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A convergent stochastic scalar auxiliary variable method

Stefan Metzger

TL;DR

An unconditionally energy stable, linear, fully discrete finite element scheme based on the augmented scalar auxiliary variable method is proposed and convergence of fully discrete solutions towards strong solutions of the stochastic Allen–Cahn equation is proved.

Abstract

We discuss an extension of the scalar auxiliary variable approach, which was originally introduced by Shen et al. ([Shen, Xu, Yang, J. Comput. Phys., 2018]) for the discretization of deterministic gradient flows. By introducing an additional scalar auxiliary variable, this approach allows to derive a linear scheme, while still maintaining unconditional stability. Our extension augments the approximation of the evolution of this scalar auxiliary variable with higher order terms, which enables its application to stochastic partial differential equations. Using the stochastic Allen--Cahn equation as a prototype for nonlinear stochastic partial differential equations with multiplicative noise, we propose an unconditionally energy stable, linear, fully discrete finite element scheme based on our augmented scalar auxiliary variable method. Recovering a discrete version of the energy estimate and establishing Nikolskii estimates with respect to time, we are able to prove convergence of discrete solutions towards pathwise unique martingale solutions by applying Jakubowski's generalization of Skorokhod's theorem. A generalization of the Gyöngy--Krylov characterization of convergence in probability to quasi-Polish spaces finally provides convergence of fully discrete solutions towards strong solutions of the stochastic Allen--Cahn equation. Finally, we present numerical simulations underlining the practicality of the scheme and the importance of the introduced augmentation terms.

A convergent stochastic scalar auxiliary variable method

TL;DR

An unconditionally energy stable, linear, fully discrete finite element scheme based on the augmented scalar auxiliary variable method is proposed and convergence of fully discrete solutions towards strong solutions of the stochastic Allen–Cahn equation is proved.

Abstract

We discuss an extension of the scalar auxiliary variable approach, which was originally introduced by Shen et al. ([Shen, Xu, Yang, J. Comput. Phys., 2018]) for the discretization of deterministic gradient flows. By introducing an additional scalar auxiliary variable, this approach allows to derive a linear scheme, while still maintaining unconditional stability. Our extension augments the approximation of the evolution of this scalar auxiliary variable with higher order terms, which enables its application to stochastic partial differential equations. Using the stochastic Allen--Cahn equation as a prototype for nonlinear stochastic partial differential equations with multiplicative noise, we propose an unconditionally energy stable, linear, fully discrete finite element scheme based on our augmented scalar auxiliary variable method. Recovering a discrete version of the energy estimate and establishing Nikolskii estimates with respect to time, we are able to prove convergence of discrete solutions towards pathwise unique martingale solutions by applying Jakubowski's generalization of Skorokhod's theorem. A generalization of the Gyöngy--Krylov characterization of convergence in probability to quasi-Polish spaces finally provides convergence of fully discrete solutions towards strong solutions of the stochastic Allen--Cahn equation. Finally, we present numerical simulations underlining the practicality of the scheme and the importance of the introduced augmentation terms.
Paper Structure (14 sections, 19 theorems, 157 equations, 10 figures, 10 tables)

This paper contains 14 sections, 19 theorems, 157 equations, 10 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Notation and assumptions
  3. The discrete scheme
  4. Main results
  5. Existence of discrete solutions
  6. Regularity results
  7. Compactness properties of discrete solutions
  8. Passage to the limit
  9. Pathwise uniqueness
  10. Proof of Theorem \ref{['thm:strongconvergence']}: Convergence towards strong solutions
  11. Numerical simulations
  12. Moderate noise
  13. Strong noise
  14. Proof of Lemma \ref{['lem:BDG']}

Key Result

Lemma 2.1

Let $\mathcal{T}_{h}$ satisfy item:spatialdisc and let $p\in[1,\infty)$ and $1\leq q\leq\infty$. Then, for $q^*=\tfrac{q}{q-1}$, if $q<\infty$, and $q^*=1$, if $q=\infty$, the estimates hold true for all $f_{h},\,g_{h}\in U_{h}$.

Figures (10)

  • Figure 11.a: Elliptical droplet used as initial condition for the numerical simulations.
  • Figure 11.b: Evolution of a droplet computed using scheme \ref{['eq:discscheme']} with $\tau=5\cdot10^{-4}$. The orange line indicates the zero level set of the deterministic evolution and the red line indicates the area depicted in the line plots in Figs. \ref{['fig:linePlotExpected']}, \ref{['fig:linePlotPath14']}, and \ref{['fig:linePlotPath16']}. The left column depicts the expected value, while the middle and the right column show two individual sample paths.
  • Figure 11.c: Line plot of expected values (based on 350 samples) of the phase-field profile using different numerical schemes (implicit scheme \ref{['eq:majeeprohl']} solid, augmented SAV \ref{['eq:discscheme']} dashed, SAV \ref{['eq:SAV']} dotted). Time-increments are color-coded ( $\blacksquare$$\tau=5\cdot10^{-4}$, $\blacksquare$$\tau=1\cdot10^{-3}$, $\blacksquare$$\tau=2\cdot10^{-3}$, $\blacksquare$$\tau=4\cdot10^{-3}$, $\blacksquare$$\tau=1\cdot10^{-2}$, $\blacksquare$$\tau=2\cdot10^{-2}$).
  • Figure 11.d: Line plots of phase-field profiles at $t=4$ for one sample path using different numerical schemes (implicit scheme \ref{['eq:majeeprohl']} solid, augmented SAV \ref{['eq:discscheme']} dashed, SAV \ref{['eq:SAV']} dotted) and different time increments ( $\blacksquare$$\tau=5\cdot10^{-4}$, $\blacksquare$$\tau=2\cdot10^{-3}$, $\blacksquare$$\tau=2\cdot10^{-2}$).
  • Figure 11.e: Line plots of phase-field profiles for one sample path using different numerical schemes (implicit scheme \ref{['eq:majeeprohl']} solid, augmented SAV \ref{['eq:discscheme']} dashed, SAV \ref{['eq:SAV']} dotted) and different time increments ( $\blacksquare$$\tau=5\cdot10^{-4}$, $\blacksquare$$\tau=2\cdot10^{-3}$, $\blacksquare$$\tau=2\cdot10^{-2}$).
  • ...and 5 more figures

Theorems & Definitions (39)

  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Remark 3.1
  • Remark 3.2
  • Lemma 4.1
  • Theorem 4.2
  • proof : Sketch of the proof
  • Remark 4.3
  • ...and 29 more