03-2. Equation Definitions

In this section we will discuss how the equations are set in the file src/equations/definitions.f90. Depending on the settings set in settings.input (primarily Eqn_number, Domain_number, Time_switch, and Non_Linear_switch), different subroutines will be called.

Explanation of File Contents

The first subroutine to consider is equation1_linear. It sets up terms of the linear equation:

$A_x(x,y) \frac{\partial^2 u}{\partial x^2} + B_x(x,y) \frac{\partial u}{\partial x } + A_y(x,y) \frac{\partial^2 u}{\partial y^2} + B_y(x,y) \frac{\partial u}{\partial y } + C(x,y) u = D(x,y) \frac{\partial u }{\partial t}$

This is subroutine will always called regardless of Eqn_number being 1 or 2; if this value is set to 2 then equation2_linear will also be called. The subroutine equation2_linear is exactly the same but solves for $v$ rather than $u$ . Ax, Bx, C and D will always be set. Ay and By will only be set if Domain_number is 2.

   Subroutine equation1_linear(x, y, Ax, Bx, Ay, By, C, D)
      real(dp), intent(in) :: x, y ! xposition in the domain
      real(dp), intent(out) :: Ax, Bx, Ay, By, C, D
      real(dp) :: epsi

      epsi = 0.05d0

      Ax = epsi
      Bx = 0.d0

      Ay = epsi
      By = 0.d0

      C = 0.d0
      D = 1.d0

   End Subroutine equation1_linear

This subroutine equation1_BC_X_Bot is a boundary condition and is identical in form to the subroutine above. However it is only called when $x$ is at the lower boundary xl. There are four boundary condition subroutines for each equation:

For $u$ :

equation1_BC_X_Bot and equation1_BC_X_Top are always called and provide conditions for the xl and xr boundaries, respectively.
equation1_BC_Y_Bot and equation1_BC_Y_Top are only called when Domain_number is 2, and provide conditions for the yl and yr boundaries, respectively.

For $v$ (only called when Eqn_number is 2):

equation2_BC_X_Bot and equation2_BC_X_Top are always called and provide conditions for the xl and xr boundaries, respectively.
equation2_BC_Y_Bot and equation2_BC_Y_Top are only called when Domain_number is 2, and provide conditions for the yl and yr boundaries, respectively.

   Subroutine equation1_BC_X_Bot(x, y, Ax, Bx, Ay, By, C, D)
      real(dp), intent(in) :: x, y ! xposition in the domain
      real(dp), intent(out) :: Ax, Bx, Ay, By, C, D

      Ax = 0.d0
      Bx = 1.d0

      Ay = 0.d0
      By = 0.d0

      C = 0.d0
      D = 0.d0

   End Subroutine equation1_BC_X_Bot

When Time_switch is 1, the following is called to set the initial condition. There is a similar subroutine for $v$ that is called if Eqn_number is 2.

   Subroutine equation1_initial_condition(x, y, IC)
      real(dp), intent(in) :: x, y ! xposition in the domain
      real(dp), intent(out) :: IC

      IC = x + y 

   End Subroutine equation1_initial_condition

When Non_linear is 1, the following is called to set the non-linear terms. There is a similar subroutine for $v$ that is called if Eqn_number is 2.
If Eqn_number is 1 then only F and Fu are called.
If Eqn_number is 2 then F, Fu and Fv are called.

   Subroutine equation1_non_linear(x, y, u, v, F, Fu, Fv)
      real(dp), intent(in) :: x, y, u, v
      real(dp), intent(out) :: F, Fu, Fv
      !!! F is a function of u *Fu is dF(u,v)/du and Fv is dF(u,v)/dv
      !!!   Non linear terms do not effect boundaries

      F = u*u*v *u
      Fu = 2.d0*u*v *1.d0 
      Fv = u*u
   End Subroutine equation1_non_linear