4.6 Article

Explicit Third-Order Unconditionally Structure-Preserving Schemes for Conservative Allen-Cahn Equations

Journal

JOURNAL OF SCIENTIFIC COMPUTING
Volume 90, Issue 1, Pages -

Publisher

SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10915-021-01691-w

Keywords

Conservative Allen-Cahn equations; Unconditionally maximum-principle-preserving; Mass-conserving; Improved stabilized integrating factor Runge-Kutta scheme

Funding

  1. National Key R&D Program of China [SQ2020YFA0709803]
  2. National Key Project [GJXM92579]
  3. National Natural Science Foundation of China [11901577, 11971481, 12071481]
  4. Natural Science Foundation of Hunan [2020JJ5652]
  5. Defense Science Foundation of China [2021-JCJQ-JJ-0538]
  6. Research Fund of National University of Defense Technology [ZK19-37, ZK18-03-49, ZZKY-JJ-21-01]

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This study focuses on the solution methods for improved conservative Allen-Cahn equations, proposing new structure-preserving schemes and isIFRK schemes. Numerical experiments confirm the advantages of isIFRK schemes, including high-order accuracy, mass conservation, and unconditional preservation of the maximum principle.
Compared with the well-known classical Allen-Cahn equation, the modified Allen-Cahn equation, which is equipped with a nonlocal Lagrange multiplier or a local-nonlocal Lagrange multiplier, enforces the mass conservation for modeling phase transitions. In this work, a class of up to third-order explicit structure-preserving schemes is proposed for solving these two modified conservative Allen-Cahn equations. Based on second-order finite-difference space discretization, we investigate the newly developed improved stabilized integrating factor Runge-Kutta (isIFRK) schemes for conservative Allen-Cahn equations. We prove that the original stabilized integrating factor Runge-Kutta schemes fail to preserve the mass conservation law when the stabilizing constant kappa > 0 and the initial mass does not equal zero, while isIFRK schemes not only preserve the maximum principle unconditionally, but also conserve the mass to machine accuracy without any restriction on the time-step size. Convergence of the proposed schemes are also presented. At last, a series of numerical experiments validate that each reformulation of the conservative Allen-Cahn equations has it own advantage, and isIFRK schemes can reach the expected high-order accuracy, conserve the mass, and preserve the maximum principle unconditionally.

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