Milstein-type Schemes for Hyperbolic SPDEs

TL;DR

This paper develops and analyzes Milstein-type numerical schemes for hyperbolic stochastic partial differential equations, achieving optimal strong convergence rates and extending previous results from parabolic to hyperbolic cases.

Contribution

It introduces Milstein schemes for hyperbolic SPDEs and establishes their strong convergence rates, extending prior work from parabolic to hyperbolic equations.

Findings

Strong convergence order one for sufficiently regular problems.

Error bounds of order h and h√log(T/h) for different Milstein schemes.

Numerical validation with stochastic Schrödinger equation.

Abstract

This article studies the temporal approximation of hyperbolic semilinear stochastic evolution equations with multiplicative Gaussian noise by Milstein-type schemes. We take the term hyperbolic to mean that the leading operator generates a contractive, not necessarily analytic $C_0$-semigroup. Optimal convergence rates are derived for the pathwise uniform strong error \[ E_h^\infty := \Big(\mathbb{E}\Big[\max_{1\le j \le M}\|U_{t_j}-u_j\|_X^p\Big]\Big)^{1/p} \] on a Hilbert space $X$ for $p\in [2,\infty)$. Here, $U$ is the mild solution and $u_j$ its Milstein approximation at time $t_j=jh$ with step size $h>0$ and final time $T=Mh>0$. For sufficiently regular nonlinearity and noise, we establish strong convergence of order one, with the error satisfying $E_h^\infty\lesssim h\sqrt{\log(T/h)}$ for rational Milstein schemes and $E_h^\infty \lesssim h$ for exponential Milstein schemes. This extends previous results from parabolic to hyperbolic SPDEs and from exponential to rational Milstein schemes. Moreover, root-mean-square error estimates are strengthened to pathwise uniform estimates. Numerical experiments validate the convergence rates for the stochastic Schr\"odinger equation. Further applications to Maxwell's and transport equations are included.