33 research outputs found
An extended Fourier pseudospectral method for the Gross-Pitaevskii equation with low regularity potential
We propose and analyze an extended Fourier pseudospectral (eFP) method for
the spatial discretization of the Gross-Pitaevskii equation (GPE) with low
regularity potential by treating the potential in an extended window for its
discrete Fourier transform. The proposed eFP method maintains optimal
convergence rates with respect to the regularity of the exact solution even if
the potential is of low regularity and enjoys similar computational cost as the
standard Fourier pseudospectral method, and thus it is both efficient and
accurate. Furthermore, similar to the Fourier spectral/pseudospectral methods,
the eFP method can be easily coupled with different popular temporal
integrators including finite difference methods, time-splitting methods and
exponential-type integrators. Numerical results are presented to validate our
optimal error estimates and to demonstrate that they are sharp as well as to
show its efficiency in practical computations.Comment: 20 pages, 7 figure
An explicit and symmetric exponential wave integrator for the nonlinear Schr\"{o}dinger equation with low regularity potential and nonlinearity
We propose and analyze a novel symmetric exponential wave integrator (sEWI)
for the nonlinear Schr\"odinger equation (NLSE) with low regularity potential
and typical power-type nonlinearity of the form ,
where is the density with the wave function and is the exponent of the nonlinearity. The sEWI is explicit and
stable under a time step size restriction independent of the mesh size. We
rigorously establish error estimates of the sEWI under various regularity
assumptions on potential and nonlinearity. For "good" potential and
nonlinearity (-potential and ), we establish an optimal
second-order error bound in -norm. For low regularity potential and
nonlinearity (-potential and ), we obtain a first-order
-norm error bound accompanied with a uniform -norm bound of the
numerical solution. Moreover, adopting a new technique of \textit{regularity
compensation oscillation} (RCO) to analyze error cancellation, for some
non-resonant time steps, the optimal second-order -norm error bound is
proved under a weaker assumption on the nonlinearity: . For
all the cases, we also present corresponding fractional order error bounds in
-norm, which is the natural norm in terms of energy. Extensive numerical
results are reported to confirm our error estimates and to demonstrate the
superiority of the sEWI, including much weaker regularity requirements on
potential and nonlinearity, and excellent long-time behavior with
near-conservation of mass and energy.Comment: 35 pages, 10 figure
Uniformly Accurate Methods for Klein-Gordon type Equations
The main contribution of this thesis is the development of a novel class of uniformly accurate methods for Klein-Gordon type equations.
Klein-Gordon type equations in the non-relativistic limit regime, i.e., , are numerically very challenging to treat, since the solutions are highly oscillatory in time. Standard Gautschi-type methods suffer from severe time step restrictions as they require a CFL-condition with time step size to resolve the oscillations. Within this thesis we overcome this difficulty by introducing limit integrators, which allows us to reduce the highly oscillatory problem to the integration of a non-oscillatory limit system. This procedure allows error bounds of order without any step size restrictions. Thus, these integrators are very efficient in the regime . However, limit integrators fail for small values of .
In order to derive numerical schemes that work well for small as well as for large , we use the ansatz of "twisted variables", which allows us to develop uniformly accurate methods with respect to . In particular, we introduce efficient and robust uniformly accurate exponential-type integrators which resolve the solution in the relativistic regime as well as in the highly oscillatory non-relativistic regime without any step size restriction. In contrast to previous works, we do not employ any asymptotic nor multiscale expansion of the solution. Compared to classical methods our new schemes allow us to reduce the regularity assumptions as they converge under the same regularity assumptions required for the integration of the corresponding limit system. In addition, the newly derived first- and second-order exponential-type integrators converge to the classical Lie and Strang splitting schemes for the limit system.
Moreover, we present uniformly accurate schemes for the Klein-Gordon-Schrödinger and the Klein-Gordon-Zakharov system. For all uniformly accurate integrators we establish rigorous error estimates and underline their uniform convergence property numerically
Exponential integrators: tensor structured problems and applications
The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed
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
Uniformly accurate exponential-type integrators for Klein-Gordon equations with asymptotic convergence to classical splitting schemes in the nonlinear Schrödinger limit
International audienceWe introduce efficient and robust exponential-type integrators for Klein-Gordon equations which resolve the solution in the relativistic regime as well as in the highly-oscillatory non-relativistic regime without any step-size restriction, and under the same regularity assumptions on the initial data required for the integration of the corresponding limit system. In contrast to previous works we do not employ any asymptotic/multiscale expansion of the solution. This allows us derive uniform convergent schemes under far weaker regularity assumptions on the exact solution. In particular, the newly derived exponential-type integrators of first-, respectively, second-order converge in the non-relativistic limit to the classical Lie, respectively, Strang splitting in the nonlinear Schrödinger limit