Semi-analytical algorithm for quasicrystal patterns

Abstract

To efficiently simulate the quasicrystal patterns, we present a multi-stage semi-analytically algorithm. Utilizing the operator splitting strategy, we first split the original equation into three subproblems. A second-order five-stage scheme consists of solving four nonlinear ordinary differential equations with half time step and solving a linear partial differential equation with full time step. Using the methods of separation of variables, the nonlinear ODEs have analytical solutions. The linear PDE can also be analytically solved by using the Fourier-spectral method in space. In this sense, our proposed is semi-analytical because we only adopt an approximation in time. In each time step, we only need to compute several analytically solutions in a step-by-step manner. Therefore, the algorithm will be highly efficient and the simulation can be easily implemented. The performance and high efficiency of our proposed algorithm are verified via several simulations. To facilitate the interested readers to develop related researches, a MATLAB code for generating 12-fold quasicrystal patterns is provided in Appendix. We also share the computational code on Code Ocean platform, please refer to https://doi.org/10.24433/CO.6028082.v1.