A new conservative vector-valued Allen-Cahn equation and its fast numerical method

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.