scholarly journals Non-asymptotic error bounds for the multilevel Monte Carlo Euler method applied to SDEs with constant diffusion coefficient

2019 ◽  
Vol 24 (0) ◽  
Author(s):  
Benjamin Jourdain ◽  
Ahmed Kebaier
Keyword(s):  
Monte Carlo ◽  
Diffusion Coefficient ◽  
Error Bounds ◽  
Euler Method ◽  
Multilevel Monte Carlo ◽  
Constant Diffusion Coefficient ◽  
Asymptotic Error ◽  
Constant Diffusion

scholarly journals Fractional stochastic heat equation with piecewise constant coefficients

2020 ◽  
Vol 21 (01) ◽  
pp. 2150002
Author(s):  
Yuliya Mishura ◽  
Kostiantyn Ralchenko ◽  
Mounir Zili ◽  
Eya Zougar
Keyword(s):  
Diffusion Coefficient ◽  
Heat Equation ◽  
Divergence Form ◽  
Stochastic Heat Equation ◽  
Piecewise Constant ◽  
Constant Diffusion Coefficient ◽  
Infinite Dimensional ◽  
Order Elliptic Operator ◽  
Constant Diffusion ◽  
Constant Coefficients

We introduce a fractional stochastic heat equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize the fundamental solution of its deterministic part, and prove the existence and the uniqueness of its solution.

scholarly journals Time-Dependent Simulation of Cosmic-Ray Shocks, Including Alfvén Transport

1994 ◽  
Vol 142 ◽  
pp. 969-973
Author(s):  
T. W. Jones
Keyword(s):  
Diffusion Coefficient ◽  
Cosmic Rays ◽  
Wave Energy ◽  
Cosmic Ray ◽  
Alfven Waves ◽  
Alfven Wave ◽  
Alfvén Waves ◽  
Constant Diffusion Coefficient ◽  
Constant Diffusion ◽  
Transport Effects

AbstractTime evolution of plane, cosmic-ray modified shocks has been simulated numerically for the case with parallel magnetic fields. Computations were done in a “three-fluid” dynamical model incorporating cosmic-ray and Alfvén-wave energy transport equations. Nonlinear feedback from the cosmic rays and Alfvén waves is included in the equation of motion for the underlying plasma, as is the finite propagation speed and energy dissipation of the Alfvén waves. Exploratory results confirm earlier, steady state analyses that found these Alfvén transport effects to be potentially important when the upstream Alfvén speed and gas sound speeds are comparable. As noted earlier, Alfvén transport effects tend to reduce the transfer of energy through a shock from gas to energetic particles. These studies show as well that the timescale for modification of the shock is altered in nonlinear ways. It is clear, however, that the consequences of Alfvén transport are strongly model dependent and that both advection of cosmic rays by the waves and dissipation of wave energy in the plasma will be important to model correctly when quantitative results are needed. Comparison is made between simulations based on a constant diffusion coefficient and more realistic diffusion models allowing the diffusion coefficient to vary in response to changes in Alfvén wave intensity. No really substantive differences were found between them.Subject headings: cosmic rays — MHD — shock waves

scholarly journals On Solving the Poisson Equation with Discontinuities on Irregular Interfaces: GFM and VIM

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Ásdís Helgadóttir ◽  
Arthur Guittet ◽  
Frédéric Gibou
Keyword(s):  
Diffusion Coefficient ◽  
Poisson Equation ◽  
Real Life ◽  
Ghost Fluid Method ◽  
Constant Diffusion Coefficient ◽  
The Poisson Equation ◽  
Constant Diffusion ◽  
Physical Phenomena ◽  
Irregular Interface ◽  
Better Than

We analyze the accuracy of two numerical methods for the variable coefficient Poisson equation with discontinuities at an irregular interface. Solving the Poisson equation with discontinuities at an irregular interface is an essential part of solving many physical phenomena such as multiphase flows with and without phase change, in heat transfer, in electrokinetics, and in the modeling of biomolecules’ electrostatics. The first method, considered for the problem, is the widely known Ghost-Fluid Method (GFM) and the second method is the recently introduced Voronoi Interface Method (VIM). The VIM method uses Voronoi partitions near the interface to construct local configurations that enable the use of the Ghost-Fluid philosophy in one dimension. Both methods lead to symmetric positive definite linear systems. The Ghost-Fluid Method is generally first-order accurate, except in the case of both a constant discontinuity in the solution and a constant diffusion coefficient, while the Voronoi Interface Method is second-order accurate in the L∞-norm. Therefore, the Voronoi Interface Method generally outweighs the Ghost-Fluid Method except in special case of both a constant discontinuity in the solution and a constant diffusion coefficient, where the Ghost-Fluid Method performs better than the Voronoi Interface Method. The paper includes numerical examples displaying this fact clearly and its findings can be used to determine which approach to choose based on the properties of the real life problem in hand.

The computer simulation of the boost diffusion oxidation process for the deep-case hardening of titanium alloys

2004 ◽  
Vol 120 ◽  
pp. 259-268
Author(s):  
J. Luo ◽  
Z. Zhang ◽  
H. Dong ◽  
T. Bell
Keyword(s):  
Diffusion Coefficient ◽  
Titanium Alloys ◽  
Concentration Dependence ◽  
Oxygen Diffusion ◽  
Oxidation Process ◽  
Case Hardening ◽  
One Dimensional ◽  
Constant Diffusion Coefficient ◽  
Constant Diffusion ◽  
Simulation Results

A one dimensional finite difference diffusion model for simulating the Boost Diffusion Oxidation (BDO) process of titanium alloys is developed and implemented as a window-based program. The program can simulate the BDO process for both constant diffusion coefficient and concentration dependent diffusion coefficient. It is found that to accurately simulate the BDO process, the concentration dependence of oxygen diffusion has to be taken into account. If the concentration dependence is taken as the Shamblen and Redden’s equation, the simulation results agree well with the experimental results.

Transport of neutral solute in articular cartilage: effects of loading and particle size

2005 ◽  
Vol 461 (2059) ◽  
pp. 2021-2042 ◽  
Author(s):  
Le Zhang ◽  
A.Z Szeri
Keyword(s):  
Particle Size ◽  
Diffusion Coefficient ◽  
Articular Cartilage ◽  
Three Dimensional ◽  
Molecular Size ◽  
Local Deformation ◽  
Static Compression ◽  
Constant Diffusion Coefficient ◽  
Constant Diffusion ◽  
The Matrix

We investigate the influence that matrix structure, size of diffusing molecules and type and intensity of mechanical loading have on the transport of neutral solutes in articular cartilage. Although this type of investigation has been performed in the past, earlier researchers assumed a constant diffusion coefficient. By contrast, our diffusion coefficient depends on the local deformation of the matrix, and thus varies both in space and in time during an experiment. We derive a three-dimensional formulation of the problem based on mixture theory and utilize the commercial finite-element code ABAQUS to study it numerically. We also make use of the Cohen–Turnbull–Yasuda model to correlate the decrease of the diffusion coefficient with the increase in tortuosity, owing to the presence of the matrix. Under appropriate circumstances, the equations derived here reduce to the classical convection/diffusion equation and the equations of the biphasic cartilage model. Even though we chose axisymmetric sample geometry for the present calculations, the model can easily be applied to irregular three-dimensional samples. Our results reinforce and refine previously published studies. The neutral solute's rate of diffusion is reduced under static compression, due to the strain dependence of the diffusion coefficient; an increase in static compression leads to a decrease in the rate of transport of solutes of all sizes. Dynamic loading, on the other hand, augments solute transport due to convection, depending on particle size. The transport of small molecular size solute is moderately enhanced, but only within the surface layer; however, the rate of transport of large molecule solute is greatly increased, even in the deep layer of the cartilage.

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