8 research outputs found
Numerical Methods for Quasicrystals
Quasicrystals are one kind of space-filling structures. The traditional
crystalline approximant method utilizes periodic structures to approximate
quasicrystals. The errors of this approach come from two parts: the numerical
discretization, and the approximate error of Simultaneous Diophantine
Approximation which also determines the size of the domain necessary for
accurate solution. As the approximate error decreases, the computational
complexity grows rapidly, and moreover, the approximate error always exits
unless the computational region is the full space. In this work we focus on the
development of numerical method to compute quasicrystals with high accuracy.
With the help of higher-dimensional reciprocal space, a new projection method
is developed to compute quasicrystals. The approach enables us to calculate
quasicrystals rather than crystalline approximants. Compared with the
crystalline approximant method, the projection method overcomes the
restrictions of the Simultaneous Diophantine Approximation, and can also use
periodic boundary conditions conveniently. Meanwhile, the proposed method
efficiently reduces the computational complexity through implementing in a unit
cell and using pseudospectral method. For illustrative purpose we work with the
Lifshitz-Petrich model, though our present algorithm will apply to more general
systems including quasicrystals. We find that the projection method can
maintain the rotational symmetry accurately. More significantly, the algorithm
can calculate the free energy density to high precision.Comment: 27 pages, 8 figures, 6 table
Spectral Deferred Correction Method for Landau-Brazovskii Model with Convex Splitting Technique
The Landau-Brazovskii model is a well-known Landau model for finding the
complex phase structures in microphase-separating systems ranging from block
copolymers to liquid crystals. It is critical to design efficient numerical
schemes for the Landau-Brazovskii model with energy dissipation and mass
conservation properties. Here, we propose a mass conservative and energy stable
scheme by combining the spectral deferred correction (SDC) method with the
convex splitting technique to solve the Landau-Brazovskii model efficiently. An
adaptive correction strategy for the SDC method is implemented to reduce the
cost time and preserve energy stability. Numerical experiments, including two-
and three-dimensional periodic crystals in the Landau-Brazovskii model, are
presented to show the efficiency of the proposed numerical method.Comment: 20 pages, 5 figure
Accurately recover global quasiperiodic systems by finite points
Quasiperiodic systems, related to irrational numbers, are space-filling
structures without decay nor translation invariance. How to accurately recover
these systems, especially for non-smooth cases, presents a big challenge in
numerical computation. In this paper, we propose a new algorithm, finite points
recovery (FPR) method, which is available for both smooth and non-smooth cases,
to address this challenge. The FPR method first establishes a homomorphism
between the lower-dimensional definition domain of the quasiperiodic function
and the higher-dimensional torus, then recovers the global quasiperiodic system
by employing interpolation technique with finite points in the definition
domain without dimensional lifting. Furthermore, we develop accurate and
efficient strategies of selecting finite points according to the arithmetic
properties of irrational numbers. The corresponding mathematical theory,
convergence analysis, and computational complexity analysis on choosing finite
points are presented. Numerical experiments demonstrate the effectiveness and
superiority of FPR approach in recovering both smooth quasiperiodic functions
and piecewise constant Fibonacci quasicrystals. While existing spectral methods
encounter difficulties in accurately recovering non-smooth quasiperiodic
functions