43 research outputs found
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
Asymptotic expansions for the linear PDEs with oscillatory input terms; Analytical form and error analysis
Partial differential equations with highly oscillatory input terms are hardly
ever solvable analytically and their numerical treatment is difficult.
Modulated Fourier expansion used as an {\it ansatz} is a well known and
extensively investigated tool in asymptotic numerical approach for this kind of
problems. Although the efficiency of this approach has been recognised, its
error analysis has not been investigated rigorously for general forms of linear
PDEs. In this paper, we start such kind of investigations for a general form of
linear PDEs with an input term characterised by a single high frequency. More
precisely we derive an analytical form of such an expansion and provide a
formula for the error of its truncation. Theoretical investigations are
illustrated by computational simulations
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
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
Solving the wave equation with multifrequency oscillations
We explore a new asymptotic-numerical solver for the time-dependent wave equation with an interaction term that is oscillating in time with a very high frequency. The method involves representing the solution as an asymptotic series in inverse powers of the oscillation frequency. Using the new scheme, high accuracy is achieved at a low computational cost. Salient features of the new approach are highlighted by a numerical example.</p