Strong Stability Preservation for Stochastic Partial Differential Equations

TL;DR

This work extends deterministic Strong Stability Preservation (SSP) to stochastic partial differential equations, enabling pathwise monotone properties to be preserved in SPDE numerical solutions. It develops SSP stochastic Runge-Kutta frameworks (SRK, SARK, SGARK) built from forward Euler and Euler–Maruyama flow maps, under three key conditions: provable monotone FE maps, bounded increments, and a nonzero radius of monotonicity, with extensions to Additive and Generalised ARK schemes. The paper proves mean-square convergence order $1/2$ for bounded-increment schemes and demonstrates, through Burgers, advection, and Euler-type SPDEs, that range-bounded and monotone solutions can be achieved when using SSP timestepping, slope limiters, and bounded stochastic increments. The results offer a principled approach to preserving nonlinear stability properties in SPDE simulations, with potential applications in data assimilation and data-driven modelling of stochastic transport phenomena.

Abstract

This paper extends deterministic notions of Strong Stability Preservation (SSP) to the stochastic setting, enabling nonlinearly stable numerical solutions to stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) with pathwise solutions that remain unconditionally bounded. This approach may offer modelling advantages in data assimilation, particularly when the signal or data is a realization of an SPDE or PDE with a monotonicity property.

Figures (6)

• Figure 3.1: We plot the final time plot, of the stochastic Burgers equation in row 1. Red indicates maxima over 1, blue indicates minima under 0 (to machine precision). We plot the maximum value as a function in time, in row 2. We plot the minimum value as a function in time, in row 3. In column 1 we plot the SSP22-Limiter-BI system, in column 2 we plot the SSP-Limiter with unbounded increments. In column 3 we plot the RK2 limited scheme with bounded increments. In column 4 we plot the SSP22 unlimited scheme with bounded increments.
• Figure 3.2: In \ref{['fig:2D_RK2_a_Limiter_BI_Final', 'fig:2D_RK2_MTE_Limiter_BI_Final']} we plot the final time solution of the $\gamma = 1/4$, $\gamma = 3/4$ respectively. In \ref{['fig:RK2_a_Limiter_BI_max', 'fig:RK2_MTE_Limiter_BI_max']} the maximum value is plotted with time for the $\gamma = 1/4$, $\gamma = 3/4$ respectively.
• Figure 3.3: We plot 16 ensemble member solutions of the final timestep on a perceptually uniform grey colour scale between [0,1] undershoots are in blue overshoots are in red (there are no over/undershoots).
• Figure 3.4: Stochastic-2D Euler. We plot 8 ensemble member solutions of the final timestep on a perceptually uniform grey colorscale between [0,1] undershoots are shown in blue and overshoots are shown in red (there are no over/undershoots). Maximums and minima of all ensemble members are plotted and remain bounded.
• Figure 3.5: Deterministic 2D Euler. We plot a 1 ensemble member solution of the deterministic incompressible Euler equation at the final timestep on a perceptually uniform grey colorscale between [0,1] undershoots are shown in blue overshoots are shown in red (there are no over/undershoots). Maximums and minima are also plotted as a function of timestep.

Theorems & Definitions (22)

• Definition 2.1: Strong Stability Preservation of Euler-Maruyama scheme.
• Definition 2.2: SSP-SRK
• Remark
• Theorem 2.1: Stochastic Runge-Kutta \ref{['method: Stochastic Runge-Kutta']} is SSP with radius of monotonicity $R(\mathbb{A})$
• proof
• Theorem 2.2: Stochastic Additive Runge-Kutta \ref{['method: Stochastic Additive Runge-Kutta']} is SSP under the usual ARK extension of the Kraiijevanger conditions.
• proof
• Example 2.1
• Example 2.2
• Definition 2.3: Mean square convergence