On the stability of alternating‐direction explicit methods for advection‐diffusion equations

Alternating‐Direction Explicit (A.D.E.) finite‐difference methods make use of two approximations that are implemented for computations proceeding in alternating directions, e.g., from left to right and from right to left, with each approximation being explicit in its respective direction of computation. Stable A.D.E. schemes for solving the linear parabolic partial differential equations that model heat diffusion are well‐known, as are stable A.D.E. schemes for solving the first‐order equations of fluid advection. Several of these are combined here to derive A.D.E. schemes for solving time‐dependent advection‐diffusion equations, and their stability characteristics are discussed. In each case, it is found that it is the advection term that limits the stability of the scheme. The most stable of the combinations presented comprises an unconditionally stable approximation for computations carried out in the direction of advection of the system, from left to right in this case, and a conditionally stable approximation for computations proceeding in the opposite direction. To illustrate the application of the methods and verify the stability conditions, they are applied to some quasi‐linear one‐dimensional advection‐diffusion problems. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

The Importance of Reasonable Numerical Schemes in Lake Circulation Simulation

Three difference methods ‐ two traditional central and upstream difference schemes (CDS and UDS) and a highresolution total variation diminishing (TVD) method ‐ are applied to discretize the one‐dimensional advection problem as well as the advection terms in the three‐dimensional advection‐diffusion lake circulation problem. It is demonstrated that the choice of reasonable numerical schemes is fairly importance. The traditional numerical treatments with respect of the advection terms in the lake dynamical equations are fairly arguable, although they are still most frequently employed in lake circulation models, and the TVD technique is most preferable and reasonable.     

A uniform estimate for the MMOC for two‐dimensional advection‐diffusion equations

We prove an optimal‐order error estimate in a weighted energy norm for the modified method of characteristics (MMOC) and the modified method of characteristics with adjusted advection (MMOCAA) for two‐dimensional time‐dependent advection‐diffusion equations, in the sense that the generic constants in the estimates depend on certain Sobolev norms of the true solution but not on the scaling diffusion parameter ε. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Polynomial spectral collocation method for space fractional advection–diffusion equation

This article discusses the spectral collocation method for numerically solving nonlocal problems: one‐dimensional space fractional advection–diffusion equation; and two‐dimensional linear/nonlinear space fractional advection–diffusion equation. The differentiation matrixes of the left and right Riemann–Liouville and Caputo fractional derivatives are derived for any collocation points within any given bounded interval. Several numerical examples with different boundary conditions are computed to verify the efficiency of the numerical schemes and confirm the exponential convergence; the physical simulations for Lévy–Feller advection–diffusion equation and space fractional Fokker–Planck equation with initial δ‐peak and reflecting boundary conditions are performed; and the eigenvalue distributions of the iterative matrix for a variety of systems are displayed to illustrate the stabilities of the numerical schemes in more general cases. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 514–535, 2014     

A non‐standard finite difference scheme for an advection‐diffusion‐reaction equation

In this note, a non‐standard finite difference (NSFD) scheme is proposed for an advection‐diffusion‐reaction equation with nonlinear reaction term. We first study the diffusion‐free case of this equation, that is, an advection‐reaction equation. Two exact finite difference schemes are constructed for the advection‐reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection‐reaction equation is combined with a finite difference space‐approximation of the diffusion term to provide a NSFD scheme for the advection‐diffusion‐reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright © 2014 JohnWiley & Sons, Ltd.     

Numerical stability of the BEM for advection‐diffusion problems

A boundary element method (BEM) approach has been developed to solve the time‐dependent 1D advection‐diffusion equation. The 1D solution is part of a 3D numerical scheme for solving advection‐diffusion (AD) problems in fractured porous media. The full 3D scheme includes a 3D solution for the porous matrix, which is coupled with a 2D solution for fractures and a 1D solution for fracture intersections. As the hydraulic conductivity of the fracture intersections is usually higher than the hydraulic conductivity of the fractures and by at least one order of magnitude higher than the hydraulic conductivity of the porous matrix, the fastest flow and solute transport occurs in the fracture intersections. Therefore it is important to have an accurate and stable 1D solution of the transient AD problems. This article presents two different 1D BEM formulations for solution of the AD problems. The particular advantage of these formulations is that they provide one of the most straightforward and simplest ways to couple multiple intersecting 2D Boundary Element problems discretized with linear discontinuous elements. Both formulations are tested and compared for accuracy, stability, and consistency. The analysis helps to select the more suitable formulations according to the properties of the problem under consideration. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Spurious solutions for the advection‐diffusion equation using wide stencils for approximating the second derivative

A one‐dimensional steady‐state advection‐diffusion problem using summation‐by‐parts operators is investigated. For approximating the second derivative, a wide stencil is used, which simplifies implementation and stability proofs. However, it also introduces spurious, oscillating, modes for all mesh sizes. We prove that the size of the spurious modes are equal to the size of the truncation error for a stable approximation and hence disappears with the convergence rate. The theoretical results are verified with numerical experiments.     

Averaging principle for multiscale stochastic reaction‐diffusion‐advection equations

Stochastic averaging principle is a powerful tool for studying qualitative analysis of multiscale stochastic dynamical systems. In this paper, we will establish an averaging principle for stochastic reaction‐diffusion‐advection equations with slow and fast time scales. Under suitable conditions, we show that the slow component strongly converges to the solution of the corresponding averaged equation.     

A NUMERICAL STUDY OF THE SPRUCE BUDWORM REACTION‐DIFFUSION EQUATION WITH HOSTILE BOUNDARIES

ABSTRACT. One of the interesting single species reaction diffusion problems is the spruce budworm model describing insect dispersal behavior. In an earlier study, Singh et al. [7] considered the two‐dimensional spruce budworm model with density dependent diffusion balanced by an artificial wind equal to the population gradient. Here we extend the model by considering more realistic density dependent diffusion and advection with hostile boundaries. We solve this model using a splitting method in which advection, diffusion and reaction processes are separated. Various hostility conditions have been used at the boundary. The numerical results show that the population moves quickly to a steady state outbreak situation when the advective components due to the density dependent diffusion are included.     

Analysis of algebraic systems arising from fourth‐order compact discretizations of convection‐diffusion equations

We study the properties of coefficient matrices arising from high‐order compact discretizations of convection‐diffusion problems. Asymptotic convergence factors of the convex hull of the spectrum and the field of values of the coefficient matrix for a one‐dimensional problem are derived, and the convergence factor of the convex hull of the spectrum is shown to be inadequate for predicting the convergence rate of GMRES. For a two‐dimensional constant‐coefficient problem, we derive the eigenvalues of the nine‐point matrix, and we show that the matrix is positive definite for all values of the cell‐Reynolds number. Using a recent technique for deriving analytic expressions for discrete solutions produced by the fourth‐order scheme, we show by analyzing the terms in the discrete solutions that they are oscillation‐free for all values of the cell Reynolds number. Our theoretical results support observations made through numerical experiments by other researchers on the non‐oscillatory nature of the discrete solution produced by fourth‐order compact approximations to the convection‐diffusion equation. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 155–178, 2002; DOI 10.1002/num.1041     

A note on second‐order finite‐difference schemes on uniform meshes for advection‐‐diffusion equations

An artificial‐viscosity finite‐difference scheme is introduced for stabilizing the solutions of advection‐diffusion equations. Although only the linear one‐dimensional case is discussed, the method is easily susceptible to generalization. Some theory and comparisons with other well‐known schemes are carried out. The aim is, however, to explain the construction of the method, rather than considering sophisticated applications. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 581–588, 1999     

Taylor‐Galerkin B‐spline finite element method for the one‐dimensional advection‐diffusion equation

The advection‐diffusion equation has a long history as a benchmark for numerical methods. Taylor‐Galerkin methods are used together with the type of splines known as B‐splines to construct the approximation functions over the finite elements for the solution of time‐dependent advection‐diffusion problems. If advection dominates over diffusion, the numerical solution is difficult especially if boundary layers are to be resolved. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show the behavior of the method with emphasis on treatment of boundary conditions. Taylor‐Galerkin methods have been constructed by using both linear and quadratic B‐spline shape functions. Results shown by the method are found to be in good agreement with the exact solution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A numerical solution for advection‐diffusion equation using dual reciprocity method

This article describes a numerical method based on the boundary integral equation and dual reciprocity method(DRM) for solving the one‐dimensional advection‐diffusion equations. The concept of DRM is used to convert the domain integral to the boundary that leads to an integration free method. The time derivative is approximated by the time‐stepping method. Numerical results are presented for some problems to demonstrate the usefulness and accuracy of the new approach. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Uniform approximation of singularly perturbed reaction‐diffusion problems by the finite element method on a Shishkin mesh

We consider the numerical approximation of singularly perturbed reaction‐diffusion problems over two‐dimensional domains with smooth boundary. Using the h version of the finite element method over appropriately designed piecewise uniform (Shishkin) meshes, we are able to uniformly approximate the solution at a quasi‐optimal rate. The results of numerical computations showing agreement with the analysis are also presented. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 89–111, 2003     

Evaluating flood extent mapping of two hydraulic models, 1D HEC‐RAS and 2D LISFLOOD‐FP in comparison with aerial imagery observations in Gorgan flood plain,Iran

We compared flood mapping techniques using a one‐dimensional (1D) hydraulic model HEC‐RAS and two‐dimensional (2D) LISFLOOD‐FP for a 10‐km reach of Gorgan River in Iran. Both models were run using the same hydrologic input data. The input into the models was a steady discharge of 90 cm, corresponds to a flood peak occurred on March 25, 2012. Flood maps generated using these two models were compared with an observed flood inundation map, using F‐statistic. The roughness coefficients of the models were calibrated by maximizing the value of the F‐statistic. Based on the F‐statistic, LISFLOOD‐FP gives a slightly better result (F = 0.69) than HEC‐RAS (F = 0.67). Visual comparison of the flood extents generated by the two models showed reasonably good agreement. Validation was done using a flood event occurred on May 31, 2014. The LISFLOOD‐FP model gave a better result for validation as well. The 2D model showed more consistency in comparison with the 1D model.     

Convergence analysis of the streamline diffusion and discontinuous Galerkin methods for the Vlasov‐Fokker‐Planck system

We prove stability estimates and derive optimal convergence rates for the streamline diffusion and discontinuous Galerkin finite element methods for discretization of the multi‐dimensional Vlasov‐Fokker‐Planck system. The focus is on the theoretical aspects, where we deal with construction and convergence analysis of the discretization schemes. Some related special cases are implemented in M. Asadzadeh [Appl Comput Meth 1(2) (2002), 158–175] and M. Asadzadeh and A. Sopasakis [Comput Meth Appl Mech Eng 191(41–42) (2002), 4641–4661]. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

An optimal‐order error estimate for MMOC and MMOCAA schemes for multidimensional advection‐reaction equations

In this article, we analyze the modified method of characteristics (MMOC) and an improved version of the MMOC, named the modified method of characteristics with adjusted advection (MMOCAA), for multidimensional advection‐reaction transport equations in a uniform manner. We derive an optimal‐order error estimate for these schemes. Numerical results are presented to verify the theoretical estimates. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 69–84, 2002     

Numerical three-dimensional free surface circulation model for the South Biscayne Bay,Florida

A three– dimensional, time dependent free surface model has been developed for predicting circulation and surface height variations in a tidal bay. An explicit finite difference numerical solution is obtained by transforming the vertical coordinate in the governing model equations. The transformed geometry consists of a fixed, flat free surface and a constant depth basin for easy computation. The ocean–bay interface open boundary condition is incorporated into this hydrodynamic model without approximation, and yields rather accurate results for the bay circulation and tide level variations. The numerical method employs a staggered grid Richardson lattice, which has the inherent property of not requiring calculation of the tangential velocity components on solid surfaces. The momentum equations ignore horizontal diffusion which is small for the South Biscayne Bay, for which vertical diffusion and advection dominate.     

An efficient high‐order algorithm for solving systems of reaction‐diffusion equations

An efficient higher‐order finite difference algorithm is presented in this article for solving systems of two‐dimensional reaction‐diffusion equations with nonlinear reaction terms. The method is fourth‐order accurate in both the temporal and spatial dimensions. It requires only a regular five‐point difference stencil similar to that used in the standard second‐order algorithm, such as the Crank‐Nicolson algorithm. The Padé approximation and Richardson extrapolation are used to achieve high‐order accuracy in the spatial and temporal dimensions, respectively. Numerical examples are presented to demonstrate the efficiency and accuracy of the new algorithm. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 340–354, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10012     

A regular perturbation analytical‐numerical method for the evolution of precancerous lesions caused by the human papillomavirus

A numerical method is developed to analyze the behavior of the evolution of the lesions at the cervical cells caused by the human papillomavirus. The model to be solved consists in a one‐dimensional nonlinear advection–diffusion‐reaction equation. Such equation is approximated by a consistent explicit difference scheme which is based on regular perturbation theory. A constructive procedure for the numerical scheme is given and finally an illustrative example of the evolution of a mild dysplasia is included. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 847–855, 2015