A comparison of four methods for simulating the diffusion process

Fortran programs for the time-dependent Gross–Pitaevskii equation in a fully anisotropic trap

A frequent configuration in computational fluid mechanics combines an explicit time advancing scheme for accuracy purposes and a computational grid with a very small portion of much smaller elements than in the remaining mesh. Examples of such situations are the traveling of a discontinuity followed by a moving mesh, and the large eddy simulation of high Reynolds number flows around bluff bodies where together very thin boundary layers and vortices of much more important size need to be captured. For such configurations, explicit time advancing schemes with global time stepping are very costly. In order to overcome this problem, the multirate time stepping approach represents an interesting improvement. The objective of such schemes, which allow to use different time steps in the computational domain, is to avoid penalizing the computational cost of the time advancement of unsteady solutions which can become large due to the use of small global time steps imposed by the smallest elements such as those constituting the boundary layers. In the present work, a new multirate scheme based on control volume agglomeration is proposed for the solution of the compressible Navier-Stokes equations possibly equipped with turbulence models. The method relies on a prediction step where large time steps are performed with an evaluation of the fluxes on macro-cells for the smaller elements for stability purpose, and on a correction step in which small time steps are employed only for the smaller elements. The efficiency of the proposed method is evaluated on several benchmarks flows: the problem of a moving contact discontinuity (inviscid flow), the computation with hybrid turbulence model of flows around bluff bodies like a tandem cylinders at Reynolds number $1.66 × 10 5$ , a circular cylinder at Reynolds number $8.4 × 10 6$ , and a flow around a space probe model at Reynolds number $1 × 10 6$ .

An accurate and simplified model of a brushless DC motor drive (BLMD) system is presented as a simulation tool for performance related studies in industrial drive applications and for system identification in controller design. The model relies on the single Fourier series expression for pulsewidth modulation (PWM), which includes the effects of inverter dead time as a substantial part of the switching period T/sub s/, in the current control loop. This simplified approach obviates the need for very small time steps (/spl sim/0.5% T/sub s/), in the search of PWM edge transitions within a switching interval, for accurate simulation. Substantial improvement in simulation speed is achieved (up to fiftyfold) with a beneficial saving in computation (CPU) resources. Model integrity is validated when simulation results are checked against actual BLMD test data for coherence. Subsequent model simulations illustrate the effect of inverter delay on the reduction of generated torque and power transfer.

Stability of semidiscrete formulations for parabolic problems at small time steps

A computer model of metabolite transport and consumption in heterogeneous brain tissue, using a combination of probabilistic and deterministic techniques is being developed. The metabolites are put into two separate classes: (I) those that have reached a membrane for the first time during a small time step, delta t, and (II) those that have not yet reached a cell membrane for the first time during that time step. The time dependent spatial distribution of class (I) molecules is determined using random walk theory, which takes into account the actual paths of the molecules. The variation of the spatial distribution of class (II) molecules with time is determined using the time dependent diffusion equation with a boundary condition of zero concentration on the enclosing membrane boundaries.

Mathematical modelling of the movement of animals, micro-organisms and cells is of great relevance in the fields of biology, ecology and medicine. Movement models can take many different forms, but the most widely used are based on the extensions of simple random walk processes. In this review paper, our aim is twofold: to introduce the mathematics behind random walks in a straightforward manner and to explain how such models can be used to aid our understanding of biological processes. We introduce the mathematical theory behind the simple random walk and explain how this relates to Brownian motion and diffusive processes in general. We demonstrate how these simple models can be extended to include drift and waiting times or be used to calculate first passage times. We discuss biased random walks and show how hyperbolic models can be used to generate correlated random walks. We cover two main applications of the random walk model. Firstly, we review models and results relating to the movement, dispersal and population redistribution of animals and micro-organisms. This includes direct calculation of mean squared displacement, mean dispersal distance, tortuosity measures, as well as possible limitations of these model approaches. Secondly, oriented movement and chemotaxis models are reviewed. General hyperbolic models based on the linear transport equation are introduced and we show how a reinforced random walk can be used to model movement where the individual changes its environment. We discuss the applications of these models in the context of cell migration leading to blood vessel growth (angiogenesis). Finally, we discuss how the various random walk models and approaches are related and the connections that underpin many of the key processes involved.

LetY(t) be a one-dimensional diffusion process anddX(t)=Y(t)dt. The process (X(t), Y(t)) is considered in the second quadrant. First, the probability that (X(t), Y(t)) will hit thex-axis before they-axis is computed explicitly whenY(t) is a standard Brownian motion or a particular case of the Bessel process. Next, an exact expression is obtained for the average time it takes (X(t), Y(t)) to exit the regionC={(x, y) ∈ ℝ2 :x < 0,0 <d 1 <y <d 2 whenY(t) is a standard Brownian motion. The solution is expressed as a generalized Fourier series.

The theory of random walks holds significance across various domains. To this end, this paper discusses the application of random walk theory in the modeling and analysis of complex systems, including the Brownian motion, Wiener process, and Pu migration. The first part mainly explains the concepts of Brownian motion and random walks, as well as their relationship. The core characteristic is the independence and normal distribution of increments, which makes it a special model of random walks. The next part is the Wiener process, which is a continuous time stochastic process. It can describe random phenomena and solve complex problems in finance, encompassing areas such as stock price prediction, option pricing, and risk management. The last part is about Pu migration, with the clear introduction of Pu migration in biology. The authors discuss why the random walk theory can be regarded as the better approach to analysis this problem. Thus, this paper highlights the importance of the random walk in solving many scientific issues.

An experimental study on the effect of mobile phone conversation on drivers' reaction time in braking response

Objective:To determine the impact of cognitive interference on foot pedal reaction time among stroke survivors with right- (RH) or left-hemiplegia (LH).Design:Cross-sectional comparison without randomization.Subjects/patients:10 patients post-stroke with RH, 10 with LH; 10 age-matched controls.Methods:Foot pedal response times were measured using three different reaction time (RT) paradigms: simple RT, dual-task RT (counting backward by serial 3 seconds), and choice RT (correct response contingent on stimuli to eliminate pre-programing). RH and LH used the non-paretic leg for all trials. Three 3 (RT task) × 3 (group) mixed-model factorial ANOVAs were used to compare RT, movement time (MT), total response time (TRT).Results:Overall controls demonstrated faster RT than RH (332 ± 73 versus 474 ± 144 ms, P < 0.001) or LH (402 ± 127 ms, P < 0.05); LH group demonstrated faster RT than those with RH (P < 0.05). Control subjects demonstrated significantly faster RT than RH for all RT conditions (P < 0.05 for all). In contrast, controls achieved significantly faster RT than LH for the choice RT condition only (P < 0.05), but not for the simple (P = 0.12) or dual-task RT conditions (P = 0.25).Conclusions:Compared to controls, response time was significantly impaired among LH and RH when the response could not be pre-programmed. While current simple RT testing commonly employed by driver rehab specialists may be sufficient for detecting RT deficits in patients with RH, simple or dual-task RT tests alone may fail to detect RT deficiencies among LH, even when testing the non-paretic limb. Choice RT should be added to post-stroke driver fitness assessment, particularly for patients with LH.

In the analysis of metastable diffusion processes, Transition Path Theory (TPT) provides a way to quantify the probability of observing a given transition between two disjoint metastable subsets of state space. However, many TPT-based methods for diffusion processes compute the primary objects from TPT, such as the committor and probability current, by solving partial differential equations. The computational performance of these methods is limited by the need for mesh-based computations, the need to estimate the coefficients of the stochastic differential equation that defines the diffusion process, and the use of Markovian processes to approximate the diffusion process. We propose a Monte Carlo method for approximating the primary objects from TPT from sample trajectory data of the diffusion process, without estimating drift or diffusion coefficients. We discretise the state space of the diffusion process using Voronoi tessellations and construct a non-Markovian jump process on the dual Delaunay graph. For the jump process, we define committors, probability currents, and streamlines, and use these to define piecewise constant approximations of the corresponding objects from TPT for diffusion processes. Rigorous error bounds and convergence theorems establish the validity of our approach. A comparison of our method with TPT for Markov chains (Metzner et al., Multiscale Model Simul. 2009) on a triple-well 2D potential provides proof of principle.

We consider non-interacting particles (or lions) performing one-dimensional random walks or Lévy flights (with Lévy index ) in the presence of a constant drift c. Initially these random walkers are uniformly distributed over the positive real line with a density . At the origin z = 0 there is an immobile absorbing trap (or a lamb), such that when a particle crosses the origin, it gets absorbed there. Our main focus is on (i) the flux of particles out of the system (the ‘Smoluchowski problem’) and (ii) the survival probability Sc(n) of the trap or lamb (the ‘lamb-lion problem’) until step n. We show that both observables can be expressed in terms of the average maximum of a single random walk or Lévy flight after n steps. This allows us to obtain the precise asymptotic behavior of both and Sc(n) analytically for large n in the two problems, for any value of and . In particular, for c > 0 and , we show that vanishes as , where is a -dependent positive constant, in contrast with the case of standard random walks (i.e. with ) for which . Our analytical results are confirmed by numerical simulations.

Exact solutions are given for a variety of models of random walks in a chain with time-dependent disorder. Dynamic disorder is modeled by white Poisson noise. Models with site-independent (global) and site-dependent (local) disorder are considered. Results are described in terms of an affective random walk in a nondisordered medium. In the cases of global disorder the effective random walk contains multistep transitions, so that the continuous limit is not a diffusion process. In the cases of local disorder the effective process is equivalent to usual random walk in the absence of disorder but with slower diffusion. Difficulties associated with the continuous-limit representation of random walk in a disordered chain are discussed. In particular, we consider explicit cases in which taking the continuous limit and averaging over disorder sources do not commute.

In transient analysis it is generally thought that small time steps can only improve the accuracy, because standard stability theorems limit the maximum time step for a given mesh size. In finite element approximations, however, small time steps may cause stability problems which lead to physically unreasonable results. It is shown that this is due to the violation of a discrete maximum principle. The influence of lumped and consistent mass matrices on a stable discretization of time and space is presented.

Currently, the mechanical dynamics of soft tissue deformation is achieved by numerical time integrations such as the explicit or implicit integration; however, the explicit integration is stable only under a small time step, whereas the implicit integration is computationally expensive in spite of the accommodation of a large time step. This paper presents a cellular neural network method for stable simulation of soft tissue deformation dynamics. The non-rigid motion equation is formulated as a cellular neural network with local connectivity of cells, and thus the dynamics of soft tissue deformation is transformed into the neural dynamics of the cellular neural network. Results show that the proposed method can achieve good accuracy at a small time step. It still remains stable at a large time step, while maintaining the computational efficiency of the explicit integration. The proposed method can achieve stable soft tissue deformation with efficiency of explicit integration for surgical simulation.