26,529 research outputs found
Mean-square stability and error analysis of implicit time-stepping schemes for linear parabolic SPDEs with multiplicative Wiener noise in the first derivative
In this article, we extend a Milstein finite difference scheme introduced in
[Giles & Reisinger(2011)] for a certain linear stochastic partial differential
equation (SPDE), to semi- and fully implicit timestepping as introduced by
[Szpruch(2010)] for SDEs. We combine standard finite difference Fourier
analysis for PDEs with the linear stability analysis in [Buckwar &
Sickenberger(2011)] for SDEs, to analyse the stability and accuracy. The
results show that Crank-Nicolson timestepping for the principal part of the
drift with a partially implicit but negatively weighted double It\^o integral
gives unconditional stability over all parameter values, and converges with the
expected order in the mean-square sense. This opens up the possibility of local
mesh refinement in the spatial domain, and we show experimentally that this can
be beneficial in the presence of reduced regularity at boundaries
On the Accuracy of Explicit Finite-Volume Schemes for Fluctuating Hydrodynamics
This paper describes the development and analysis of finite-volume methods for the Landau–Lifshitz Navier–Stokes (LLNS) equations and related stochastic partial differential equations in fluid dynamics. The LLNS equations incorporate thermal fluctuations into macroscopic hydrodynamics by the addition of white noise fluxes whose magnitudes are set by a fluctuation-dissipation relation. Originally derived for equilibrium fluctuations, the LLNS equations have also been shown to be accurate for nonequilibrium systems. Previous studies of numerical methods for the LLNS equations focused primarily on measuring variances and correlations computed at equilibrium and for selected nonequilibrium flows. In this paper, we introduce a more systematic approach based on studying discrete equilibrium structure factors for a broad class of explicit linear finite-volume schemes. This new approach provides a better characterization of the accuracy of a spatiotemporal discretization as a function of wavenumber and frequency, allowing us to distinguish between behavior at long wavelengths, where accuracy is a prime concern, and short wavelengths, where stability concerns are of greater importance. We use this analysis to develop a specialized third-order Runge–Kutta scheme that minimizes the temporal integration error in the discrete structure factor at long wavelengths for the one-dimensional linearized LLNS equations.Together with a novel method for discretizing the stochastic stress tensor in dimension larger than one, our improved temporal integrator yields a scheme for the three-dimensional equations that satisfies a discrete fluctuation-dissipation balance for small time steps and is also sufficiently accurate even for time steps close to the stability limit
Mean-square stability analysis of approximations of stochastic differential equations in infinite dimensions
The (asymptotic) behaviour of the second moment of solutions to stochastic
differential equations is treated in mean-square stability analysis. This
property is discussed for approximations of infinite-dimensional stochastic
differential equations and necessary and sufficient conditions ensuring
mean-square stability are given. They are applied to typical discretization
schemes such as combinations of spectral Galerkin, finite element,
Euler-Maruyama, Milstein, Crank-Nicolson, and forward and backward Euler
methods. Furthermore, results on the relation to stability properties of
corresponding analytical solutions are provided. Simulations of the stochastic
heat equation illustrate the theory.Comment: 22 pages, 4 figures; deleted a section; shortened the presentation of
results; corrected typo
Numerical stability analysis of the Euler scheme for BSDEs
In this paper, we study the qualitative behaviour of approximation schemes
for Backward Stochastic Differential Equations (BSDEs) by introducing a new
notion of numerical stability. For the Euler scheme, we provide sufficient
conditions in the one-dimensional and multidimensional case to guarantee the
numerical stability. We then perform a classical Von Neumann stability analysis
in the case of a linear driver and exhibit necessary conditions to get
stability in this case. Finally, we illustrate our results with numerical
applications
Stochastic fiber dynamics in a spatially semi-discrete setting
We investigate a spatially discrete surrogate model for the dynamics of a
slender, elastic, inextensible fiber in turbulent flows. Deduced from a
continuous space-time beam model for which no solution theory is available, it
consists of a high-dimensional second order stochastic differential equation in
time with a nonlinear algebraic constraint and an associated Lagrange
multiplier term. We establish a suitable framework for the rigorous formulation
and analysis of the semi-discrete model and prove existence and uniqueness of a
global strong solution. The proof is based on an explicit representation of the
Lagrange multiplier and on the observation that the obtained explicit drift
term in the equation satisfies a one-sided linear growth condition on the
constraint manifold. The theoretical analysis is complemented by numerical
studies concerning the time discretization of our model. The performance of
implicit Euler-type methods can be improved when using the explicit
representation of the Lagrange multiplier to compute refined initial estimates
for the Newton method applied in each time step.Comment: 20 pages; typos removed, references adde
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