35 research outputs found
Staggered Schemes for Fluctuating Hydrodynamics
We develop numerical schemes for solving the isothermal compressible and
incompressible equations of fluctuating hydrodynamics on a grid with staggered
momenta. We develop a second-order accurate spatial discretization of the
diffusive, advective and stochastic fluxes that satisfies a discrete
fluctuation-dissipation balance, and construct temporal discretizations that
are at least second-order accurate in time deterministically and in a weak
sense. Specifically, the methods reproduce the correct equilibrium covariances
of the fluctuating fields to third (compressible) and second (incompressible)
order in the time step, as we verify numerically. We apply our techniques to
model recent experimental measurements of giant fluctuations in diffusively
mixing fluids in a micro-gravity environment [A. Vailati et. al., Nature
Communications 2:290, 2011]. Numerical results for the static spectrum of
non-equilibrium concentration fluctuations are in excellent agreement between
the compressible and incompressible simulations, and in good agreement with
experimental results for all measured wavenumbers.Comment: Submitted. See also arXiv:0906.242
Adaptive mesh refinement for the Landau–Lifshitz–Gilbert equation
We propose a new adaptive algorithm for the approximation of the Landau–Lifshitz–Gilbert equation via a higher-order tangent plane scheme. We show that the adaptive approximation satisfies an energy inequality and demonstrate numerically, that the adaptive algorithm outperforms uniform approaches
Unconditional well-posedness and IMEX improvement of a family of predictor-corrector methods in micromagnetics
Recently, Kim & Wilkening (Convergence of a mass-lumped finite element method for the Landau-Lifshitz equation, Quart. Appl. Math., 76, 383-405, 2018) proposed two novel predictor-corrector methods for the Landau-Lifshitz-Gilbert equation (LLG) in micromagnetics, which models the dynamics of the magnetization in ferromagnetic materials. Both integrators are based on the so-called Landau-Lifshitz form of LLG, use mass-lumped variational formulations discretized by first-order finite elements, and only require the solution of linear systems, despite the nonlinearity of LLG. The first(-order in time) method combines a linear update with an explicit projection of an intermediate approximation onto the unit sphere in order to fulfill the LLG-inherent unit-length constraint at the discrete level. In the second(-order in time) integrator, the projection step is replaced by a linear constraint-preserving variational formulation. In this paper, we extend the analysis of the integrators by proving unconditional well-posedness and by establishing a close connection of the methods with other approaches available in the literature. Moreover, the new analysis also provides a well-posed integrator for the Schrödinger map equation (which is the limit case of LLG for vanishing damping). Finally, we design an implicit-explicit strategy for the treatment of the lower-order field contributions, which significantly reduces the computational cost of the schemes, while preserving their theoretical properties
Analysis of a splitting scheme for a class of nonlinear stochastic Schr\uf6dinger equations
We analyze the qualitative properties and the order of convergence of a splitting scheme for a class of nonlinear stochastic Schr\uf6dinger equations driven by additive It\uf4 noise. The class of nonlinearities of interest includes nonlocal interaction cubic nonlinearities. We show that the numerical solution is symplectic and preserves the expected mass for all times. On top of that, for the convergence analysis, some exponential moment bounds for the exact and numerical solutions are proved. This enables us to provide strong orders of convergence as well as orders of convergence in probability and almost surely. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme