Third order, uniform in low to high oscillatory coefficients, exponential integrators for Klein-Gordon equations

Allowing for space- and time-dependence of mass in Klein--Gordon equations resolves the problem of negative probability density and violation of Lorenz covariance of interaction in quantum mechanics. Moreover, it extends their applicability to the domain of quantum cosmology, where the variation in mass may be accompanied by high oscillations. In this paper, we propose a third-order exponential integrator, where the main idea lies in embedding the oscillations triggered by the possibly highly oscillatory component intrinsically into the numerical discretisation. While typically high oscillation requires appropriately small time steps, an application of Filon methods allows implementation with large time steps even in the presence of very high oscillation. This greatly improves the efficiency of the time-stepping algorithm. Proof of the convergence and its rate are nontrivial and require alternative representation of the equation under consideration. We derive careful bounds on the growth of global error in time discretisation and prove that, contrary to standard intuition, the error of time integration does not grow once the frequency of oscillations increases. Several of numerical simulations are presented to confirm the theoretical investigations and the robustness of the method in all oscillatory regimes.Comment: 14 pages, 3 figure

Efficient computation of delay differential equations with highly oscillatory terms.

This paper is concerned with the asymptotic expansion and numerical solution of systems of linear delay differential equations with highly oscillatory forcing terms. The computation of such problems using standard numerical methods is exceedingly slow and inefficient, indeed standard software is practically useless for this purpose. We propose an alternative, consisting of an asymptotic expansion of the solution, where each term can be derived either by recursion or by solving a non-oscillatory problem. This leads to methods which, counter-intuitively to those developed according to standard numerical reasoning, exhibit improved performance with growing frequency of oscillation

Effective highly accurate integrators for linear Klein-Gordon equations from low to high frequency regimes

We introduce an efficient class of numerical schemes for the Klein--Gordon equation which are highly accurate from slowly varying up to highly oscillatory regimes. Their construction is based on Magnus expansions tailored to the structure of the input term which allows us to resolve the oscillations in the system up to second order convergence in time uniformly in all frequencies $\omega_n$. Depending on the nature of the oscillatory term, the proposed methods even show superior convergence, reaching up to fourth-order convergence, while maintaining high efficiency and small error constants. Numerical experiments highlight our theoretical findings and underline the efficiency of the new schemes

Magnus-Lanczos methods with simplified commutators for the Schr\"odinger equation with a time-dependent potential

The computation of the Schr\"odinger equation featuring time-dependent potentials is of great importance in quantum control of atomic and molecular processes. These applications often involve highly oscillatory potentials and require inexpensive but accurate solutions over large spatio-temporal windows. In this work we develop Magnus expansions where commutators have been simplified. Consequently, the exponentiation of these Magnus expansions via Lanczos iterations is significantly cheaper than that for traditional Magnus expansions. At the same time, and unlike most competing methods, we simplify integrals instead of discretising them via quadrature at the outset -- this gives us the flexibility to handle a variety of potentials, being particularly effective in the case of highly oscillatory potentials, where this strategy allows us to consider significantly larger time steps.Comment: 27 pages, 5 figure