A conservative Allen–Cahn equation with a space–time dependent Lagrange multiplier

We present a new numerical scheme for solving a conservative Allen–Cahn equation with a space–time dependent Lagrange multiplier. Since the well-known classical Allen–Cahn equation does not have mass conservation property, Rubinstein and Sternberg introduced a nonlocal Allen–Cahn equation with a time dependent Lagrange multiplier to enforce conservation of mass. However, with their model it is difficult to keep small features since they dissolve into the bulk region. One of the reasons for this is that mass conservation is realized by a global correction using the time-dependent Lagrange multiplier. To resolve the problem, we use a space–time dependent Lagrange multiplier to preserve the volume of the system and propose a practically unconditionally stable hybrid scheme to solve the model. The numerical results indicate a potential usefulness of our proposed numerical scheme for accurately calculating geometric features of interfaces.