89 research outputs found
Invariant Discretization Schemes Using Evolution-Projection Techniques
Finite difference discretization schemes preserving a subgroup of the maximal
Lie invariance group of the one-dimensional linear heat equation are
determined. These invariant schemes are constructed using the invariantization
procedure for non-invariant schemes of the heat equation in computational
coordinates. We propose a new methodology for handling moving discretization
grids which are generally indispensable for invariant numerical schemes. The
idea is to use the invariant grid equation, which determines the locations of
the grid point at the next time level only for a single integration step and
then to project the obtained solution to the regular grid using invariant
interpolation schemes. This guarantees that the scheme is invariant and allows
one to work on the simpler stationary grids. The discretization errors of the
invariant schemes are established and their convergence rates are estimated.
Numerical tests are carried out to shed some light on the numerical properties
of invariant discretization schemes using the proposed evolution-projection
strategy
A low complexity algorithm for non-monotonically evolving fronts
A new algorithm is proposed to describe the propagation of fronts advected in
the normal direction with prescribed speed function F. The assumptions on F are
that it does not depend on the front itself, but can depend on space and time.
Moreover, it can vanish and change sign. To solve this problem the Level-Set
Method [Osher, Sethian; 1988] is widely used, and the Generalized Fast Marching
Method [Carlini et al.; 2008] has recently been introduced. The novelty of our
method is that its overall computational complexity is predicted to be
comparable to that of the Fast Marching Method [Sethian; 1996], [Vladimirsky;
2006] in most instances. This latter algorithm is O(N^n log N^n) if the
computational domain comprises N^n points. Our strategy is to use it in regions
where the speed is bounded away from zero -- and switch to a different
formalism when F is approximately 0. To this end, a collection of so-called
sideways partial differential equations is introduced. Their solutions locally
describe the evolving front and depend on both space and time. The
well-posedness of those equations, as well as their geometric properties are
addressed. We then propose a convergent and stable discretization of those
PDEs. Those alternative representations are used to augment the standard Fast
Marching Method. The resulting algorithm is presented together with a thorough
discussion of its features. The accuracy of the scheme is tested when F depends
on both space and time. Each example yields an O(1/N) global truncation error.
We conclude with a discussion of the advantages and limitations of our method.Comment: 30 pages, 12 figures, 1 tabl
Treatment of complex interfaces for Maxwell's equations with continuous coefficients using the correction function method
We propose a high-order FDTD scheme based on the correction function method
(CFM) to treat interfaces with complex geometry without increasing the
complexity of the numerical approach for constant coefficients. Correction
functions are modeled by a system of PDEs based on Maxwell's equations with
interface conditions. To be able to compute approximations of correction
functions, a functional that is a square measure of the error associated with
the correction functions' system of PDEs is minimized in a divergence-free
discrete functional space. Afterward, approximations of correction functions
are used to correct a FDTD scheme in the vicinity of an interface where it is
needed. We perform a perturbation analysis on the correction functions' system
of PDEs. The discrete divergence constraint and the consistency of resulting
schemes are studied. Numerical experiments are performed for problems with
different geometries of the interface. A second-order convergence is obtained
for a second-order FDTD scheme corrected using the CFM. High-order convergence
is obtained with a corrected fourth-order FDTD scheme. The discontinuities
within solutions are accurately captured without spurious oscillations.Comment: 29 pages, 12 figures, modification of Acknowledgment
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