# Delay Equations

One can obtain a very important speedup by using the option of mesh decimation. Decimation consists in replacing D points of the mesh by the analytical solution of the wave equation with source. This is done starting from the first point on the left of the cavity toward the last one on the right side. This transformation is done by replacing the traveling wave equations by coupled Delay Algebraic Equations (DAE), see  for details.

For the sake of simplicity, we consider here a situation similar to the one described in the previous example of Passive Mode-Locking, namely a two section Fabry-Perot laser with a short saturable absorber section. As we are free to choose different decimation factors in each section, we denote in the following D1 and D2 the decimation factors in the gain and the absorber section, respectively.

 The discretization for the two sections is (N1,N2) = (513, 17). One can see on the left the time trace for the output intensity on the right facet of the absorber section. From top to bottom, the decimation factors are: (D1, D2) = (1, 1) (no decimation), (D1, D2) = (16, 2), (D1, D2) = (32, 4) and (D1, D2) = (32, 8), respectively. Notice that although the transient are, and must be, different, the steady state regime and the predicted pulsewidth are identical. The speedup in the bottom case as compared to the top one is close to 40. This approach allows to bridge time consuming spatially distributed model with the much less resource hungry delay equation approach.

The option of decimation is available only if the mesh lengths N is odd and such that (N − 1)⁄D is a natural number . Finally, if one use power of two for N − 1 and D, each mesh is a subset of the previous one which allows restarting the simulation from the previous ones to study the accuracy of the truncation for increasing values of D.