Borel control and efficient numerical techniques to solve the Allen–Cahn equation governed by temporal multiplicative noise

Abstract

In this manuscript, we investigate a stochastic version of the Allen–Cahn equation, which is a nonlinear partial differential equation from mathematical physics. The stochastic model considers temporal multiplicative noise in the form of the derivative of the standard Wiener process. The existence and the uniqueness of solutions for this stochastic model is established rigorously using the theory of distributions. As a corollary from these analytical results, some a priori optimal estimates for the solutions of this model are constructed. In this work, we develop reliable numerical schemes which possess similar features as those of the solutions for the analytical model. Moreover, we establish mathematically the von Neumann stability and the consistency for the schemes proposed in this paper. Both the analytical and numerical results derived in this work are computationally verified through some simulations.

Keywords:

• Stochastic Allen–Cahn equation
• unique existence
• proposed schemes
• stability and consistency analyses
• numerical simulations

AMS Subject Classification:

• 65C30

Acknowledgements

The authors would like to thank the anonymous reviewers and the associate editor in charge of handling this manuscript for their time and efforts. Their suggestions and criticisms contributed enormously to improve the quality of the present work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The corresponding author (J.E.M.-D.) wishes to acknowledge the financial support from the National Council for Science and Technology of Mexico (CONACYT) through grant A1-S-45928.