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Register a new userOptimal non-reversible linear drift for the convergence to equilibrium of a diffusion
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We consider non-reversible perturbations of reversible diffusions that do not alter the invariant distribution and we ask whether there exists an optimal perturbation such that the rate of convergence to equilibrium is maximized. We solve this problem for the case of linear drift by proving the existence of such optimal perturbations and by providing an easily implementable algorithm for constructing them. We discuss in particular the role of the prefactor in the exponential convergence estimate. Our rigorous results are illustrated by numerical experiments.
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The radiation magnetohydrodynamics (RMHD) system couples the ideal magnetohydrodynamics equations with a gray radiation transfer equation. The main challenge is that the radiation travels at the speed of light while the magnetohydrodynamics changes with the time scale of the fluid. The time scales of these two processes can vary dramatically. In order to use mesh sizes and time steps that are independent of the speed of light, asymptotic preserving (AP) schemes in both space and time are desired. In this paper, we develop an AP scheme in both space and time for the RMHD system. Two different scalings are considered. One results in an equilibrium diffusion limit system, while the other results in a non-equilibrium system. The main idea is to decompose the radiative intensity into three parts, each part is treated differently with suitable combinations of explicit and implicit discretizations guaranteeing the favorable stability conditionand computational efficiency. The performance of the AP method is presented, for both optically thin and thick regions, as well as for the radiative shock problem.
We investigate a recombination-drift-diffusion model coupled to Poissons equation modelling the transport of charge within certain types of semiconductors. In more detail, we study a two-level system for electrons and holes endowed with an intermediate energy level for electrons occupying trapped states. As our main result, we establish an explicit functional inequality between relative entropy and entropy production, which leads to exponential convergence to equilibrium. We stress that our approach is applied uniformly in the lifetime of electrons on the trap level assuming that this lifetime is sufficiently small.
Irreversible drift-diffusion processes are very common in biochemical reactions. They have a non-equilibrium stationary state (invariant measure) which does not satisfy detailed balance. For the corresponding Fokker-Planck equation on a closed manifold, via Voronoi tessellation, we propose two upwind finite volume schemes with or without the information of the invariant measure. Both two schemes enjoy stochastic $Q$-matrix structures and can be decomposed as a gradient flow part and a Hamiltonian flow part, which enable us to prove unconditional stability, ergodicity and error estimates. Based on two upwind schemes, several numerical examples - including sampling accelerated by a mixture flow, image transformations and simulations for stochastic model of chaotic system - are conducted. These two structure-preserving schemes also give a natural random walk approximation for a generic irreversible drift-diffusion process on a manifold. Thus they can be adapted to manifold-related computations induced from high dimensional molecular dynamics.
We propose an efficient numerical strategy for simulating fluid flow through porous media with highly oscillatory characteristics. Specifically, we consider non-linear diffusion models. This scheme is based on the classical homogenization theory and uses a locally mass-conservative formulation. In addition, we discuss some properties of the standard non-linear solvers and use an error estimator to perform a local mesh refinement. The main idea is to compute the effective parameters in such a way that the computational complexity is reduced without affecting the accuracy. We perform some numerical examples to illustrate the behaviour of the adaptive scheme and of the non-linear solvers. Finally, we discuss the advantages of the implementation of the numerical homogenization in a periodic media and the applicability of the same scheme in non-periodic test cases such as SPE10th project.
This paper presents stability and convergence analysis of a finite volume scheme (FVS) for solving aggregation, breakage and the combined processes by showing Lipschitz continuity of the numerical fluxes. It is shown that the FVS is second order convergent independently of the meshes for pure breakage problem while for pure aggregation and coupled equations, it shows second order convergent on uniform and non-uniform smooth meshes. Furthermore, it gives only first order convergence on non-uniform grids. The mathematical results of convergence analysis are also demonstrated numerically for several test problems.
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