TY - JOUR

T1 - An efficient and accurate numerical algorithm for the vector-valued Allen-Cahn equations

AU - Lee, Hyun Geun

AU - Kim, Junseok

N1 - Funding Information:
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0023794 ). The authors thank the reviewers for their constructive and valuable comments.

PY - 2012/10

Y1 - 2012/10

N2 - 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.

AB - 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.

KW - Grain growth

KW - Linear geometric multigrid

KW - Multiple crystals growth

KW - Operator splitting

KW - Vector-valued Allen-Cahn equations

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U2 - 10.1016/j.cpc.2012.05.013

DO - 10.1016/j.cpc.2012.05.013

M3 - Article

AN - SCOPUS:84863108804

VL - 183

SP - 2107

EP - 2115

JO - Computer Physics Communications

JF - Computer Physics Communications

SN - 0010-4655

IS - 10

ER -