# 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:

### Second partial time derivative

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

### 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)

or

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

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.

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

Eqn:A.6 |

Eqn:A.7 |

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

## 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

- Lecture Notes: The FDTD Method – Part I, Dr. Nikolova pp.4-9 (pdf)
- Fdtd Main Index Page (with example code)
- Maxwell’s Equations
- The Yee Algorithm
- 2D UPML Algorithm and Theory
- The 2D FDTD Wave Algorithm and Theory
- The 2D FDTD Wave Hz Algorithm
- Analytical Solution: Infinite Magnetic Line-source
- Spherical and Cartesian coordinates
- Vector Algebra
- Fundamental and Derived Units
- Glossary
- FDTD References
- FDTD Resources

## References

- Taflove2005 – Computational Electrodynamics, 2005

## External Links

- Five Point Stencil wikipedia
- Finite Difference wikipedia