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