A note on exponential Rosenbrock–Euler method for the finite element discretization of a semilinear parabolic partial differential equation

Stochastic Processes and their Applications

2020

Self Cite

This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part, usually called stochastic dominated transport equations. Most standard numerical schemes lose their good stability properties on such equations, including the current linear implicit Euler method. We discretise the SPDE in space by the finite element method and propose a new scheme in time appropriate for such equations, called stochastic Rosenbrock-Type scheme, which is based on the local linearisation of the semi-discrete problem obtained after space discretisation. We provide a strong convergence of the new fully discrete scheme toward the exact solution for multiplicative and additive noise. Our convergence rates are in agreement with results in the literature. Numerical experiments to sustain our theoretical results are provided.

Section: Novel Fully Discrete Scheme and Main Resultsmentioning

confidence: 99%

“…The method is based on the continuous linearization of (27). More precisely we linearize (27) at each time step as follows…”

Section: Novel Fully Discrete Scheme and Main Resultsmentioning

confidence: 99%

“…Let us recall that the linear implicit Euler scheme applied to the semi-discrete problem (27) leads to…”

Section: Standard Linear Implicit Euler Methods and Stability Propertiesmentioning

confidence: 99%

Section: Main Assumptions and Well Posedness Problemmentioning

confidence: 99%

See 2 more Smart Citations

Strong convergence of a stochastic Rosenbrock-type scheme for the finite element discretization of semilinear SPDEs driven by multiplicative and additive noise

Stochastic Processes and their Applications

2020

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“…Note that similar schemes for stochastic differential equation in finite dimensional have been proposed in [2,3]. Using some deterministic tools from [23], we propose a strong convergence proof of the new schemes where the linear operator A is not necessarily self adjoint. Note that the orders of convergence are the same with stochastic exponential schemes proposed in [22].…”

Section: Introductionmentioning

confidence: 96%

Strong Convergence Analysis of the Stochastic Exponential Rosenbrock Scheme for the Finite Element Discretization of Semilinear SPDEs Driven by Multiplicative and Additive Noise

2017

J Sci Comput

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In this paper, we consider the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part also called stochastic reactive dominated transport equations. Most numerical techniques, including current stochastic exponential integrators lose their good stability properties on such equations. Using finite element for space discretization, we propose a new scheme appropriated on such equations, called stochastic exponential Rosenbrock scheme (SERS) based on local linearization at every time step of the semi-discrete equation obtained after space discretization. We consider noise that is in a trace class and give a strong convergence proof of the new scheme toward the exact solution in the root-mean-square L 2 norm. Numerical experiments to sustain theoretical results are provided.

Higher order stable schemes for stochastic convection–reaction–diffusion equations driven by additive Wiener noise

Math Methods in App Sciences

2021

Self Cite

In this paper, we investigate the numerical approximation of stochastic convection-reaction-diffusion equations using two explicit exponential integrators. The stochastic partial differential equation (SPDE) is driven by additive Wiener process. The approximation in space is done via a combination of the standard finite element method and the Galerkin projection method. Using the linear functional of the noise, we construct two accelerated numerical methods, which achieve higher convergence orders. In particular, we achieve convergence rates approximately 1 for trace class noise and 1 2 for space-time white noise. These convergence orders are obtained under less regularity assumptions on the nonlinear drift function than those used in the literature for stochastic reaction-diffusion equations. Numerical experiments to illustrate our theoretical results are provided.