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Convergence of an exponential method for the stochastic Schrödinger equation with power-law nonlinearityPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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(English)Manuscript (preprint) (Other academic)
##### Abstract [en]

##### National Category

Mathematics
##### Identifiers

URN: urn:nbn:se:umu:diva-146946OAI: oai:DiVA.org:umu-146946DiVA, id: diva2:1200350
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt472",{id:"formSmash:j_idt472",widgetVar:"widget_formSmash_j_idt472",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt478",{id:"formSmash:j_idt478",widgetVar:"widget_formSmash_j_idt478",multiple:true}); Available from: 2018-04-24 Created: 2018-04-24 Last updated: 2018-06-09
##### In thesis

A temporal approximation of the stochastic Schrödinger equation with a power-law nonlinearity and driven by multiplicative Ito noise is considered. Observe that the nonlinearity is not globally Lipschitz continuous and, in general, the exact solution cannot be assumed to remain bounded for all times. The first of these issues is handled by considering a truncated version of the equation. The second issue is handled by working with stopping times and random time intervals, on which the solution is almost surely bounded. For the time integration we use a stochastic exponential method. This exponential method has the advantage of being explicit and it does not suffer any CFL-type step size restrictions in general. In this work we prove almost sure convergence and convergence in probability, accompanied by convergence orders of 1/2- and 1/2, respectively. We also find that the regularity assumptions required on the noise and initial value are less restrictive for the convergence of the exponential scheme, compared to the Crank-Nicolson scheme. In addition, we provide numerical experiments that illustrate our theoretical results.

1. Exponential integrators for stochastic partial differential equations$(function(){PrimeFaces.cw("OverlayPanel","overlay1200403",{id:"formSmash:j_idt757:0:j_idt761",widgetVar:"overlay1200403",target:"formSmash:j_idt757:0:parentLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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