TY - JOUR
T1 - A new conservative vector-valued Allen–Cahn equation and its fast numerical method
AU - Kim, Junseok
AU - Lee, Hyun Geun
N1 - Funding Information:
The authors thank the reviewers for the constructive and helpful comments on the revision of this article. The first author (J.S. Kim) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( 2014R1A2A2A01003683 ). The corresponding author (H.G. Lee) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education ( 2017R1D1A1B03034619 ).
Publisher Copyright:
© 2017 Elsevier B.V.
PY - 2017/12
Y1 - 2017/12
N2 - The scalar Allen–Cahn (AC) equation does not conserve the total mass, and its conservative forms have been studied analytically and numerically. Compared to the conservative scalar AC equations, a conservative form of the vector-valued AC equation is less studied. In this study, we introduce a new conservative vector-valued AC equation that conserves total mass and keeps the bulk phase values (away from the interfacial transition region) close to local minima. To solve the equation, we propose a fast numerical method that is based on the operator splitting method. In the proposed method, we split the equation into three subequations, and each subequation is solved in a component-wise manner. As a result, the conservative vector-valued AC equation is solved quickly, and the average CPU time is nearly linear with respect to the number of components. Numerical experiments with three and more components are presented to demonstrate the usefulness of the proposed method.
AB - The scalar Allen–Cahn (AC) equation does not conserve the total mass, and its conservative forms have been studied analytically and numerically. Compared to the conservative scalar AC equations, a conservative form of the vector-valued AC equation is less studied. In this study, we introduce a new conservative vector-valued AC equation that conserves total mass and keeps the bulk phase values (away from the interfacial transition region) close to local minima. To solve the equation, we propose a fast numerical method that is based on the operator splitting method. In the proposed method, we split the equation into three subequations, and each subequation is solved in a component-wise manner. As a result, the conservative vector-valued AC equation is solved quickly, and the average CPU time is nearly linear with respect to the number of components. Numerical experiments with three and more components are presented to demonstrate the usefulness of the proposed method.
KW - Linear multigrid
KW - Mass conservation
KW - Operator splitting
KW - Vector-valued Allen–Cahn equation
UR - http://www.scopus.com/inward/record.url?scp=85029594964&partnerID=8YFLogxK
U2 - 10.1016/j.cpc.2017.08.006
DO - 10.1016/j.cpc.2017.08.006
M3 - Article
AN - SCOPUS:85029594964
SN - 0010-4655
VL - 221
SP - 102
EP - 108
JO - Computer Physics Communications
JF - Computer Physics Communications
ER -