| 1. |
- Berg, Andre, et al.
(författare)
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Approximated exponential integrators for the stochastic Manakov equation
- 2021
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- This article presents and analyses an exponential integrator for the stochastic Manakov equation, a system arising in the study of pulse propagation in randomly birefringent optical fibers. We first prove that the strong order of the numerical approximation is 1/2 if the nonlinear term in the system is globally Lipschitz-continuous. Then, we use this fact to prove that the exponential integrator has convergence order 1/2 in probability and almost sure order 1/2, in the case of the cubic nonlinear coupling which is relevant in optical fibers. Finally, we present several numerical experiments in order to support our theoretical findings and to illustrate the efficiency of the exponential integrator as well as a modified version of it.
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| 2. |
- Berg, André, 1990-
(författare)
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Numerical analysis and simulation of stochastic partial differential equations with white noise dispersion
- 2023
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This doctoral thesis provides a comprehensive numerical analysis and exploration of several stochastic partial differential equations (SPDEs). More specifically, this thesis investigates time integrators for SPDEs with white noise dispersion. The thesis begins by examining the stochastic nonlinear Schrödinger equation with white noise dispersion (SNLSE), see Paper 1. The investigation probes the performance of different numerical integrators for this equation, focusing on their convergences, L2-norm preservation, and computational efficiency. Further, this thesis thoroughly investigates a conjecture on the critical exponent of the SNLSE, related to a phenomenon known as blowup, through numerical means. The thesis then introduces and studies exponential integrators for the stochastic Manakov equation (SME) by presenting two new time integrators - the explicit and symmetric exponential integrators - and analyzing their convergence properties, see Paper 2. Notably, this study highlights the flexibility and efficiency of these integrators compared to traditional schemes. The narrative then turns to the Lie-Trotter splitting integrator for the SME, see Paper 3, comparing its performance to existing time integrators. Theoretical proofs for convergence in various senses, alongside extensive numerical experiments, shed light on the efficacy of the proposed numerical scheme. The thesis also deep dives into the critical exponents of the SME, proposing a conjecture regarding blowup conditions for this SPDE.Lastly, the focus shifts to the stochastic generalized Benjamin-Bona-Mahony equation, see Paper 4. The study introduces and numerically assesses four novel exponential integrators for this equation. A primary finding here is the superior performance of the symmetric exponential integrator. This thesis also offers a succinct and novel method to depict the order of convergence in probability.
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| 3. |
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| 4. |
- Cohen, David, et al.
(författare)
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Energy-preserving integrators for stochastic Poisson systems
- 2014
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Ingår i: Communications in Mathematical Sciences. - : International Press of Boston. - 1539-6746 .- 1945-0796. ; 12:8, s. 1523-1539
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Tidskriftsartikel (refereegranskat)abstract
- A new class of energy-preserving numerical schemes for stochastic Hamiltonian systems with non-canonical structure matrix (in the Stratonovich sense) is proposed. These numerical integrators are of mean-square order one and also preserve quadratic Casimir functions. In the deterministic setting, our schemes reduce to methods proposed in [E. Hairer, JNAIAM. J. Numer. Anal. Ind. Appl. Math., 5(1-2), 73–84, 2011] and [D. Cohen, and E. Hairer, BIT, 51(1), 91–101, 2011].
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| 5. |
- Cohen, David, et al.
(författare)
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Exponential integrators for nonlinear Schrödinger equations with white noise dispersion
- 2017
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Ingår i: Stochastics and Partial Differential Equations: Analysis and Computations. - New York : Springer. - 2194-0401 .- 2194-041X. ; 5:4, s. 592-613
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Tidskriftsartikel (refereegranskat)abstract
- This article deals with the numerical integration in time of the nonlinear Schrödinger equation with power law nonlinearity and random dispersion. We introduce a new explicit exponential integrator for this purpose that integrates the noisy part of the equation exactly. We prove that this scheme is of mean-square order 1 and we draw consequences of this fact. We compare our exponential integrator with several other numerical methods from the literature. We finally propose a second exponential integrator, which is implicit and symmetric and, in contrast to the first one, preserves the L2-norm of the solution.
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