### Principal investigators

Prof. Dr. Marlis Hochbruck | (7/2015 - ) | |

Prof. Dr. Tobias Jahnke | (7/2015 - ) | |

Prof. Dr. Christian Lubich | (7/2015 - ) |

### Project summary

Evolution equations with highly oscillatory solutions are abundant in the simulation of wave phenomena. When standard numerical integrators are applied to such problems, an acceptable accuracy can only be achieved if the size of the time steps is small compared to the inverse of the highest frequency arising in the system. As a consequence, many time steps have to be carried out, which incurs huge computational costs and thus makes traditional methods prohibitively inefficient. The central goals of this project are the construction, analysis, and efficient implementation of numerical integrators for highly oscillatory problems arising in simulations of wave phenomena. Our main objective is to prove error bounds which guarantee convergence of the method under realistic regularity assumptions and without any severe step size restrictions.

- Adiabatic integrators for dispersion-managed nonlinear Schrödinger equations. Continue reading. Collapse content.
Data transmission through an optical fiber with strong dispersion management is modeled by the semilinear Schrödinger equation \begin{equation}\label{DMNLS} \partial_t u(t,x) = \tfrac{\mathrm{i}}{\varepsilon} \gamma\left(\tfrac{t}{\varepsilon}\right) \partial_x^2 u(t,x)+\mathrm{i} |u(t,x)|^2 u(t,x)\,, \quad t\in[0,T]\,\end{equation} with a small parameter \(0\lt\varepsilon\ll T\);[TBF12]. The coefficient function \(\gamma\) is given by \[\begin{equation}\label{gamma} \gamma(t) = \chi(t)+\varepsilon\alpha\,,\end{equation}\] where \(\alpha\gt0\) is the mean dispersion, and where \[\begin{equation}\label{chi} \chi(t) = \left\{\begin{array}{ll}-\delta\,, \text{ if } t \in [m, m+1) \text{ for even } m \in \mathbb{N}, \\[2mm] \,\,\,\,\delta\,, \text{ if } t \in [m, m+1) \text{ for odd }\, m \in \mathbb{N} \end{array}\right.\end{equation}\] is a periodic, piecewise constant function which alternates between \(-\delta\) and \(+\delta\). A short introduction to dispersion management from a mathematical perspective can be found in the introduction of [JM18] and in the dissertation [Mik17].

Approximating the solution of \eqref{DMNLS} numerically is a challenging task. The factor \(\mathrm{i}/\varepsilon\) in the right-hand side causes a highly oscillatory behavior of typical solutions. As a consequencs, traditional numerical time-integrators (e.g. Runge-Kutta or multistep methods) yield a very poor accuracy unless a huge number of time-steps with a tiny step-size \(\tau\ll\varepsilon\) is made, which causes excessive numerical costs. Additional difficulties are caused by the fact that the time-derivative \(t \mapsto \partial_t u(t,\cdot)\) discontinuous due to the piecewise constant coefficient function \(\gamma\). This is in contradiction to standard assumptions which are typically made to prove higher order convergence of many time integrators. Last but not least, the nonlinearity \(\mathrm{i} |u(t,x)|^2 u(t,x)\) makes implicit methods prohibitively costly and complicates the construction of new integrators.

In [JM18] we have devised and analyzed a novel time-integrator for the dispersion-managed nonlinear Schrödinger equation \eqref{DMNLS} on the torus \(\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}\). This new method, called the adiabatic midpoint rule, is based on a transformation of \eqref{DMNLS} to an evolution equation which has several advantageous properties, and on the fact that certain highly oscillatory exponential functions which occur in the transformed equation can be integrated analytically. This approach is related to the ideas developed in [Jah04, JL03]. It was shown that under suitable regularity assumptions the adiabatic midpoint rule converges with order 1 with an error constant which does not depend on \(\varepsilon\) and without any \(\varepsilon\)-induced step-size restriction. Moreover, we proved that the accuracy increases if the step-size \(\tau\) is chosen in a special way: The global error reduces to \(\mathcal{O}(\varepsilon^2 + \tau^2)\) if \(\tau = \varepsilon k \) with \(k\in \mathbb{N}\), and to \(\mathcal{O}(\varepsilon\tau)\) if \(\tau = \varepsilon/k\), respectively. In both cases the error constant remains independent on \(\varepsilon\), which explains the excellent accuracy of the adiabatic midpoint rule in numerical experiments; cf. Figure 1.

Unfortunately, the \(L_2\) norm of numerical solutions provided by the adiabatic midpoint rule is in general not constant, although it can easily be shown that the norm \(\|u(t,\cdot)\|_{L_2}\) of every solution of \eqref{DMNLS} is preserved. This was our motivation to construct an exponential version of the adiabatic midpoint rule which conserves the norm of the numerical solution exactly and, in addition, has the same favourable convergence behaviour as its non-exponential counterpart. The exponential adiabatic midpoint rule presented in [JM19] is closely related to our approach in [JM18], but in the error analysis we had to cope with new challenges arising from the exponential structure of the new method.

In mathematical physics the dispersion-managed nonlinear Schrödinger equation \eqref{DMNLS} is often considered on \(\mathbb{R}\) instead of the torus \(\mathbb{T}\). Unfortunately, extending the techniques developed in [JM19, Mik17, TBF12] from \(\mathbb{R}\) to \(\mathbb{T}\) is impossible, such that the construction of efficient numerical methods for \eqref{DMNLS} on \(\mathbb{R}\) requires completely new ideas. Julian Baumstark (PhD student since 10/2017) and Tobias Jahnke are currently working on this problem. Continue reading. Collapse content.

- Splitting methods for highly oscillatory differential equations Continue reading. Collapse content.
Splitting methods have become an essential tool in solving (partial) differential equations numerically. For oscillatory differential equations, though, classical schemes are inefficient and suffer from order reduction since their solution does not fulfill the required high regularity assumptions. Hence we need to come up with different approaches that take the oscillatory nature of the problem into account.

Simulating phenomena like vibration and light is an omnipresent task in modern technology and research. From physics we have for example the discrete wave equation \[\begin{align} \label{eq: wave_discr} u''(t) = - \Omega^2 u(t) + g(u). \end{align} \] with a matrix \(\Omega\) of extremely large norm while the nonlinearity \(g\) is suitably smooth. Such solutions can be approximated by many well established numerical algorithms. For the analysis and construction the behaviour of the wave, so the solution \(u\), needs to be somewhat “nice”, which means for example that higher derivatives of the solution have to exist and are suitably bounded.

From an analytic point of view we often cannot guarantee such regularity. Also numerical experiments show that for non smooth solutions resonances in the error can be observed if the step size times a high frequency of the system happens to hit a multiple of \(2 \pi\).We now raise the question if there is a way to overcome these problems and still get reasonable approximations of the real wave?

In [GH06], [HLW06] several trigonometric integrators are proposed to solve such problems. They also note that those can be understood as a Splitting method for a filtered equation, where the nonlinearity \(g\) is modified by filters depending on the oscillating part \(\Omega\) and the step size \(\tau\).

To gain a better insight, in [BGGHJ18] the analysis was done from this point of view. We considered the discrete wave equation (\ref{eq: wave_discr}) with a linear \(g\) and integrated it using the Strang Splitting applied to the modified equation. Thereby it was proven that the error still is of order \(\mathcal{O}(\tau^2)\). In a first step one estimates the error obtained by the perturbation through the filter. In a second step the error of the splitting is analysed. Because of the filters some critical terms can be handled and one can partly use the classical techniques for Splitting methods. Arguments like the lady windermere's fan, though, need to be carefully adapted to this special situation using for example summation by parts. The results can also be extended to nonlinear functions \(g\).

We now want to generalize the results. We take a step back from the ODE level to the abstract evolution equation arising from \[ \begin{align*} u''(t) = \Delta u(t) + f(u) \end{align*}. \] On the one hand this will widen the range of possible equations we can treat, on the other hand we will gain deeper insights in the behaviour of the numerical scheme which will hopefully lead to the right ideas for analysing the full discretisation error. Continue reading. Collapse content.