diff options
| -rw-r--r-- | mixed_domain/chapter.tex | 12 |
1 files changed, 7 insertions, 5 deletions
diff --git a/mixed_domain/chapter.tex b/mixed_domain/chapter.tex index 4c4445e..7328651 100644 --- a/mixed_domain/chapter.tex +++ b/mixed_domain/chapter.tex @@ -880,7 +880,7 @@ y(t_0+\Delta) &\approx& y(t_0) + \Delta\frac{d}{dt}y(t_0) \\ We have used the operator $\mathpzc{f}(t_0, y(t_0)) \equiv -Py(t_0)+Q(t_0)$ for simplicity.
-We used the Heun method (also known as the improved Euler method \cite{BlanchardP2006.000}), which
+We used the Heun method (also known as the improved Euler method), which
includes the second order of \autoref{eq:taylor_expansion}.
By doing this we can increase the time step while maintaining the same error tolerance.
For this case, a computationally cheap way to increase the order of the Taylor expansion without
@@ -904,11 +904,11 @@ Note that since $y(t_0+\Delta t) = y(t_0)+\Delta\mathpzc{f}(t_0, y(t_0))+O(\Delt \frac{\Delta^2}{2}\frac{d^2}{dt^2}y(t_0) = \frac{\Delta}{2}\mathpzc{f}\left(t_0, y(t_0)+\Delta\mathpzc{f}(t_0,y(t_0))\right) + O(\Delta^3)
\end{equation}
-which has the same error scaling as truncating the Taylor series at second order. With our second derivative substitution, the second-order Taylor expansion becomes
+which has the same error scaling as truncating the Taylor series at second order. With our second
+derivative substitution, the second-order Taylor expansion becomes
\begin{eqnarray}
y(t_0+\Delta) &\approx& y(t_0) + \frac{\Delta}{2}\left[\mathpzc{f}\left(t_0, y(t_0)\right) + \mathpzc{f}\left(t_0+\Delta, y(t_0)+\Delta\mathpzc{f}\left(t_0,y(t_0)\right)\right)\right] \\
-&=& y(t_0) + \frac{\Delta}{2}\left[-Py(t_0)+Q(t_0)-P\left(y(t_0)+\Delta\left(-Py(t_0)+Q(t_0)\right)\right)+Q(t_0+\Delta)\right] \\
&=& y(t_0) + \frac{\Delta}{2}\left[Q(t_0) + Q(t_0+\Delta) - 2Py(t_0) + P^2\Delta y(t_0)\right]
\end{eqnarray}
@@ -975,7 +975,9 @@ the Scipy stack) to easily decompress and import. \subsection{Simulation package}
-We have built an open source Python package for simulating CMDS. We call it Numerical Integration of Schrödinger's Equation (NISE). It is built on top of the open source SciPy library \cite{JonesEric2011.000}, and is compatible with all versions of Python 2.7 and newer.
+We have built an open source Python package for simulating CMDS. We call it Numerical Integration
+of Schrödinger's Equation (NISE). It is built on top of the open source SciPy library \cite{SciPy},
+and is compatible with all versions of Python 2.7 and newer.
We have included the NISE package in this supplementary information. To use NISE first install Python and SciPy if you haven't already, see \href{https://www.scipy.org/install.html}{https://www.scipy.org/install.html}. Then place the NISE package into a directory within your \texttt{PYTHONPATH}. You should be able to \texttt{import NISE} to run and interact with simulations.
@@ -2026,4 +2028,4 @@ Driven character gives rise to pathway overlap peak shifting in the 2D delay res artificially produces rephasing near pulse overlap. %
Driven character also produces resonances that depend on $\omega_1-\omega_2$ near pulse overlap. %
Determination of the homogeneous and inhomogeneous broadening at ultrashort times is only possible
-by performing correlation analysis in both the frequency and time domain. %
\ No newline at end of file +by performing correlation analysis in both the frequency and time domain. %
|