An efficient and accurate numerical algorithm for the vector-valued Allen-Cahn equations

In this paper, we consider the vector-valued Allen-Cahn equations which model phase separation in N-component systems. The considerations of solving numerically the vector-valued Allen-Cahn equations are as follows: (1) the use of a small time step is appropriate to obtain a stable solution and (2) a sufficient number of phase-field variables is required to capture the correct dynamics. However, stability restrictions on the time step and a large number of phase-field variables cause huge computational costs and make the calculation very inefficient. To overcome this problem, we present an efficient and accurate numerical algorithm which is based on an operator splitting technique and is solved by a fast solver such as a linear geometric multigrid method. The algorithm allows us to convert the vector-valued Allen-Cahn equations with N components into a system of N-1 binary Allen-Cahn equations and drastically reduces the required computational time and memory. We demonstrate the efficiency and accuracy of the algorithm with various numerical experiments. Furthermore, using our algorithm, we can simulate the growth of multiple crystals with different orientation angles and fold numbers on a single domain. Finally, the efficiency of our algorithm is validated with an example that includes the growth of multiple crystals with consideration of randomness effects.