Low regularity integrators for semilinear parabolic equations with maximum bound principles

by   Cao-Kha Doan, et al.

This paper is concerned with conditionally structure-preserving, low regularity time integration methods for a class of semilinear parabolic equations of Allen-Cahn type. Important properties of such equations include maximum bound principle (MBP) and energy dissipation law; for the former, that means the absolute value of the solution is pointwisely bounded for all the time by some constant imposed by appropriate initial and boundary conditions. The model equation is first discretized in space by the central finite difference, then by iteratively using Duhamel's formula, first- and second-order low regularity integrators (LRIs) are constructed for time discretization of the semi-discrete system. The proposed LRI schemes are proved to preserve the MBP and the energy stability in the discrete sense. Furthermore, their temporal error estimates are also successfully derived under a low regularity requirement that the exact solution of the semi-discrete problem is only assumed to be continuous in time. Numerical results show that the proposed LRI schemes are more accurate and have better convergence rates than classic exponential time differencing schemes, especially when the interfacial parameter approaches zero.


Stabilized exponential-SAV schemes preserving energy dissipation law and maximum bound principle for the Allen-Cahn type equations

It is well-known that the Allen-Cahn equation not only satisfies the ene...

Maximum bound principle preserving integrating factor Runge-Kutta methods for semilinear parabolic equations

A large class of semilinear parabolic equations satisfy the maximum boun...

Maximum bound principles for a class of semilinear parabolic equations and exponential time differencing schemes

The ubiquity of semilinear parabolic equations has been illustrated in t...

Unconditionally stable exponential time differencing schemes for the mass-conserving Allen-Cahn equation with nonlocal and local effects

It is well known that the classic Allen-Cahn equation satisfies the maxi...

Stabilized integrating factor Runge-Kutta method and unconditional preservation of maximum bound principle

Maximum bound principle (MBP) is an important property for a large class...

Effective maximum principles for spectral methods

Many physical problems such as Allen-Cahn flows have natural maximum pri...

Fourier integrator for periodic NLS: low regularity estimates via discrete Bourgain spaces

In this paper, we propose a new scheme for the integration of the period...

Please sign up or login with your details

Forgot password? Click here to reset