Sharp convergence to steady states of Allen-Cahn

06/16/2021
by   Dong Li, et al.
0

In our recent work we found a surprising breakdown of symmetry conservation: using standard numerical discretization with very high precision the computed numerical solutions corresponding to very nice initial data may converge to completely incorrect steady states due to the gradual accumulation of machine round-off error. We solved this issue by introducing a new Fourier filter technique for solutions with certain band gap properties. To further investigate the attracting basin of steady states we classify in this work all possible bounded nontrivial steady states for the Allen-Cahn equation. We characterize sharp dependence of nontrivial steady states on the diffusion coefficient and prove strict monotonicity of the associated energy. In particular, we establish a certain self-replicating property amongst the hierarchy of steady states and give a full classification of their energies and profiles. We develop a new modulation theory and prove sharp convergence to the steady state with explicit rates and profiles.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
06/12/2021

On a parabolic sine-Gordon model

We consider a parabolic sine-Gordon model with periodic boundary conditi...
research
09/22/2022

Aggregation Methods for Computing Steady-States in Statistical Physics

We give a new proof of local convergence of a multigrid method called it...
research
11/16/2020

Steady States and Well-balanced Schemes for Shallow Water Moment Equations with Topography

In this paper, we investigate steady states of shallow water moment equa...
research
06/05/2020

Augmented saddle point formulation of the steady-state Stefan–Maxwell diffusion problem

We investigate structure-preserving finite element discretizations of th...
research
07/20/2021

Effective maximum principles for spectral methods

Many physical problems such as Allen-Cahn flows have natural maximum pri...
research
12/22/2022

Fully well balanced entropy controlled DGSEM for shallow water flows: global flux quadrature and cell entropy correction

In this paper we propose a high order DGSEM formulation for balance laws...

Please sign up or login with your details

Forgot password? Click here to reset