Magnus-type integrator for non-autonomous SPDEs driven by multiplicative noise

Antoine Tambue; Jean Daniel Mukam

doi:10.3934/dcds.2020194

Article Contents

2020, Volume 40, Issue 8: 4597-4624. Doi: 10.3934/dcds.2020194

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Magnus-type integrator for non-autonomous SPDEs driven by multiplicative noise

Antoine Tambue^1,2, , and
Jean Daniel Mukam^3,

1.
Department of Computer science, Electrical engineering and Mathematical sciences, Western Norway University of Applied Sciences, Inndalsveien 28, 5063 Bergen, Norway, Center for Research in Computational and Applied Mechanics (CERECAM)
2.
Department of Mathematics and Applied Mathematics, University of Cape Town, 7701 Rondebosch, South Africa, The African Institute for Mathematical Sciences(AIMS) of South Africa
3.
Fakultät für Mathematik, Technische Universität Chemnitz, 09126 Chemnitz, Germany

^* Corresponding author: Antoine Tambue, email: antonio@aims.ac.za
^* Corresponding author: Antoine Tambue, email: antonio@aims.ac.za

Received: January 2019

Revised: February 2020

Published: May 2020

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Abstract

This paper aims to investigate numerical approximation of a general second order non-autonomous semilinear parabolic stochastic partial differential equation (SPDE) driven by multiplicative noise. Numerical approximations of autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet understood. We discretize the non-autonomous SPDE by the finite element method in space and the Magnus-type integrator in time. We provide a strong convergence proof of the fully discrete scheme toward the mild solution in the root-mean-square $ L^2 $ norm. The result reveals how the convergence orders in both space and time depend on the regularity of the noise and the initial data. In particular, for multiplicative trace class noise we achieve convergence order $ \mathcal{O}\left(h^2\left(1+\max(0, \ln\left(t_m/h^2\right)\right)\right. \left.+\Delta t^{\frac{1}{2}}\right) $. Numerical simulations to illustrate our theoretical finding are provided.

Keywords:

Magnus-type integrator,
stochastic partial differential equations,
multiplicative noise,
strong convergence,
non-autonomous equations,
finite element method.

Mathematics Subject Classification: Primary: 65C30, 65J08, 65M60, 65M12, 65M15; Secondary: 65J15.

Citation:

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Figure 1. Convergence of the Magnus integrator for $ \beta = 1 $, and $ \beta = 2 $ in (154). The order of convergence in time is $ 0.57 $ for $ \beta = 1 $, $ 0.54 $ for $ \beta = 2 $. The total number of samples used is $ 100 $

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