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pro vyhledávání: '"Schratz A"'
The regularity of solutions to the stochastic nonlinear wave equation plays a critical role in the accuracy and efficiency of numerical algorithms. Rough or discontinuous initial conditions pose significant challenges, often leading to a loss of accu
Externí odkaz:
http://arxiv.org/abs/2412.14644
In this paper, we design non-resonant low-regularity numerical integrators for the cubic nonlinear stochastic Schr$\ddot{\rm o}$dinger equations (SNLSE). First, we begin with a mean-square convergence analysis for an explicit low-regularity stochasti
Externí odkaz:
http://arxiv.org/abs/2410.22201
Autor:
Rowbottom, James, Maierhofer, Georg, Deveney, Teo, Schratz, Katharina, Liò, Pietro, Schönlieb, Carola-Bibiane, Budd, Chris
We present a novel, and effective, approach to the long-standing problem of mesh adaptivity in finite element methods (FEM). FE solvers are powerful tools for solving partial differential equations (PDEs), but their cost and accuracy are critically d
Externí odkaz:
http://arxiv.org/abs/2407.04516
The use of implicit time-stepping schemes for the numerical approximation of solutions to stiff nonlinear time-evolution equations brings well-known advantages including, typically, better stability behaviour and corresponding support of larger time
Externí odkaz:
http://arxiv.org/abs/2407.03945
Autor:
Rousset, Frédéric, Schratz, Katharina
A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation in
Externí odkaz:
http://arxiv.org/abs/2405.10572
Publikováno v:
SIAM J. Numer. Anal., 62(5), 2071--2086 (2024)
For the numerical solution of the cubic nonlinear Schr\"{o}dinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting scheme in time is considered. This scheme is shown to converge even
Externí odkaz:
http://arxiv.org/abs/2311.14366
We introduce a unified framework of symmetric resonance based schemes which preserve central symmetries of the underlying PDE. We extend the resonance decorated trees approach introduced in arXiv:2005.01649 to a richer framework by exploring novel wa
Externí odkaz:
http://arxiv.org/abs/2305.16737
We introduce a new non-resonant low-regularity integrator for the cubic nonlinear Schr\"odinger equation (NLSE) allowing for long-time error estimates which are optimal in the sense of the underlying PDE. The main idea thereby lies in treating the ze
Externí odkaz:
http://arxiv.org/abs/2302.00383
A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schr\"odinger equation on the two-dimensional torus $\mathbb{T}^2$. The scheme is analyzed in a framework of discrete Bourgain spaces, which allows us to cons
Externí odkaz:
http://arxiv.org/abs/2301.10639
Autor:
Feng, Yue, Schratz, Katharina
We establish the improved uniform error bounds on a Lawson-type exponential integrator Fourier pseudospectral (LEI-FP) method for the long-time dynamics of sine-Gordon equation where the amplitude of the initial data is $O(\varepsilon)$ with $0 < \va
Externí odkaz:
http://arxiv.org/abs/2211.09402