output.open ( output_filename.c_str ( ) );

  if ( !output )
  {
    cerr << "\n";
    cerr << "R8MAT_WRITE - Fatal error!\n";
    cerr << "  Could not open the output file.\n";
    exit ( 1 );
  }
//
//  Write the data.
//
  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      output << "  " << setw(24) << setprecision(16) << table[i+j*m];
    }
    output << "\n";
  }
//
//  Close the file.
//
  output.close ( );

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

double *r8vec_linspace_new ( int n, double a_first, double a_last )

//****************************************************************************80
//
//  Purpose:
//
//    R8VEC_LINSPACE_NEW creates a vector of linearly spaced values.
//
//  Discussion:
//
//    An R8VEC is a vector of R8's.
//
//  Licensing:
//
//    This code is distributed under the MIT license.
//
//  Modified:
//
//    29 March 2011
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Input, double A_FIRST, A_LAST, the first and last entries.
//
//    Output, double R8VEC_LINSPACE_NEW[N], a vector of linearly spaced data.
//
{
  double *a;
  int i;

  a = new double[n];

  if ( n == 1 )
  {
    a[0] = ( a_first + a_last ) / 2.0;
  }
  else
  {
    for ( i = 0; i < n; i++ )
    {
      a[i] = ( ( double ) ( n - 1 - i ) * a_first 
             + ( double ) (         i ) * a_last ) 
             / ( double ) ( n - 1     );
    }
  }
  return a;
}
//****************************************************************************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 MIT license.
//
//  Modified:
//
//    08 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    None
//
{
# define TIME_SIZE 40

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

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

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

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

  return;
# undef TIME_SIZE
}