Two‐grid methods for mixed finite‐element solution of coupled reaction‐diffusion systems

We develop 2‐grid schemes for solving nonlinear reaction‐diffusion systems: where p = (p, q) is an unknown vector‐valued function. The schemes use discretizations based on a mixed finite‐element method. The 2‐grid approach yields iterative procedures for solving the nonlinear discrete equations. The idea is to relegate all the Newton‐like iterations to grids much coarser than the final one, with no loss in order of accuracy. The iterative algorithms examined here extend a method developed earlier for single reaction‐diffusion equations. An application to prepattern formation in mathematical biology illustrates the method's effectiveness. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 589–604, 1999     

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.     

A new residual posteriori error estimates of mixed finite element methods for convection‐diffusion‐reaction equations

We propose and analyze a new technique for developing residual‐based a posteriori error estimates over the stress and scalar displacement error for the lowest‐order Raviart–Thomas mixed finite element discretizations of convection‐diffusion‐reaction equations in two‐dimension space. The new technique is based on the abstract error estimates, the postprocessed approximation of the scalar displacement, and on the construction of an auxiliary problem. We consider the centered and upwind‐weighted mixed schemes, and concentrate the attention on the presence of an inhomogeneous and an anisotropic diffusion‐dispersion tensor and on a possible convection dominance. Global upper bounds can be directly computed on the base of the solution of the mixed schemes without any additional cost. Local lower bounds without any saturation assumption, hold from the case where convection or reaction are not present to convection‐ or reaction‐dominated equations, and their local efficiency depends on local or global variations in coefficients similar to Péclect number. Numerical experiments are reported to show the competitive behavior of the proposed posteriori error estimates, and to confirm the theoretical findings. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 593–624, 2014     

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     

Two‐grid mixed finite‐element methods for nonlinear Schrödinger equations

Two‐grid mixed finite element schemes are developed for solving both steady state and unsteady state nonlinear Schrödinger equations. The schemes use discretizations based on a mixed finite‐element method. The two‐grid approach yields iterative procedures for solving the nonlinear discrete equations. The idea is to relegate all of the Newton‐like iterations to grids much coarser than the final one, with no loss in order of accuracy. Numerical tests are performed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 63‐73, 2012     

Nonstandard finite difference schemes for Michaelis–Menten type reaction‐diffusion equations

We compare and investigate the performance of the exact scheme of the Michaelis–Menten (M–M) ordinary differential equation with several new nonstandard finite difference (NSFD) schemes that we construct using Mickens' rules. Furthermore, the exact scheme of the M–M equation is used to design several dynamically consistent NSFD schemes for related reaction‐diffusion equations, advection‐reaction equations, and advection‐reaction‐diffusion equations. Numerical simulations that support the theory and demonstrate computationally the power of NSFD schemes are presented. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Implicit‐explicit multistep finite element methods for nonlinear convection‐diffusion problems

Implicit‐explicit multistep finite element methods for nonlinear convection‐diffusion equations are presented and analyzed. In space we discretize by finite element methods. The discretization in time is based on linear multistep schemes. The linear part of the equation is discretized implicitly and the nonlinear part of the equation explicitly. The schemes are stable and very efficient. We derive optimal order error estimates. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:93–104, 2001     

SERK2v2: A new second‐order stabilized explicit Runge‐Kutta method for stiff problems

Traditionally, explicit numerical algorithms have not been used with stiff ordinary differential equations (ODEs) due to their stability. Implicit schemes are usually very expensive when used to solve systems of ODEs with very large dimension. Stabilized Runge‐Kutta methods (also called Runge–Kutta–Chebyshev methods) were proposed to try to avoid these difficulties. The Runge–Kutta methods are explicit methods with extended stability domains, usually along the negative real axis. They can easily be applied to large problem classes with low memory demand, they do not require algebra routines or the solution of large and complicated systems of nonlinear equations, and they are especially suited for discretizations using the method of lines of two and three dimensional parabolic partial differential equations. In Martín‐Vaquero and Janssen [Comput Phys Commun 180 (2009), 1802–1810], we showed that previous codes based on stabilized Runge–Kutta algorithms have some difficulties in solving problems with very large eigenvalues and we derived a new code, SERK2, based on sixth‐order polynomials. Here, we develop a new method based on second‐order polynomials with up to 250 stages and good stability properties. These methods are efficient numerical integrators of very stiff ODEs. Numerical experiments with both smooth and nonsmooth data support the efficiency and accuracy of the new algorithms when compared to other well‐known second‐order methods such as RKC and ROCK2. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

In this study, new high‐order backward semi‐Lagrangian methods are developed to solve nonlinear advection–diffusion type problems, which are realized using high‐order characteristic‐tracking strategies. The proposed characteristic‐tracking strategies are second‐order L‐stable and third‐order L(α)‐stable methods, which are based on a classical implicit multistep method combined with a error‐correction method. We also use backward differentiation formulas and the fourth‐order finite‐difference scheme for diffusion problem discretization in the temporal and spatial domains, respectively. To demonstrate the adaptability and efficiency of these time‐discretization strategies, we apply these methods to nonlinear advection–diffusion type problems such as the viscous Burgers' equation. Through simulations, not only the temporal and spatial accuracies are numerically evaluated but also the proposed methods are shown to be superior to the compared existing characteristic‐tracking methods under the same rates of convergence in terms of accuracy and efficiency. Finally, we have shown that the proposed method well preserves the energy and mass when the viscosity coefficient becomes zero.     

An upwind finite volume scheme and its maximum‐principle‐preserving ADI splitting for unsteady‐state advection‐diffusion equations

We develop an upwind finite volume (UFV) scheme for unsteady‐state advection‐diffusion partial differential equations (PDEs) in multiple space dimensions. We apply an alternating direction implicit (ADI) splitting technique to accelerate the solution process of the numerical scheme. We investigate and analyze the reason why the conventional ADI splitting does not satisfy maximum principle in the context of advection‐diffusion PDEs. Based on the analysis, we propose a new ADI splitting of the upwind finite volume scheme, the alternating‐direction implicit, upwind finite volume (ADFV) scheme. We prove that both UFV and ADFV schemes satisfy maximum principle and are unconditionally stable. We also derive their error estimates. Numerical results are presented to observe the performance of these schemes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 211–226, 2003     

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     

An adaptive wavelet space‐time SUPG method for hyperbolic conservation laws

This article concerns with incorporating wavelet bases into existing streamline upwind Petrov‐Galerkin (SUPG) methods for the numerical solution of nonlinear hyperbolic conservation laws which are known to develop shock solutions. Here, we utilize an SUPG formulation using continuous Galerkin in space and discontinuous Galerkin in time. The main motivation for such a combination is that these methods have good stability properties thanks to adding diffusion in the direction of streamlines. But they are more expensive than explicit semidiscrete methods as they have to use space‐time formulations. Using wavelet bases we maintain the stability properties of SUPG methods while we reduce the cost of these methods significantly through natural adaptivity of wavelet expansions. In addition, wavelet bases have a hierarchical structure. We use this property to numerically investigate the hierarchical addition of an artificial diffusion for further stabilization in spirit of spectral diffusion. Furthermore, we add the hierarchical diffusion only in the vicinity of discontinuities using the feature of wavelet bases in detection of location of discontinuities. Also, we again use the last feature of the wavelet bases to perform a postprocessing using a denosing technique based on a minimization formulation to reduce Gibbs oscillations near discontinuities while keeping other regions intact. Finally, we show the performance of the proposed combination through some numerical examples including Burgers’, transport, and wave equations as well as systems of shallow water equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2062–2089, 2017     

Robust spectral method for numerical valuation of european options under Merton's jump‐diffusion model

We propose a novel numerical method based on rational spectral collocation and Clenshaw–Curtis quadrature methods together with the “” transformation for pricing European vanilla and butterfly spread options under Merton's jump‐diffusion model. Under certain assumptions, such model leads to a partial integro‐differential equation (PIDE). The differential and integral parts of the PIDE are approximated by the rational spectral collocation and the Clenshaw–Curtis quadrature methods, respectively. The application of spectral collocation method to the PIDE leads to a system of ordinary differential equations, which is solved using the implicit–explicit predictor–corrector (IMEX‐PC) schemes in which the diffusion term is integrated implicitly, whereas the convolution integral, reaction, advection terms are integrated explicitly. Numerical experiments illustrate that our approach is highly accurate and efficient for pricing financial options.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1169–1188, 2014     

Extension of high‐order compact schemes to time‐dependent problems

In this article, we present an extension of our previous approaches for steady‐state higher‐order compact (HOC) difference methods to time‐dependent problems. The formulation also provides a framework for similar treatment of other HOC spatial schemes. A stability analysis is provided for transient convection‐diffusion in 1D and transient diffusion in 2D. Supporting numerical experiments are included to illustrate stability and accuracy as well as oscillatory and dissipative behavior. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 657–672, 2001     

Rapid error reduction for block Gauss–Seidel based on p‐hierarchical basis

We consider a two‐level block Gauss–Seidel iteration for solving systems arising from finite element discretizations employing higher‐order elements. A p‐hierarchical basis is used to induce this block structure. Using superconvergence results normally employed in the analysis of gradient recovery schemes, we argue that a massive reduction of the H¹‐error occurs in the first iterate, so that the discrete solution is adequately resolved in very few iterates—sometimes a single iteration is sufficient. Numerical experiments on uniform and adapted meshes support these claims. Copyright © 2012 John Wiley & Sons, Ltd.     

Time discretizations for Maxwell‐Bloch equations

In this article we derive new time discretizations for the numerical simulation of Maxwell‐Bloch equations. These discretizations decouple the equations, thus leading to improved efficiency. This approach may be combined with the fulfilment of physical properties, such as positiveness properties, which are not accounted for by classical schemes. Our time discretizations are moreover proved to be nonlinearly stable. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 284–300, 2003.     

Nonstandard finite difference schemes for reaction‐diffusion equations

We construct finite difference schemes for a particular class of one‐space dimension, nonlinear reaction‐diffusion PDEs. The use of nonstandard finite difference methods and the imposition of a positivity condition constrain the schemes to be explicit and allow the determination of functional relations between the space and time step‐sizes. The general procedure is illustrated by applying it to several important model systems of PDEs © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 201–214, 1999     

Parameter uniform approximations for time‐dependent reaction‐diffusion problems

Using discrete Green's functions techniques, we present a classification of fitted mesh methods for time‐dependent reaction diffusion problems, based on the analyses of Linß (Linß, Numer Algor 40 (2005), 23–32) for the analogous steady‐state problem and of Kopteva (Kopteva, Computing 66 (2001), 179–197) for time‐dependent convection‐diffusion problems. As examples of how to apply the analysis, we derive error estimates for the fitted meshes of Shishkin and Bakhvalov, and provide supporting numerical results. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Stability of a colocated finite volume scheme for the Navier‐Stokes equations

The aim of this article is to describe a colocated finite volume approximation of the incompressible Navier‐Stokes equation and study its stability. One of the advantages of colocated finite volume space discretizations over staggered space discretizations is that all the variables share the same location; hence, the possibility to more easily use complex geometries and hierarchical decompositions of the unknowns. The time discretization used in the scheme studied here is a projection method. First, we give the full discretization of the incompressible Navier‐Stokes equations, then, we state the stability result and prove it following the methods of Marion and Temam. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005