Numerical Methods

The FDTD algorithm is one in a general class of grid-based differential time-domain numerical modeling methods.

Maxwell’s equations (in partial differential form) are discretized using central-difference approximations to the space and time partial derivatives.

The numerical methods used in FDTD are shown below.

Contents

Finite Difference Equations

Taflove2005, pages 23-26 (especially page 24)

Second partial space derivative

The finite difference equations for the second order accurate central-difference approximation
to the second partial space derivative:

0b130348e5f7b2611c0e5ef7cebddc78.png Eqn:A.1

Second partial time derivative

The finite difference equations for the second order accurate central-difference approximation to the second partial time derivative:

0a18b6228404196bd3d1d682496a3f1e.png Eqn:A.2

First partial space derivative

The finite difference equations for the second order accurate central-difference approximation
to the first partial space derivative (Taflove2005, page 62):
and Lecture Notes: The FDTD Method – Part I, Dr. Nikolova p.4 (pdf)

9aeb87367ed4d3035e2764ea7802bc3f.png Eqn:A.3

or

f469b94075213e218cbc912d9ed94e8f.png Eqn:A.4

Second partial space derivative with respect to x and y

The finite difference equations for the second order accurate central-difference approximation
to the second partial space derivative with respect to x and y:

Using Ey as an example. Calculate Ey at x=0.5, y=0, z=0

wdata/yee6.jpg Calculate the Second Partial derivative of Ey (at x=0.5,y=0,z=0) with respect to X and Y

Using Equation A.3 first evaluate with respect to X.

577ae1092eb5a712af3201d28de1c5df.png Eqn:A.5

And again using Equation A.3 evaluate Equation A.5 with respect to Y.

646f0ff157a0065ba7b27a32f212660f.png Eqn:A.6
3cad6efd03d5a5bb6c6c210b251326a1.png Eqn:A.7

Which gives the final equation (assume dy = dx):

35ce97671442d51bcfbc0c6b8e566323.png Eqn:A.8

Stencils

       o x,y+1                      o        o        o       
       |                            x-1,y+1  |x,y+1   x+1,y+1 
       |                                     |                
x-1,y  |      x+1,y                 x-1,y    |        x+1,y   
o------o------o                     o--------o--------o       
       |x,y                                  |x,y             
       |                                     |                
       |                            x-1,y-1  |x,y-1   x+1,y-1 
       o x,y-1                      o        o        o       
                                                              
   5 point (2-D)                          9 point (2-D)       
   (7 point, 3-D)                      (27 point, 3-D)

See Also

References

External Links