TY - JOUR
AU1 - Yang, Jiang
AU2 - Yuan, Zhaoming
AU3 - Zhou, Zhi
AB - 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(τk+hr+1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(\tau ^k+h^{r+1})$$\end{document} 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.
TI - Arbitrarily High-Order Maximum Bound Preserving Schemes with Cut-off Postprocessing for Allen–Cahn Equations
JF - Journal of Scientific Computing
DO - 10.1007/s10915-021-01746-y
DA - 2022-02-01
UR - https://www.deepdyve.com/lp/springer-journals/arbitrarily-high-order-maximum-bound-preserving-schemes-with-cut-off-qIRgiJSgx6
VL - 90
IS - 2
DP - DeepDyve
ER -