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

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.

Automated summary

This paper presents a class of maximum bound preserving schemes for approximately solving Allen-Cahn equations. The developed schemes include kth-order single-step schemes in time and lumped mass finite element methods in space. By using a cut-off post-processing technique, the numerical solutions satisfy the maximum bound principle, and an optimal error bound is theoretically proved for a certain class of schemes. Furthermore, the combination of the cut-off strategy and the scalar auxiliary value technique leads to the development of energy-stable and arbitrarily high-order schemes in time.