Strong Convergence Analysis of the Stochastic Exponential Rosenbrock 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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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: Main Assumptions and Well Posedness Problemmentioning

confidence: 99%

Section: Assumption 4 [Diffusion Term ]mentioning

confidence: 99%

“…When dealing with additive noise, to achieve higher convergence order in time, we assume that the nonlinear function satisfies the following assumption, also used in [26,43,44].…”

Section: Assumptionmentioning

confidence: 99%

Section: Introductionmentioning

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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“…When we turn our attention to the case of semilinear SPDEs, still with constant operator A(t) = A, but not necessary self-adjoint, the list of references becomes remarkably short, see e.g. [24,31]. Note that modelling real world phenomena with time dependent linear operator is more realistic than modelling with time independent linear 2 operator (see e.g.[4] and references therein).…”

mentioning

confidence: 99%

Strong convergence of the linear implicit Euler method for the finite element discretization of semilinear non-autonomous SPDEs driven by multiplicative or additive noise

Applied Numerical Mathematics

2020

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This paper aims to investigate the numerical approximation of semilinear non-autonomous stochastic partial differential equations (SPDEs) driven by multiplicative or additive noise. Such equations are more realistic than the autonomous ones when modeling real world phenomena. Such equations find applications in many fields such as transport in porous media, quantum fields theory, electromagnetism and nuclear physics. Numerical approximations of autonomous SPDEs are thoroughly investigated in the literature, while the non-autonomous case is not yet well understood. Here, a non-autonomous SPDE is discretized in space by the finite element method and in time by the linear implicit Euler method. We break the complexity in the analysis of the time depending, not necessarily self-adjoint linear operators with the corresponding semigroup and provide the strong convergence result of the fully discrete scheme toward the mild solution. The results indicate how the converge order depends on the regularity of the initial solution and the noise. Additionally, for additive noise we achieve convergence order in time approximately 1 under less regularity assumptions on the nonlinear drift term than required in the current literature, even in the autonomous case.Numerical simulations motivated from realistic porous media flow are provided to illustrate our theoretical finding.We consider numerical approximation of the following non-autonomous SPDE defined in Λ ⊂on the Hilbert space L 2 (Λ, R), T > 0 is the final time, F and B are nonlinear functions and X 0 is the initial data, which is random. The family of linear operators A(t) are unbounded, not necessarily self-adjoint, and for all s ∈ [0, T ], −A(s) is the generator of an analytic semigroup S s (t) =: e −tA(s) , t ≥ 0. The noise W (t) is a Q−Wiener process defined in a filtered probability space (Ω, F , P, {F t } t≥0 ). The filtration is assumed to fulfill the usual conditions (see e.g. [38,Definition 2.1.11]). Note that the noise can be represented as follows (see e.g. [38,37])where q i , e i , i ∈ N d are respectively the eigenvalues and the eigenfunctions of the covariance operator Q, and β i , i ∈ N, are independent and identically distributed standard Brownian motions. Precise assumptions on F , B, X 0 and A(t) will be given in the next section to ensure the existence of the unique mild solution X of (1). In many situations it is hard to exhibit explicit solutions of SPDEs. Therefore, numerical algorithms are good tools to provide realistic approximations. Strong approximations of autonomous SPDEs with constant linear self-adjoint operator A(t) = A are widely investigated in the literature, see e.g. [47,46,20,17,25,48] and references therein. When we turn our attention to the case of semilinear SPDEs, still with constant operator A(t) = A, but not necessary self-adjoint, the list of references becomes remarkably short, see e.g. [24,31]. Note that modelling real world phenomena with time dependent linear operator is more realistic than modelling with time independent lin...

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

Math Methods in App Sciences

2021

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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.