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Next: Extension to 2 dimensions Up: A new FDTD scheme Previous: Introduction

The new scheme in one dimension

The system to discretize


is replaced by the extended system


where tex2html_wrap_inline497 , tex2html_wrap_inline499 , tex2html_wrap_inline501 , and tex2html_wrap_inline503 are four free additional parameters which will be used in the discretized approximation, either to model a physical effect (some dispersion with tex2html_wrap_inline497 for example), either to adjust some numerical corrections or to model a specific boundary condition. The variables E and tex2html_wrap_inline509 are defined by tex2html_wrap_inline511 where tex2html_wrap_inline513 and tex2html_wrap_inline515 are the components propagating respectively forward and backward along the axis, with identical definitions for B.

The discretized system takes the form




In practice, a numerical correction of the form tex2html_wrap_inline519 . tex2html_wrap_inline521 must be added to the first equation of the set (2)in order to satisfy the static limit tex2html_wrap_inline523 in vacuum.

For tex2html_wrap_inline525 tex2html_wrap_inline483 , the system (2) reduces to (1) because, in vacuum, we have tex2html_wrap_inline529 and tex2html_wrap_inline531 :

For tex2html_wrap_inline545 at a boundary, the system reduces to the usual one-dimensional Sommerfeld outgoing-wave boundary condition and is extended to the second order approximation of the Engquist and Majda outgoing-wave boundary condition in higher dimension.

Figure 1: tex2html_wrap_inline483 (when nonequal to zero) has an effect when using the discretized form of the equations. On the left, it is shown that the short wavelength waves ( tex2html_wrap_inline485 is the mesh size) are damped when tex2html_wrap_inline541 . A benefit of this damping is a tunable reduction of numerical noise, has shown on the right where the response of the system to a heavyside signal is displayed for tex2html_wrap_inline489 (top) and for tex2html_wrap_inline491 (bottom).

next up previous
Next: Extension to 2 dimensions Up: A new FDTD scheme Previous: Introduction

Jean-luc Vay
Tue Jan 13 15:57:00 PST 1998