Arbitrarily High-Order Maximum Bound Preserving Schemes with Cut-off Postprocessing for Allen–Cahn Equations

Abstract

We develop and analyze a class of maximum bound preserving schemes for approximately solving Allen–Cahn equations. We apply a kth-order single-step scheme in time (where the nonlinear term is linearized by multi-step extrapolation), and a lumped mass finite element method in space with piecewise rth-order polynomials and Gauss–Lobatto quadrature. At each time level, a cut-off post-processing is proposed to eliminate extra values violating the maximum bound principle at the finite element nodal points. As a result, the numerical solution satisfies the maximum bound principle (at all nodal points), and the optimal error bound \(O(\tau ^k+h^{r+1})\) is theoretically proved for a certain class of schemes. These time stepping schemes include algebraically stable collocation-type methods, which could be arbitrarily high-order in both space and time. Moreover, combining the cut-off strategy with the scalar auxiliary value (SAV) technique, we develop a class of energy-stable and maximum bound preserving schemes, which is arbitrarily high-order in time. Numerical results are provided to illustrate the accuracy of the proposed method.