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

Mukam, Jean Daniel; Tambue, Antoine

doi:10.48550/arxiv.1610.05525

Cited by 2 publications

(5 citation statements)

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“…For the time discretization, we consider the one-step method which provides the numerical approximated solution X h m of X h (t m ) at discrete time t m = m∆t, m = 0, • • • , M . 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%

“…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%

“…As in the current literature for deterministic Rosenbrock-type methods, [28,29], deterministic exponential Rosemnbrock-type method [10,27] and stochastic exponential Rosenbrock-type methods [26], we make the following assumption on the nonlinear drift term.…”

Section: Assumption 1 [Linear Operatormentioning

confidence: 99%

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

Mukam

Tambue

2020

Stochastic Processes and their Applications

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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.

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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%

See 1 more Smart Citation

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

Mukam

Tambue

2020

Stochastic Processes and their Applications

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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: Mathematics Subject Classification (2010) Msc 65c30 • Msc 74...mentioning

confidence: 96%

“…As in the current literature on deterministic exponential Rosenbrock-Type methods [10,11,23,29,30], we make the following assumption on the nonlinear term.…”

Section: Assumption 1 [Linear Operatormentioning

confidence: 99%

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

Mukam

Tambue

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.

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

Cited by 2 publications

References 0 publications

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

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

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

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