Discrete Diffusion Equation Research Articles

Time-fractional discrete diffusion equation for Schrödinger operator

Time-fractional discrete diffusion equation for Schrödinger operator

Banded preconditioning with shift compensation for solving discrete Riesz space-fractional diffusion equations

Banded preconditioning with shift compensation for solving discrete Riesz space-fractional diffusion equations

Longtime behavior for solutions to a temporally discrete diffusion equation with a free boundary

Longtime behavior for solutions to a temporally discrete diffusion equation with a free boundary

Finite Difference Scheme and Finite Volume Scheme for Fractional Laplacian Operator and Some Applications

The fractional Laplacian operator is a very important fractional operator that is often used to describe several anomalous diffusion phenomena. In this paper, we develop some numerical schemes, including a finite difference scheme and finite volume scheme for the fractional Laplacian operator, and apply the resulting numerical schemes to solve some fractional diffusion equations. First, the fractional Laplacian operator can be characterized as the weak singular integral by an integral operator with zero boundary condition. Second, because the solutions of fractional diffusion equations are usually singular near the boundary, we use a fractional interpolation function in the region near the boundary and use a classical interpolation function in other intervals. Then, we apply a finite difference scheme to the discrete fractional Laplacian operator and fractional diffusion equation with the above fractional interpolation function and classical interpolation function. Moreover, it is found that the differential matrix of the above scheme is a symmetric matrix and strictly row-wise diagonally dominant in special fractional interpolation functions. Third, we show a finite volume scheme for a discrete fractional diffusion equation by fractional interpolation function and classical interpolation function and analyze the properties of the differential matrix. Finally, the numerical experiments are given, and we verify the correctness of the theoretical results and the efficiency of the schemes.

Diffusion-aware voltage source: An equivalent circuit network to resolve lithium concentration gradients in active particles

Traditional equivalent circuit models (ECMs) have difficulties in estimating battery internal states due to the lack of relevant physics, such as the lithium diffusion in active particles. Here we configure a circuit network to describe the lithium diffusion and define it as a new high-level circuit element called diffusion-aware voltage source. The circuit representation is proven equivalent to the discretized diffusion equation. The new voltage source gives the electrode potential as a function of the surface concentration and thus automatically incorporates the diffusion overpotential. We show that an ECM with the proposed diffusion-aware voltage sources (called “shell ECM”) can reproduce the single particle model simulation results, making it a trustworthy easy-to-implement substitute. Furthermore, the simplest shell ECM consisting of a single diffusion-aware voltage source and a resistor is validated against experimental constant-current discharges at various rates. The diffusion-aware voltage source can be used to measure diffusivity by fitting the diffusion resistance against experimental data. The viability of the shell ECM for onboard usage is confirmed by implementation into a battery management system of WAE Technologies. By tracking the internal concentration states, the shell ECM demonstrates robustness to dynamic applied-current profiles.

Generalized discrete and ultradiscrete Burgers equations derived through the correlated random walk

The correlated random walk is known as a generalization of the well-known random walk. In this study, we present that a generalized discrete Burgers equation corresponding to the correlated random walk can be obtained through a Cole–Hopf transformation to a generalized discrete diffusion equation. By applying a technique called ultradiscretization, the generalized ultradiscrete diffusion equation, the ultradiscrete Cole–Hopf transformation, and a variant of the ultradiscrete Burgers equation are obtained. Additionally, this study shows that the resulting ultradiscrete Burgers equation yields cellular automata that can be interpreted as a traffic flow model with controllability of vehicle flow.

A hybrid epidemiological model based on individual dynamics

This article proposes a hybrid susceptible–infected–removed model which takes into consideration the spatiotemporal dynamics of the individuals. The model is based on a system of discrete stochastic diffusion equations. To build these equations, the two-dimensional diffusion equations coming from a balanced method are coupled with the human displacement probability law pattern through a discretization made by the finite volume method for complex geometries. The validation of this model is applied to COVID-19 spread. Since it is actual and the statistics are available elsewhere. The case of a developed country is used for simulation under some assumptions. Firstly to fit the chosen displacement pattern, then the accuracy of the statistics provided helps to analyze the sensitivity of the parameters of the model. The results explain the influence of the population movement on the evolution of the spread.

A lopsided scaled DTS preconditioning method for the discrete space-fractional diffusion equations

The Grünwald–Letnikov type formulas are applied to discretize the space-fractional diffusion equation and the discretization result is got as a system of linear equations. A lopsided scaled diagonal and Toeplitz splitting (LSDTS) iteration method is constructed to solve this linear system and the convergence is also discussed. An LSDTSτ preconditioner is proposed based on the LSDTS iteration method and the GMRES method combined with this preconditioner is applied to solve the linear system. We theoretically show that the eigenvalues of the LSDTSτ preconditioned matrix are clustered. The numerical experiments illustrate that the LSDTSτ preconditioner can significantly improve the convergence properties of the GMRES method.

Spatial maxima, unimodality, and asymptotic behaviour of solutions to discrete diffusion-type equations

We consider a class of partial difference equations which generalizes the discrete diffusion (heat) equation. In the first part of the paper, we study their fundamental solutions, and focus on the location of their spatial maxima and unimodality. In the second part, we describe the asymptotic behaviour of solutions with arbitrary bounded initial data. Throughout the paper, we take advantage of the relation between discrete diffusion-type equations and random walks on the set of integers.

Discrete heat kernel, UV modified Green’s function, and higher derivative theories

We perform the UV deformation of the Green’s function in free scalar field theory using a discrete heat kernel method. It is found that the simplest UV deformation based on the discretized diffusion equation leads to the well-known Pauli–Villars effective Lagrangian. Furthermore, by extending assumptions on the discretized equation, we find that the general higher derivative theory is derived from the present UV deformation. In some specific cases, we also calculate the vacuum expectation values of the scalar field squared in nontrivial background spaces and examine their dependence on the UV cutoff constant.

Bulk-edge correspondence of classical diffusion phenomena

We elucidate that diffusive systems, which are widely found in nature, can be a new platform of the bulk-edge correspondence, a representative topological phenomenon. Using a discretized diffusion equation, we demonstrate the emergence of robust edge states protected by the winding number for one- and two-dimensional systems. These topological edge states can be experimentally accessible by measuring diffusive dynamics at edges. Furthermore, we discover a novel diffusion phenomenon by numerically simulating the distribution of temperatures for a honeycomb lattice system; the temperature field with wavenumber pi cannot diffuse to the bulk, which is attributed to the complete localization of the edge state.

Diffusion equations from master equations—A discrete geometric approach

In this paper, continuous-time master equations with finite states employed in nonequilibrium statistical mechanics are formulated in the language of discrete geometry. In this formulation, chains in algebraic topology are used, and master equations are described on graphs that consist of vertices representing states and of directed edges representing transition matrices. It is then shown that master equations under the detailed balance conditions are equivalent to discrete diffusion equations, where the Laplacians are defined as self-adjoint operators with respect to introduced inner products. An isospectral property of these Laplacians is shown for non-zero eigenvalues, and its applications are given. The convergence to the equilibrium state is shown by analyzing this class of diffusion equations. In addition, a systematic way to derive closed dynamical systems for expectation values is given. For the case that the detailed balance conditions are not imposed, master equations are expressed as a form of a continuity equation.

Exact Spatiotemporal Dynamics of Confined Lattice Random Walks in Arbitrary Dimensions: A Century after Smoluchowski and Pólya

An exact solution to the discrete diffusion equation allows for accurate predictions of how the probabilities of reaction diffusion processes evolve over time.

Personalized Tumor Growth Prediction Using Multiscale Modeling

Purpose: Cancer is one of the most complex phenomena in biology and medicine. Extensive attempts have been made to work around this complexity. In this study, we try to take a selective approach; not modeling each particular facet in detail but rather only the pertinent and essential parts of the tumor system are simulated and followed by optimization, revealing specific traits. This leads us to a pellucid personalized model which is noteworthy as it closely approximates existing experimental results. Methods: In the present study, a hybrid modeling approach which consists of cellular automata for discrete cell state representation and diffusion equations to calculate distribution of relevant substances in the tumor microenvironment is favored. Moreover, naive Bayesian decision making with weighted stochastic equations and a Bayesian network to model the temporal order of mutations is presented. The model is personalized according to the evidence using Markov Chain Monte Carlo. To validate the tumor model, a data set belonging to the A549 cell line is used. The data represents the growth of a tumor for 30 days. We optimize the coefficients of the stochastic decision-making equations using the first half of the timeline. Results: Simulation results of the developed model are promising with their low error margin all correlation coefficients are over 0.8 under different microenvironment conditions and simulated growth data is in line with laboratory results r=0.97, p

Global Stability of Non-monotone Noncritical Traveling Waves for a Discrete Diffusion Equation with a Convolution Type Nonlinearity

This paper is concerned with the global stability of non-monotone traveling waves for a discrete diffusion equation with a monostable convolution type nonlinearity. It has been proved by Yang and Zhang (Sci. China Math. 61 (2018), 1789--1806) that all noncritical traveling waves (waves with speeds $c \gt c_*$, $c_*$ is minimal speed) are time-exponentially stable, when the initial perturbations around the waves are small. In this paper, we further prove that all traveling waves with large speed are globally stable, when the initial perturbations around the waves in a weighted Sobolev space can be arbitrarily large. The approaches adopted are the nonlinear Halanay's inequality, the technical weighted energy method and Fourier's transform.

Diffusion synthetic acceleration with the fine mesh rebalance of the subcell balance method with tetrahedral meshes for SN transport calculations

Abstract A diffusion synthetic acceleration (DSA) technique for the SN transport equation discretized with the linear discontinuous expansion method with subcell balance (LDEM-SCB) on unstructured tetrahedral meshes is presented. The LDEM-SCB scheme solves the transport equation with the discrete ordinates method by using the subcell balances and linear discontinuous expansion of the flux. Discretized DSA equations are derived by consistently discretizing the continuous diffusion equation with the LDEM-SCB method, however, the discretized diffusion equations are not fully consistent with the discretized transport equations. In addition, a fine mesh rebalance (FMR) method is devised to accelerate the discretized diffusion equation coupled with the preconditioned conjugate gradient (CG) method. The DSA method is applied to various test problems to show its effectiveness in speeding up the iterative convergence of the transport equation. The results show that the DSA method gives small spectral radii for the tetrahedral meshes having various minimum aspect ratios even in highly scattering dominant mediums for the homogeneous test problems. The numerical tests for the homogeneous and heterogeneous problems show that DSA with FMR (with preconditioned CG) gives significantly higher speedups and robustness than the one with the Gauss-Seidel-like iteration.

A three-dimensional Fourier analysis of fine mesh rebalance acceleration of linear discontinuous sub-cell balance method for diffusion equation on tetrahedral meshes

This paper presents a three-dimensional Fourier analysis of the fine mesh rebalance (FMR) acceleration of the Linear Discontinuous Expansion Method with Subcell Balance (LDEM-SCB) discretization method for solving the neutron diffusion equation on three-dimensional unstructured tetrahedral meshes in order to theoretically understand the convergence characteristics. The three-dimensional Fourier analysis is performed on one and two cubes called the basic elements in which each element is divided into six tetrahedral meshes for translation and reflective boundary conditions on the external boundaries. The Fourier analysis shows that the FMR method significantly accelerates the Gauss-Seidel (GS)-like iteration to solve the LDEM-SCB discretized diffusion equation. The analytical results of the homogeneous test problem taking different aspect ratios show that the convergence of FMR with the translation boundary conditions is less sensitive on the aspect ratio than the one with the reflective boundary conditions. In addition, we identified the convergence characteristics of the GS-like iteration and its FMR acceleration of the heterogeneous problem.

Change agents and internal communications in organizational networks

The adoption of change in organizational networks is conditioned by the engagement of the so-called “change agents” initiating “cascades of change”, as well as by the internal communication among the members of the network. We investigate how the dynamics of the adoption of change is influenced by the engagement policy of the change agents and by the internal communication. In this perspective, we compare central engagements (high degree, high closeness, high betweenness, high eigen-centrality) with random engagement. We also compare three selection rules for communication, namely: selection by randomness, selection by high link weight, and selection by high link weight and degree centrality. The dynamics of change adoption is modeled by generalizing the discrete diffusion equation in order to incorporate both the engagement policies of the change agents and the internal communication rules. Results are obtained by simulating the solutions (change management scenaria) on 4 real organizational networks. Agents with high degree (hubs) are found to be most suitable to act as change agents. The most suitable change agents however, are not always the senior employees. The adoption of change is much faster, if agents communicate with “local hubs”, avoiding random contacts. Change agents feed the members of the network with “thin slices” of influence, in order to avoid crossing the “confidence bound”. Small increase of the size of “slices”, results in significant acceleration of the adoption of change. We also estimate the value of “planning” versus “no planning”, in the context of Change Management.

Population dynamic caused by war involvement via fractional derivative on time scales

This work suggests a model for a population dynamic caused by an enemy attack to a domain of residential areas. With the help of a local non-integer order rate of change and a new structure induced on the real line, we derive a spatial discrete diffusion equation of fractional order. Then making use of the d'Alembert's change of variable we obtain a time scale which is made of union of disjoint compact intervals. These considerations lead us to a non-homogeneous second order nonlinear differential equation. The existence of a positive solution is discussed and through a numerical example the theory is illustrated.

Special Solutions to a Nonlinear Coarsening Model with Local Interactions

We consider a class of mass transfer models on a one-dimensional lattice with nearest-neighbour interactions. The evolution is given by the discrete backward fast diffusion equation, with exponent $\beta$ in the regime $(-\infty,0) \cup (0,1]$. Sites with mass zero are deleted from the system, which leads to a coarsening of the mass distribution. The rate of coarsening suggested by scaling is $t^\frac{1}{1-\beta}$ if $\beta \neq 1$ and exponential if $\beta = 1$. We prove that such solutions actually exist by an analysis of the time-reversed evolution. In particular we establish positivity estimates and long-time equililibrium properties for discrete parabolic equations with bounded initial data.