Multi-dimensional asymptotically stable finite difference schemes for the advection–diffusion equation

An algorithm is presented which solves the multi-dimensional advection–diffusion equation on complex shapes to second-order accuracy and is asymptotically stable in time. This bounded-error result is achieved by constructing, on a rectangular grid, a differentiation matrix whose symmetric part is negative definite. The differentiation matrix accounts for the Dirichlet boundary condition by imposing penalty-like terms. Numerical examples in two dimensions show that the method is effective even where standard schemes, stable by traditional definitions, fail. It gives accurate, non oscillatory results even when boundary layers are not resolved.     

Fractional Jacobi Galerkin spectral schemes for multi-dimensional time fractional advection–diffusion–reaction equations

One of the ongoing issues with time fractional diffusion models is the design of efficient high-order numerical schemes for the solutions of limited regularity. We construct in this paper two efficient Galerkin spectral algorithms for solving multi-dimensional time fractional advection–diffusion–reaction equations with constant and variable coefficients. The model solution is discretized in time with a spectral expansion of fractional-order Jacobi orthogonal functions. For the space discretization, the proposed schemes accommodate high-order Jacobi Galerkin spectral discretization. The numerical schemes do not require imposition of artificial smoothness assumptions in time direction as is required for most methods based on polynomial interpolation. We illustrate the flexibility of the algorithms by comparing the standard Jacobi and the fractional Jacobi spectral methods for three numerical examples. The numerical results indicate that the global character of the fractional Jacobi functions makes them well-suited to time fractional diffusion equations because they naturally take the irregular behavior of the solution into account and thus preserve the singularity of the solution.

     

Asymptotic preserving schemes for the Wigner–Poisson–BGK equations in the diffusion limit

This work focuses on the numerical simulation of the Wigner–Poisson–BGK equation in the diffusion asymptotics. Our strategy is based on a “micro–macro” decomposition, which leads to a system of equations that couple the macroscopic evolution (diffusion) to a microscopic kinetic contribution for the fluctuations. A semi-implicit discretization provides a numerical scheme which is stable with respect to the small parameter ε$\epsilon$ (mean free path) and which possesses the following properties: (i) it enjoys the asymptotic preserving property in the diffusive limit; (ii) it recovers a standard discretization of the Wigner–Poisson equation in the collisionless regime. Numerical experiments confirm the good behavior of the numerical scheme in both regimes. The case of a spatially dependent ε(x)$\epsilon \left(x\right)$ is also investigated.     

Analysis of running waves stability in the Ginzburg–Landau equation with small diffusion

The local dynamics of the Ginzburg–Landau equation with small diffusion and periodic boundary conditions in the neighborhood of running waves is analyzed. Sufficient conditions for the instability of running waves and sufficient conditions for their stability are found.     

Controllability and observability of an artificial advection–diffusion problem

In this paper we study the controllability of an artificial advection?Cdiffusion system through the boundary. Suitable Carleman estimates give us the observability of the adjoint system in the one dimensional case. We also study some basic properties of our problem such as backward uniqueness and we get an intuitive result on the control cost for vanishing viscosity.     

On the stabilization parameter in the subgrid scale approximation of scalar convection–diffusion–reaction equations on distorted meshes

In this paper we revisit the definition of the stabilization parameter in the finite element approximation of the convection–diffusion–reaction equation. The starting point is the decomposition of the unknown into its finite element component and a subgrid scale that needs to be approximated. In order to incorporate the distortion of the mesh into this approximation, we transform the equation for the subgrid scale within each element to the shape-regular reference domain. The expression for the subgrid scale arises from an approximate Fourier analysis and the identification of the wave number direction where instabilities are most likely to occur. The final outcome is an expression for the stabilization parameter that accounts for anisotropy and the dominance of either convection or reaction terms in the equation.     

A higher-order moment method of the lattice Boltzmann model for the Korteweg–de Vries equation

In this paper, a lattice Boltzmann model for the Korteweg–de Vries (KdV) equation with higher-order accuracy of truncation error is presented by using the higher-order moment method. In contrast to the previous lattice Boltzmann model, our method has a wide flexibility to select equilibrium distribution function. The higher-order moment method bases on so-called a series of lattice Boltzmann equation obtained by using multi-scale technique and Chapman–Enskog expansion. We can also control the stability of the scheme by modulating some special moments to design the dispersion term and the dissipation term. The numerical example shows the higher-order moment method can be used to raise the accuracy of truncation error of the lattice Boltzmann scheme.     

Lattice Boltzmann simulation of some nonlinear convection–diffusion equations

In this work we proposed a lattice Boltzmann model for the nonlinear convection–diffusion equation (NCDE) with anisotropic diffusion. The constraints on the model for correctly recovering macroscopic equation are also carefully analyzed, which are ignored in some existing work. Detailed simulations of some 1D/2D NCDEs, including the nonlinear Schrödinger equation (NLSE), Buckley–Leverett equation with discontinuous initial data, NCDE with anisotropic diffusion, and generalized Zakharov system, are performed. The numerical results obtained by the proposed model agree well with the analytical solutions and/or the numerical solutions reported in previous studies. It is also found that, for complex-valued NLSE, the model using a complex distribution function is superior to that using two real distribution functions for the real and imaginary parts of the NLSE separately.     

The meshless approach for solving 2D variable-order time-fractional advection–diffusion equation arising in anomalous transport

In this paper, with the aim of extending an elegant and straightforward numerical approximation to describe one of the most common physical phenomena has been undertaken. In this regard, the generalization of advection–diffusion equation, namely, the time-fractional advection–diffusion equation with understanding sense variable-order fractional derivative, is taken into consideration. An efficient and accurate approach is relying on the Kansa scheme and finite difference method to provide a mathematical framework to treat the spatial discretization and temporal term, respectively. The meshless collocation approach is utilized for interior scattered points and those on the boundary. Thus, the problem under consideration is reduced to a system of linear algebraic equations. The use of the radial basis function as shape function brings many advantages for proposal numerical method in terms of improved accuracy by setting an appropriate shape parameter and applied for solving high-dimensional models without extra cost. The validity and accuracy of the proposed approach is investigated by four various examples involving three benchmark examples and a practical application of pollution transfer phenomena.

     

Numerical efficiency of some exponential methods for an advection–diffusion equation

In this paper, we investigate several modified exponential finite-difference methods to approximate the solution of the one-dimensional viscous Burgers' equation. Burgers' equation admits solutions that are positive and bounded under appropriate conditions. Motivated by these facts, we propose nonsingular exponential methods that are capable of preserving some structural properties of the solutions of Burgers' equation. The fact that some of the techniques preserve structural properties of the solutions is thoroughly established in this work. Rigorous analyses of consistency, stability and numerical convergence of these schemes are presented for the first time in the literature, together with estimates of the numerical solutions. The methods are computationally improved for efficiency using the Padé approximation technique. As a result, the computational cost is substantially reduced in this way. Comparisons of the numerical approximations against the exact solutions of some initial-boundary-value problems for different Reynolds numbers show a good agreement between them.     

A unified operator splitting approach for multi-scale fluid–particle coupling in the lattice Boltzmann method

A unified framework to derive discrete time-marching schemes for the coupling of immersed solid and elastic objects to the lattice Boltzmann method is presented. Based on operator splitting for the discrete Boltzmann equation, second-order time-accurate schemes for the immersed boundary method, viscous force coupling and external boundary force are derived. Furthermore, a modified formulation of the external boundary force is introduced that leads to a more accurate no-slip boundary condition. The derivation also reveals that the coupling methods can be cast into a unified form, and that the immersed boundary method can be interpreted as the limit of force coupling for vanishing particle mass. In practice, the ratio between fluid and particle mass determines the strength of the force transfer in the coupling. The integration schemes formally improve the accuracy of first-order algorithms that are commonly employed when coupling immersed objects to a lattice Boltzmann fluid. It is anticipated that they will also lead to superior long-time stability in simulations of complex fluids with multiple scales.     

Dynamical behavior of a general reaction–diffusion–advection model for two competing species

It is well known that the studies of the evolution of biased movement along a resource gradient could create very interesting phenomena. This paper deals with a general two-species Lotka–Volterra competition model for the same resources in an advective nonhomogeneous environment, where the individuals are exposed to unidirectional flow (advection) but no individuals are lost through the boundary. It is assumed that the two species have the same population dynamics but different diffusion and advection rates. It is shown that at least five scenarios can occur (i) If one with a very strong biased movement relative to diffusion and the other with a more balanced approach, the species with much larger advection dispersal rate is driven to extinction; (ii) If one with a very strong biased movement and the other is smaller compare to its diffusion, the two species can coexist since one species mainly pursues resources at places of locally most favorable environments while the other relies on resources from other parts of the habitat; (iii) If both of the species random dispersal rates are sufficiently large (respectively small), two competing species coexist; (iv) If one with a sufficiently large random dispersal rate and the other with a sufficiently small one, two competing species still coexist; (v) If one with a sufficiently small random dispersal rate and the other with a suitable diffusion, which causes the extinction of the species with smaller random movement. Where (iii), (iv) and (v) show the global dynamics of (5) when both of the species dispersal rates are sufficiently large or sufficiently small. These results provide a new mechanism for the coexistence of competing species, and they also imply that selection is against excessive advection along environmental gradients (respectively, random dispersal rate), and an intermediate biased movement rate (respectively, random dispersal rate) may evolve. Finally, we also apply a perturbation argument to illustrate the evolution of these rates.     

Large-scale parallel lattice Boltzmann–cellular automaton model of two-dimensional dendritic growth

An extremely scalable lattice Boltzmann (LB)–cellular automaton (CA) model for simulations of two-dimensional (2D) dendritic solidification under forced convection is presented. The model incorporates effects of phase change, solute diffusion, melt convection, and heat transport. The LB model represents the diffusion, convection, and heat transfer phenomena. The dendrite growth is driven by a difference between actual and equilibrium liquid composition at the solid–liquid interface. The CA technique is deployed to track the new interface cells. The computer program was parallelized using the Message Passing Interface (MPI) technique. Parallel scaling of the algorithm was studied and major scalability bottlenecks were identified. Efficiency loss attributable to the high memory bandwidth requirement of the algorithm was observed when using multiple cores per processor. Parallel writing of the output variables of interest was implemented in the binary Hierarchical Data Format 5 (HDF5) to improve the output performance, and to simplify visualization. Calculations were carried out in single precision arithmetic without significant loss in accuracy, resulting in 50% reduction of memory and computational time requirements. The presented solidification model shows a very good scalability up to centimeter size domains, including more than ten million of dendrites.     

The local discontinuous Galerkin method for 2D nonlinear time-fractional advection–diffusion equations

This paper presents a numerical solution of time-fractional nonlinear advection–diffusion equations (TFADEs) based on the local discontinuous Galerkin method. The trapezoidal quadrature scheme (TQS) for the fractional order part of TFADEs is investigated. In TQS, the fractional derivative is replaced by the Volterra integral equation which is computed by the trapezoidal quadrature formula. Then the local discontinuous Galerkin method has been applied for space-discretization in this scheme. Additionally, the stability and convergence analysis of the proposed method has been discussed. Finally some test problems have been investigated to confirm the validity and convergence of the proposed method.

     

Family of quadratic spline difference schemes for a convection–diffusion problem

We consider the finite difference approximation of a singularly perturbed one-dimensional convection–diffusion two-point boundary value problem. It is discretized using quadratic splines as approximation functions, equations with various piecewise constant coefficients as collocation equations and a piecewise uniform mesh of Shishkin type. The family of schemes is derived using the collocation method. The numerical methods developed here are non-monotone and therefore apart from the consistency error we use Green's grid function analysis to prove uniform convergence. We prove the almost first order of convergence and furthermore show that some of the schemes have almost second-order convergence. Numerical experiments presented in the paper confirm our theoretical results.     

Numerical solutions of fractional advection–diffusion equations with a kind of new generalized fractional derivative

In the current paper, the numerical solutions for a class of fractional advection–diffusion equations with a kind of new generalized time-fractional derivative proposed last year are discussed in a bounded domain. The fractional derivative is defined in the Caputo type. The numerical solutions are obtained by using the finite difference method. The stability of numerical scheme is also investigated. Numerical examples are solved with different fractional orders and step sizes, which illustrate that the numerical scheme is stable, simple and effective for solving the generalized advection–diffusion equations. The order of convergence of the numerical scheme is evaluated numerically, and the first-order convergence rate has been observed.     

The intention–behaviour gap in technology usage: the moderating role of attitude strength

Extant theories of information technology (IT) usage present users' behavioural intention as the primary predictor of their IT usage behaviour. However, empirical evidence reveals only a low-to-medium effect size for this association. We call this inconsistency the ‘intention–behaviour gap’, and argue that a clearer understanding of this gap requires a deeper theoretical examination of the conditions under which intentions may or may not influence behaviour. Drawing on recent attitude theoretic research in social psychology, we distinguish between two types of attitudes – strong versus weak – and suggest that the intention–behaviour association may hold for users with strong attitudes but is likely to be weaker for those with weak attitudes. Using the elaboration-likelihood model, we propose two dimensions of attitude strength relevant to the IT usage context – personal relevance and related expertise – and theorise them to moderate the intention–behaviour association in a positive manner. Results from a longitudinal field survey of document management system usage among governmental employees at L'viv City Hall, Ukraine support our theoretical hypotheses. Theoretical and practical implications of our findings are discussed.     

Re-assessment of diffusion mobilities in the face-centered cubic Cu–Sn alloys

Based on the available thermodynamic information and diffusion coefficient data of the Cu–Sn binary system, the atomic mobilities of Cu and Sn in face-centered cubic (fcc) Cu–Sn alloys have been assessed as a function of temperature and composition in terms of the CALPHAD method using the DICTRA^® software package. Optimized mobility parameters are presented. To testify the reliability of the optimized mobility parameters, the composition–distance profiles of the Cu/Cu-7 at.% Sn diffusion couples at different temperatures (923 K, 1023 K and 1053 K) have been measured in the present work by means of electronic probe micro-analysis (EPMA). Comparisons between the calculated and measured diffusion coefficients show that most of the experimental information can be reproduced reasonably. The obtained mobility parameters can also predict satisfactorily composition–distance profiles of the diffusion zone in binary Cu–Sn diffusion couples determined in the present work and reported in published literature.     

A meshless method for solving the time fractional advection–diffusion equation with variable coefficients

In this paper, an efficient and accurate meshless method is proposed for solving the time fractional advection–diffusion equation with variable coefficients which is based on the moving least square (MLS) approximation. In the proposed method, firstly the time fractional derivative is approximated by a finite difference scheme of order $O\left({\left(\delta t\right)}^{2?\alpha }\right),0<\alpha \le 1$ and then the MLS approach is employed to approximate the spatial derivative where time fractional derivative is expressed in the Caputo sense. Also, the validity of the proposed method is investigated in error analysis discussion. The main aim is to show that the meshless method based on the MLS shape functions is highly appropriate for solving fractional partial differential equations (FPDEs) with variable coefficients. The efficiency and accuracy of the proposed method are verified by solving several examples.