Unconditionally stable exponential time differencing schemes for the mass?conserving <scp> Allen <b>–</b> Cahn </scp> equation with nonlocal and local effects

It is well known that the classic Allen–Cahn equation satisfies maximum bound principle (MBP), is, absolute value of its solution uniformly bounded for all time by certain constant under suitable initial and boundary conditions. In this paper, we consider numerical solutions modified with a Lagrange multiplier nonlocal local effects, which not only shares same MBP as original but also conserves mass exactly. We reformulate model linear stabilizing technique, then construct first- second-order exponential differencing schemes integration. prove unconditional preservation conservation proposed in discrete sense derive their error estimates some regularity assumptions. Various experiments two three dimensions are conducted to verify theoretical results.