Time-Splitting Methods to Solve the Stochastic Incompressible Stokes Equation

Carelli, Erich; Hausenblas, Erika; Prohl, Andreas

doi:10.1137/100819436

Cited by 29 publications

(41 citation statements)

References 17 publications

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“…The splitting methods form an important group of methods which are quite accurate and efficient [57]. Actually, they have been widely applied for dealing with highly oscillatory systems such as the Schrödinger/nonlinear Schrödinger equations [1,8,9,22,23,55,67], the Dirac/nonlinear Dirac equations [5,6,14,54], the Maxwell-Dirac system [10,49], the Zakharov system [12,13,41,50], the Gross-Pitaevskii equation for Bose-Einstein condensation (BEC) [11], the Stokes equation [21], and the Enrenfest dynamics [32], etc.…”

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confidence: 99%

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

et al. 2016

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Super-resolution of the Lie-Trotter splitting (S 1 ) and Strang splitting (S 2 ) is rigorously analyzed for the nonlinear Dirac equation without external magnetic potentials in the nonrelativistic limit regime with a small parameter 0 < ε ≤ 1 inversely proportional to the speed of light. In this limit regime, the solution highly oscillates in time with wavelength at O(ε 2 ) in time. The splitting methods surprisingly show super-resolution, in the sense of breaking the resolution constraint under the Shannon's sampling theorem, i.e. the methods can capture the solution accurately even if the time step size τ is much larger than the sampled wavelength at O(ε 2 ). Similar to the linear case, S 1 and S 2 both exhibit 1/2 order convergence uniformly with respect to ε. Moreover, if τ is non-resonant, i.e. τ is away from certain region determined by ε, S 1 would yield an improved uniform first order O(τ ) error bound, while S 2 would give improved uniform 3/2 order convergence. Numerical results are reported to confirm these rigorous results. Furthermore, we note that super-resolution is still valid for higher order splitting methods.

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confidence: 99%

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

et al. 2016

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“…We shall now prove the inequality (15). To start with we will apply Itô's formula to ϕ(u) = u 2 1 4 .…”

Section: Proposition 26 If the Assumptions (B1) To (B3) Hold And (Gmentioning

confidence: 99%

“…Remark 2.9. Note that owing to (15) and (30) 1 4 ). More precisely, for any small number ε > 0, any θ 0 ∈ 0, 1 4 − β − ε and θ 1 ∈ (0, 1 4 − β) we have…”

Section: Proposition 26 If the Assumptions (B1) To (B3) Hold And (Gmentioning

confidence: 99%

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Numerical approximation of stochastic evolution equations: Convergence in scale of Hilbert spaces

Bessaih

Hausenblas

Randrianasolo

et al. 2018

Journal of Computational and Applied Mathematics

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The present paper is devoted to the numerical approximation of an abstract stochastic nonlinear evolution equation in a separable Hilbert space H. Examples of equations which fall into our framework include the GOY and Sabra shell models and a class of nonlinear heat equations. The spacetime numerical scheme is defined in terms of a Galerkin approximation in space and a semi-implicit Euler-Maruyama scheme in time. We prove the convergence in probability of our scheme by means of an estimate of the error on a localized set of arbitrary large probability. Our error estimate is shown to hold in a more regular space V β ⊂ H with β ∈ [0, 1 4 ) and that the explicit rate of convergence of our scheme depends on this parameter β.

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“…In the Hamiltonian system and general ordinary differential equations (ODEs), the splitting approach has been shown to preserve the structural/geometric properties [31,47] and are superior in many applications. Developments of splitting type methods in solving partial differential equations (PDEs) include utilization in Schrödinger/nonlinear Schrödinger equations [2,9,10,19,20,37,45], Dirac/nonlinear Dirac equations [7,8,14,36], Maxwell-Dirac system [11,32], Zakharov system [12,13,28,34,35], Stokes equation [18], and Enrenfest dynamics [25], etc.…”

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confidence: 99%

Super-resolution of time-splitting methods for the Dirac equation in the nonrelativistic regime

2020

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We establish error bounds of the Lie-Trotter splitting (S 1 ) and Strang splitting (S 2 ) for the Dirac equation in the nonrelativistic limit regime in the absence of external magnetic potentials, with a small parameter 0 < ε ≤ 1 inversely proportional to the speed of light. In this limit regime, the solution propagates waves with O(ε 2 ) wavelength in time. Surprisingly, we find out that the splitting methods exhibit super-resolution, in the sense of breaking the resolution constraint under the Shannon's sampling theorem, i.e. the methods can capture the solutions accurately even if the time step size τ is much larger than the sampled wavelength at O(ε 2 ). S 1 shows 1/2 order convergence uniformly with respect to ε, by establishing that there are two independent error bounds τ + ε and τ + τ /ε. Moreover, if τ is non-resonant, i.e. τ is away from certain region determined by ε, S 1 would yield an improved uniform first order O(τ ) error bound. In addition, we show S 2 is uniformly convergent with 1/2 order rate for general time step size τ and uniformly convergent with 3/2 order rate for non-resonant time step size. Finally, numerical examples are reported to validate our findings.

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Time-Splitting Methods to Solve the Stochastic Incompressible Stokes Equation

Cited by 29 publications

References 17 publications

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

Error estimates of numerical methods for the nonlinear Dirac equation in the nonrelativistic limit regime

Numerical approximation of stochastic evolution equations: Convergence in scale of Hilbert spaces

Super-resolution of time-splitting methods for the Dirac equation in the nonrelativistic regime

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