Strong convergence rates of tamed exponential Euler schemes for superlinear hyperbolic SPDEs
Abstract
In this paper, we prove pathwise uniform convergence at rates up to 1/2 for tamed exponential Euler schemes for semilinear hyperbolic stochastic evolution equations with superlinearly growing nonlinearities and multiplicative noise. We take the term hyperbolic to mean that the leading operator generates a contractive C0-semigroup but no parabolic smoothing occurs. Under local Lipschitz, polynomial growth, coercivity, and monotonicity conditions on the nonlinearities, we establish pathwise uniform strong error estimates of the form (𝔼max0≤ j ≤ N |U(tj)−Uj|Xp)1/p ≲ √k on a Hilbert space X for p∈ [2,∞). Here, U is the mild solution and Uj is the tamed exponential Euler approximation at time tj=jk with step size k>0. This extends previous convergence results for non-parabolic SPDEs from globally to locally Lipschitz nonlinearities, allowing both drift and diffusion to grow polynomially. In a stochastic Kato framework, we further establish local and global well-posedness as well as uniform a priori estimates for the mild solution and its approximation. Applications to nonlinear stochastic transport, Airy, wave-type, and dissipatively damped nonlinear Schrödinger equations are included, covering different nonlinearities-stopped and fractionally tamed schemes. For the Klein-Gordon equation with cubic velocity damping, this complements previous results obtained for additive noise.
Related
- Numerical Ergodicity and Optimal Strong Error Estimates for a Class of Novel Tamed Schemes to Superlinear SPDEs
- Strong convergence of Euler-Maruyama schemes for additive fractional SDEs with Lipschitz drift
- Exponential Contraction for Underdamped Langevin Diffusions with Superlinear Forces
- Uniform-in-time Strong Error Estimates of Tamed-FEM to Superlinear SPDEs driven by Multiplicative Noise
- A Proximal Newton Framework with Local Superlinear Convergence for Simple Bilevel Optimization
BibTeX
@misc{klioba2026strong,
title = {{Strong convergence rates of tamed exponential Euler schemes for superlinear hyperbolic SPDEs}},
author = {Katharina Klioba},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08872},
eprint = {2609.08872},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08872},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Strong convergence rates of tamed exponential Euler schemes for superlinear hyperbolic SPDEs
AU - Katharina Klioba
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08872
UR - https://arxiv.org/abs/2609.08872
AB - In this paper, we prove pathwise uniform convergence at rates up to 1/2 for tamed exponential Euler schemes for semilinear hyperbolic stochastic evolution equations with superlinearly growing nonlinearities and multiplicative noise. We take the term hyperbolic to mean that the leading operator generates a contractive C_0-semigroup but no parabolic smoothing occurs. Under local Lipschitz, polynomial growth, coercivity, and monotonicity conditions on the nonlinearities, we establish pathwise uniform strong error estimates of the form (𝔼max_0≤ j ≤ N |U(t_j)−U^j|_X^p)^1/p ≲ √k on a Hilbert space X for p∈ [2,∞). Here, U is the mild solution and U^j is the tamed exponential Euler approximation at time t_j=jk with step size k>0. This extends previous convergence results for non-parabolic SPDEs from globally to locally Lipschitz nonlinearities, allowing both drift and diffusion to grow polynomially. In a stochastic Kato framework, we further establish local and global well-posedness as well as uniform a priori estimates for the mild solution and its approximation. Applications to nonlinear stochastic transport, Airy, wave-type, and dissipatively damped nonlinear Schrödinger equations are included, covering different nonlinearities-stopped and fractionally tamed schemes. For the Klein-Gordon equation with cubic velocity damping, this complements previous results obtained for additive noise.
ER -