  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.
