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Numerical analysis and simulation of stochastic partial differential equations with white noise dispersion
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
2023 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)Alternativ titel
Numerisk analys och simulering av stokastiska partiella differentialekvationer med dispersion av vitt brus (Svenska)
Abstract [en]

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.

Ort, förlag, år, upplaga, sidor
Umeå: Umeå University , 2023. , s. 60
Serie
Research report in mathematics, ISSN 1653-0810 ; 75/23
Nyckelord [en]
stochastic partial differential equation, mathematics, numerical analysis, numerical scheme, time integrator, convergence analysis, blowup, critical exponent, nonlinear Schrödinger equation, Manakov equation, Benjamin-Bona-Mahony equation, BBM equation, stochastic, random, dispersion, white noise dispersion, finite difference, pseudospectral, code, matlab, exponential integrator, splitting integrator, convergence in probability
Nationell ämneskategori
Beräkningsmatematik
Forskningsämne
matematik; numerisk analys
Identifikatorer
URN: urn:nbn:se:umu:diva-213773ISBN: 9789180701419 (digital)ISBN: 9789180701402 (tryckt)OAI: oai:DiVA.org:umu-213773DiVA, id: diva2:1792315
Disputation
2023-09-25, Hörsal MIT.A.121, Umeå, 09:15 (Engelska)
Opponent
Handledare
Tillgänglig från: 2023-09-04 Skapad: 2023-08-29 Senast uppdaterad: 2023-08-30Bibliografiskt granskad
Delarbeten
1. Numerical study of nonlinear Schrödinger equations with white noise dispersion
Öppna denna publikation i ny flik eller fönster >>Numerical study of nonlinear Schrödinger equations with white noise dispersion
(Engelska)Manuskript (preprint) (Övrigt vetenskapligt)
Nationell ämneskategori
Beräkningsmatematik
Identifikatorer
urn:nbn:se:umu:diva-213770 (URN)
Tillgänglig från: 2023-08-29 Skapad: 2023-08-29 Senast uppdaterad: 2023-08-29
2. Approximated exponential integrators for the stochastic Manakov equation
Öppna denna publikation i ny flik eller fönster >>Approximated exponential integrators for the stochastic Manakov equation
2023 (Engelska)Ingår i: Journal of Computational Dynamics, ISSN 2158-2491, Vol. 10, nr 2, s. 323-344Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

This article presents and analyzes an approximated exponential integrator for the (inhomogeneous) stochastic Manakov system. This system of SPDE occurs in the study of pulse propagation in randomly birefringent optical fibers. For a globally Lipschitz-continuous nonlinearity, we prove that the strong order of the time integrator is 1/2. This is then used to prove that the approximated 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 approximated exponential integrator as well as a modified version of it.

Ort, förlag, år, upplaga, sidor
American Institute of Mathematical Sciences (AIMS), 2023
Nationell ämneskategori
Beräkningsmatematik
Identifikatorer
urn:nbn:se:umu:diva-212451 (URN)10.3934/jcd.2023002 (DOI)000954598600001 ()2-s2.0-85165988777 (Scopus ID)
Forskningsfinansiär
Vetenskapsrådet, 2018-04443
Tillgänglig från: 2023-07-31 Skapad: 2023-07-31 Senast uppdaterad: 2024-02-19Bibliografiskt granskad
3. Lie–Trotter Splitting for the Nonlinear Stochastic Manakov System
Öppna denna publikation i ny flik eller fönster >>Lie–Trotter Splitting for the Nonlinear Stochastic Manakov System
2021 (Engelska)Ingår i: Journal of Scientific Computing, ISSN 0885-7474, E-ISSN 1573-7691, Vol. 88, nr 1, artikel-id 6Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

This article analyses the convergence of the Lie–Trotter splitting scheme for the stochastic Manakov equation, a system arising in the study of pulse propagation in randomly birefringent optical fibers. First, we prove that the strong order of the numerical approximation is 1/2 if the nonlinear term in the system is globally Lipschitz. Then, we show that the splitting scheme has convergence order 1/2 in probability and almost sure order 12- in the case of a cubic nonlinearity. We provide several numerical experiments illustrating the aforementioned results and the efficiency of the Lie–Trotter splitting scheme. Finally, we numerically investigate the possible blowup of solutions for some power-law nonlinearities.

Ort, förlag, år, upplaga, sidor
Springer, 2021
Nyckelord
Almost sure convergence, Blowup, Convergence in probability, Convergence rates, Coupled system of stochastic nonlinear Schrödinger equations, Lie–Trotter scheme, Numerical schemes, Splitting scheme, Stochastic Manakov equation, Stochastic partial differential equations, Strong convergence
Nationell ämneskategori
Beräkningsmatematik
Identifikatorer
urn:nbn:se:umu:diva-184202 (URN)10.1007/s10915-021-01514-y (DOI)000653198800001 ()2-s2.0-85106869214 (Scopus ID)
Forskningsfinansiär
Vetenskapsrådet, 2018−04443
Tillgänglig från: 2021-06-14 Skapad: 2021-06-14 Senast uppdaterad: 2023-08-29Bibliografiskt granskad
4. Numerical simulations of stochastic generalized Benjamin-Bona-Mahony equations
Öppna denna publikation i ny flik eller fönster >>Numerical simulations of stochastic generalized Benjamin-Bona-Mahony equations
(Engelska)Manuskript (preprint) (Övrigt vetenskapligt)
Nationell ämneskategori
Beräkningsmatematik
Identifikatorer
urn:nbn:se:umu:diva-213771 (URN)
Tillgänglig från: 2023-08-29 Skapad: 2023-08-29 Senast uppdaterad: 2023-08-29

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Berg, André

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