Improved error estimates for splitting methods applied to highly-oscillatory nonlinear Schrödinger equations

International audienceIn this work, the error behavior of operator splitting methods is analyzed for highly-oscillatory differential equations. The scope of applications includes time-dependent nonlinear Schrödinger equations, where the evolution operator associated with the principal linear part is highly-oscillatory and periodic in time. In a first step, a known convergence result for the second-order Strang splitting method applied to the cubic Schrödinger equation is adapted to a wider class of nonlinearities. In a second step, the dependence of the global error on the decisive parameter 0 < ε < < 1, defining the length of the period, is examined. The main result states that, compared to established error estimates, the Strang splitting method is more accurate by a factor ε, provided that the time stepsize is chosen as an integer fraction of the period. This improved error behavior over a time interval of fixed length, which is independent of the period, is due to an averaging effect. The extension of the convergence result to higher-order splitting methods and numerical illustrations complement the investigations

Time-integration methods for a dispersion-managed nonlinear Schrödinger equation

Modeling dispersion-managed optical fibers leads to a nonlinear Schrödinger equation where the linear part is multiplied by a rapidly changing piecewise constant coefficient function. Typically, the occurring oscillations of the solution and the discontinuous coefficients impose severe problems for traditional time-integrators. In this thesis, we present and analyze tailor-made numerical methods for this equation which attain a desired accuracy with significantly larger step-sizes than traditional methods. The construction of the methods is based on a favorable transformation of problem and the explicit computation of certain integrals over highly oscillatory phases. In the error analysis, we deviate from the classical concept “stability and consistency yield convergence”. Instead, we utilize recursion formulas for the global error to exploit cancellation effects of various oscillatory error terms allowing us to prove higher accuracy for special step-sizes

High-frequency wave-propagation: error analysis for analytical and numerical approximations

In this thesis we investigate a specific type of semilinear hyperbolic systems with highly oscillatory initial data. This type of systems is numerically very challenging to treat since the solutions are highly oscillatory in space and time. The goal is to derive suitable analytical and numerical approximations. Based on the classical slowly varying envelope approximation (SVEA), an improved error estimate is proven for this analytical approximation. The envelope equation avoids oscillations in space, making this approximation attractive for numerical computations. Furthermore, more accurate analytical approximations are obtained by extending the ansatz of the SVEA. In addition to the analytical study of the SVEA two numerical time integrators are constructed and analyzed without any step-size restrictions. Numerical examples are provided to illustrate the theoretical results. Finally, a complementary approach is presented which address both problems, the oscillations in space and time, simultaneously

On numerical methods for the semi-nonrelativistic system of the nonlinear Dirac equation

Solving the nonlinear Dirac equation in the nonrelativistic limit regime numerically is difficult, because the solution oscillates in time with frequency of $\mathcal{O}(\varepsilon^{-2})$, where $0 < ε \ll 1$ is inversely proportional to the speed of light. It was shown in [7], however, that such solutions can be approximated up to an error of $\mathcal{O}(\varepsilon^{-2})$ by solving the semi-nonrelativistic limit system, which is a non-oscillatory problem. For this system, we construct a two-step method, called the exponential explicit midpoint rule, and prove second-order convergence of the semi-discretization in time. Furthermore, we construct a benchmark method based on standard techniques and compare the efficiency of both methods. Numerical experiments show that the new integrator reduces the computational costs per time step to 40% and within a given runtime improves the accuracy by a factor of six

Adiabatic Midpoint Rule for the dispersion-managed nonlinear Schrödinger Equation

The dispersion-managed nonlinear Schrödinger equation contains a rapidly changing discontinuous coefficient function. Approximating the solution numerically is a challenging task because typical solutions oscillate in time which imposes severe step-size restrictions for traditional methods. We present and analyze a tailor-made time integrator which attains the desired accuracy with a significantly larger step-size than traditional methods. The construction of this method is based on a favorable transformation to an equivalent problem and the explicit computation of certain integrals over highly oscillatory phases. The error analysis requires the thorough investigation of various cancellation effects which result in improved accuracy for special step-sizes

Geometric Integrators for Schrödinger Equations

The celebrated Schrödinger equation is the key to understanding the dynamics of quantum mechanical particles and comes in a variety of forms. Its numerical solution poses numerous challenges, some of which are addressed in this work. Arguably the most important problem in quantum mechanics is the so-called harmonic oscillator due to its good approximation properties for trapping potentials. In Chapter 2, an algebraic correspondence-technique is introduced and applied to construct efficient splitting algorithms, based solely on fast Fourier transforms, which solve quadratic potentials in any number of dimensions exactly - including the important case of rotating particles and non-autonomous trappings after averaging by Magnus expansions. The results are shown to transfer smoothly to the Gross-Pitaevskii equation in Chapter 3. Additionally, the notion of modified nonlinear potentials is introduced and it is shown how to efficiently compute them using Fourier transforms. It is shown how to apply complex coefficient splittings to this nonlinear equation and numerical results corroborate the findings. In the semiclassical limit, the evolution operator becomes highly oscillatory and standard splitting methods suffer from exponentially increasing complexity when raising the order of the method. Algorithms with only quadratic order-dependence of the computational cost are found using the Zassenhaus algorithm. In contrast to classical splittings, special commutators are allowed to appear in the exponents. By construction, they are rapidly decreasing in size with the semiclassical parameter and can be exponentiated using only a few Lanczos iterations. For completeness, an alternative technique based on Hagedorn wavepackets is revisited and interpreted in the light of Magnus expansions and minor improvements are suggested. In the presence of explicit time-dependencies in the semiclassical Hamiltonian, the Zassenhaus algorithm requires a special initiation step. Distinguishing the case of smooth and fast frequencies, it is shown how to adapt the mechanism to obtain an efficiently computable decomposition of an effective Hamiltonian that has been obtained after Magnus expansion, without having to resolve the oscillations by taking a prohibitively small time-step. Chapter 5 considers the Schrödinger eigenvalue problem which can be formulated as an initial value problem after a Wick-rotating the Schrödinger equation to imaginary time. The elliptic nature of the evolution operator restricts standard splittings to low order, ¿ < 3, because of the unavoidable appearance of negative fractional timesteps that correspond to the ill-posed integration backwards in time. The inclusion of modified potentials lifts the order barrier up to ¿ < 5. Both restrictions can be circumvented using complex fractional time-steps with positive real part and sixthorder methods optimized for near-integrable Hamiltonians are presented. Conclusions and pointers to further research are detailed in Chapter 6, with a special focus on optimal quantum control.Bader, PK. (2014). Geometric Integrators for Schrödinger Equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/38716TESISPremios Extraordinarios de tesis doctorale

Randomized exponential integrators for modulated nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation with dispersion modulated by a (formal) derivative of a $\alpha$-Hölder continuous time-dependent function. Due to the highly oscillatory nature of the problem classical numerical methods face severe order reduction in non-smooth regimes $\alpha < 1$. In this work, we develop a new randomized exponential integrator based on a stratified Monte Carlo approximation which allows us to average the high oscillations in the problem and obtain improved error bounds of order $\alpha + 1/2$. In addition, the new approach allows us to treat a far more general class of modulations than the available literature. Numerical results underline our theoretical findings and show the favorable error behavior of our new scheme compared to classical methods