#! /usr/bin/env python
#
def fd1d_heat_implicit_cfl ( k, t_num, t_min, t_max, x_num, x_min, x_max ):

#*****************************************************************************80
#
## FD1D_HEAT_IMPLICIT_CFL: compute the Courant-Friedrichs-Loewy coefficient.
#
#  Discussion:
#
#    The equation to be solved has the form:
#
#      dUdT - k * d2UdX2 = F(X,T)
#
#    over the interval [X_MIN,X_MAX] with boundary conditions
#
#      U(X_MIN,T) = U_X_MIN(T),
#      U(X_MIN,T) = U_X_MAX(T),
#
#    over the time interval [T_MIN,T_MAX] with initial conditions
#
#      U(X,T_MIN) = U_T_MIN(X)
#
#    The code uses the finite difference method to approximate the
#    second derivative in space, and an explicit forward Euler approximation
#    to the first derivative in time.
#
#    The finite difference form can be written as
#
#      U(X,T+dt) - U(X,T)                  ( U(X-dx,T) - 2 U(X,T) + U(X+dx,T) )
#      ------------------  = F(X,T) + k *  ------------------------------------
#               dt                                   dx * dx
#
#    or, assuming we have solved for all values of U at time T, we have
#
#      U(X,T+dt) = U(X,T) + cfl * ( U(X-dx,T) - 2 U(X,T) + U(X+dx,T) ) + dt * F(X,T) 
#
#    Here "cfl" is the Courant-Friedrichs-Loewy coefficient:
#
#      cfl = k * dt / dx / dx
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license. 
#
#  Modified:
#
#    14 April 2017
#
#  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, real K, the heat conductivity coefficient.
#
#    Input, integer T_NUM, the number of time values, including the initial
#    value.
#
#    Input, real T_MIN, T_MAX, the minimum and maximum times.
#
#    Input, integer X_NUM, the number of nodes.
#
#    Input, real X_MIN, X_MAX, the minimum and maximum spatial coordinates.
#
#    Output, real CFL, the Courant-Friedrichs-Loewy coefficient.
#
  x_step = ( x_max - x_min ) / float ( x_num - 1 )
  t_step = ( t_max - t_min ) / float ( t_num - 1 )
  cfl = k * t_step / x_step / x_step

  return cfl