140 research outputs found
On leapfrog-Chebyshev schemes for second-order differential equations
In this thesis the efficient time integration of semilinear second-order ordinary differential equations is investigated. Based on the leapfrog (Störmer, Verlet) scheme a new class of explicit two-step schemes is constructed by utilizing Chebyshev polynomials. For deriving rigorous error bounds of these leapfrog-Chebyshev (LFC) schemes a more general class of two-step schemes is introduced. Precise conditions are stated for this general class guaranteeing stability as well as second-order convergence in time. In addition, the influence of the starting value is analyzed in detail. Furthermore, by combining the leapfrog scheme with this general class of schemes a class of multirate two-step methods is constructed. Sufficient conditions for the stability of these schemes are derived as well as error bounds showing the second-order convergence in time. For both the LFC schemes and the multirate schemes if equipped with the LFC schemes it is shown that in specific situations they outperform the leapfrog scheme. Numerical examples are provided to illustrate the theoretical results
A Numerical Study of Methods for Moist Atmospheric Flows: Compressible Equations
We investigate two common numerical techniques for integrating reversible
moist processes in atmospheric flows in the context of solving the fully
compressible Euler equations. The first is a one-step, coupled technique based
on using appropriate invariant variables such that terms resulting from phase
change are eliminated in the governing equations. In the second approach, which
is a two-step scheme, separate transport equations for liquid water and vapor
water are used, and no conversion between water vapor and liquid water is
allowed in the first step, while in the second step a saturation adjustment
procedure is performed that correctly allocates the water into its two phases
based on the Clausius-Clapeyron formula. The numerical techniques we describe
are first validated by comparing to a well-established benchmark problem.
Particular attention is then paid to the effect of changing the time scale at
which the moist variables are adjusted to the saturation requirements in two
different variations of the two-step scheme. This study is motivated by the
fact that when acoustic modes are integrated separately in time (neglecting
phase change related phenomena), or when sound-proof equations are integrated,
the time scale for imposing saturation adjustment is typically much larger than
the numerical one related to the acoustics
An extension of A-stability to alternating direction implicit methods
An alternating direction implicit (ADI) scheme was constructed by the method of approximate factorization. An A-stable linear multistep method (LMM) was used to integrate a model two-dimensional hyperbolic-parabolic partial differential equation. Sufficient conditions for the A-stability of the LMM were determined by applying the theory of positive real functions to reduce the stability analysis of the partial differential equations to a simple algebraic test. A linear test equation for partial differential equations is defined and then used to analyze the stability of approximate factorization schemes. An ADI method for the three-dimensional heat equation is also presented
Plane wave stability of some conservative schemes for the cubic Schr\"{o}dinger equation
The plane wave stability properties of the conservative schemes of Besse and
Fei et al. for the cubic Schr\"{o}dinger equation are analysed. Although the
two methods possess many of the same conservation properties, we show that
their stability behaviour is very different. An energy preserving
generalisation of the Fei method with improved stability is presented.Comment: 12 pages, 6 figure
Multiple-grid convergence acceleration of viscous and inviscid flow computations
A multiple-grid algorithm for use in efficiently obtaining steady solution to the Euler and Navier-Stokes equations is presented. The convergence of a simple, explicit fine-grid solution procedure is accelerated on a sequence of successively coarser grids by a coarse-grid information propagation method which rapidly eliminates transients from the computational domain. This use of multiple-gridding to increase the convergence rate results in substantially reduced work requirements for the numerical solution of a wide range of flow problems. Computational results are presented for subsonic and transonic inviscid flows and for laminar and turbulent, attached and separated, subsonic viscous flows. Work reduction factors as large as eight, in comparison to the basic fine-grid algorithm, were obtained. Possibilities for further performance improvement are discussed
Broadcast Caching Networks with Two Receivers and Multiple Correlated Sources
The correlation among the content distributed across a cache-aided broadcast
network can be exploited to reduce the delivery load on the shared wireless
link. This paper considers a two-user three-file network with correlated
content, and studies its fundamental limits for the worst-case demand. A class
of achievable schemes based on a two-step source coding approach is proposed.
Library files are first compressed using Gray-Wyner source coding, and then
cached and delivered using a combination of correlation-unaware cache-aided
coded multicast schemes. The second step is interesting in its own right and
considers a multiple-request caching problem, whose solution requires coding in
the placement phase. A lower bound on the optimal peak rate-memory trade-off is
derived, which is used to evaluate the performance of the proposed scheme. It
is shown that for symmetric sources the two-step strategy achieves the lower
bound for large cache capacities, and it is within half of the joint entropy of
two of the sources conditioned on the third source for all other cache sizes.Comment: in Proceedings of Asilomar Conference on Signals, Systems and
Computers, Pacific Grove, California, November 201
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