333 research outputs found
Linear multistep methods for integrating reversible differential equations
This paper studies multistep methods for the integration of reversible
dynamical systems, with particular emphasis on the planar Kepler problem. It
has previously been shown by Cano & Sanz-Serna that reversible linear
multisteps for first-order differential equations are generally unstable. Here,
we report on a subset of these methods -- the zero-growth methods -- that evade
these instabilities. We provide an algorithm for identifying these rare
methods. We find and study all zero-growth, reversible multisteps with six or
fewer steps. This select group includes two well-known second-order multisteps
(the trapezoidal and explicit midpoint methods), as well as three new
fourth-order multisteps -- one of which is explicit. Variable timesteps can be
readily implemented without spoiling the reversibility. Tests on Keplerian
orbits show that these new reversible multisteps work well on orbits with low
or moderate eccentricity, although at least 100 steps/radian are required for
stability.Comment: 31 pages, 9 figures, in press at The Astronomical Journa
Computational Engineering
This Workshop treated a variety of finite element methods and applications in computational engineering and expanded their mathematical foundation in engineering analysis. Among the 53 participants were mathematicians and engineers with focus on mixed and nonstandard finite element schemes and their applications
Wavelet-based Edge Multiscale Parareal Algorithm for subdiffusion equations with heterogeneous coefficients in a large time domain
We present the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm,
recently proposed in [Li and Hu, {\it J. Comput. Phys.}, 2021], for efficiently
solving subdiffusion equations with heterogeneous coefficients in long time.
This algorithm combines the benefits of multiscale methods, which can handle
heterogeneity in the spatial domain, and the strength of parareal algorithms
for speeding up time evolution problems when sufficient processors are
available. Our algorithm overcomes the challenge posed by the nonlocality of
the fractional derivative in previous parabolic problem work by constructing an
auxiliary problem on each coarse temporal subdomain to completely uncouple the
temporal variable. We prove the approximation properties of the correction
operator and derive a new summation of exponential to generate a single-step
time stepping scheme, with the number of terms of
independent of the final time, where
is the fine-scale time step size. We establish the convergence rate of our
algorithm in terms of the mesh size in the spatial domain, the level parameter
used in the multiscale method, the coarse-scale time step size, and the
fine-scale time step size. Finally, we present several numerical tests that
demonstrate the effectiveness of our algorithm and validate our theoretical
results.Comment: arXiv admin note: text overlap with arXiv:2003.1044
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The Application of Improved Numerical Techniques to 1-D Micellar/Polymer Flooding Simulation
Three examples of three phase flow models which have been developed are compared under various conditions. Although the dif-ference in oil recovery and surfactant trapping among the models was rather large with constant. salinity, a salinity gradient produced high oil recovery and low surfactant trapping with all three models. Since surfactant trapping is important and it is highly uncertain, this is another reason for designing a micellar flood with a salinity gradient, or something equivalent to a salinity gradient. The semi-discrete method was applied to a 1-D micellar/polymer flooding simulator. By using a semi-discrete method, the time step size can be controlled and varied to be as large as pos-sible without sacrificing accuracy. The stability limit can also be detected with this method. The method is tested and compared with the fully discrete method in various conditions such as differ-ent phase behavior environments and with or without adsorption. In the application of the semi-discrete method, four different ODE in-tegrators were used. Two of them are explicit methods while the other two are implicit methods. Although the implicit methods did not work as well as the explicit methods, there may be some improve-ment possible. With respect to the computation time, one of the explicit methods which is based on the· Runge-Kutta approximation worked best. Although the method can save 20 to 30% computation
time under some conditions, compared with the fully-discrete method, the results are highly problem-dependent. To improve the computation time, two methods are suggested. One is to check the error only in the oil or water component rather than all components or any other one component such as surfactant. The other is to check absolute error instead of relative error and multiply by a small conservative factor to the calculated time step size. The stability was analyzed for the oil bank, and for the surfactant front. The former imposes a rather constant limitation on the time step size continuously until the plateau of the oil bank is completely produced: Although approximate, the stability analysis for the surfactant front suggests an unconditional local instability, which is caused by the change in the fractional flow curve due to the surfactant.Petroleum and Geosystems Engineerin
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
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