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Reaction Term (reaction + term)
Selected AbstractsA comparative study of efficient iterative solvers for generalized Stokes equationsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2008Maxim Larin Abstract We consider a generalized Stokes equation with problem parameters ,,0 (size of the reaction term) and ,>0 (size of the diffusion term). We apply a standard finite element method for discretization. The main topic of the paper is a study of efficient iterative solvers for the resulting discrete saddle point problem. We investigate a coupled multigrid method with Braess,Sarazin and Vanka-type smoothers, a preconditioned MINRES method and an inexact Uzawa method. We present a comparative study of these methods. An important issue is the dependence of the rate of convergence of these methods on the mesh size parameter and on the problem parameters , and ,. We give an overview of the main theoretical convergence results known for these methods. For a three-dimensional problem, discretized by the Hood,Taylor ,,2,,,1 pair, we give results of numerical experiments. Copyright © 2007 John Wiley & Sons, Ltd. [source] Numerical studies of a nonlinear heat equation with square root reaction termNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2009Ron Buckmire Abstract Interest in calculating numerical solutions of a highly nonlinear parabolic partial differential equation with fractional power diffusion and dissipative terms motivated our investigation of a heat equation having a square root nonlinear reaction term. The original equation occurs in the study of plasma behavior in fusion physics. We begin by examining the numerical behavior of the ordinary differential equation obtained by dropping the diffusion term. The results from this simpler case are then used to construct nonstandard finite difference schemes for the partial differential equation. A variety of numerical results are obtained and analyzed, along with a comparison to the numerics of both standard and several nonstandard schemes. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source] A fourth-order compact algorithm for nonlinear reaction-diffusion equations with Neumann boundary conditionsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2006Wenyuan Liao Abstract In this article, we discuss a scheme for dealing with Neumann and mixed boundary conditions using a compact stencil. The resulting compact algorithm for solving systems of nonlinear reaction-diffusion equations is fourth-order accurate in both the temporal and spatial dimensions. We also prove that the standard second-order approximation to zero Neumann boundary conditions provides fourth-order accuracy when the nonlinear reaction term is independent of the spatial variables. Numerical examples, including an application of this algorithm to a mathematical model describing frontal polymerization process, are presented in the article to demonstrate the accuracy and efficiency of the scheme. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005 [source] A simulation of the non-isothermal resin transfer molding processPOLYMER ENGINEERING & SCIENCE, Issue 12 2000Vincenza Antonucci A simulation of the non-isothermal resin transfer molding manufacturing process accounting for both the filling and the consolidation stage has been developed. The flow of an exothermally reactive resin through a porous medium has been analyzed with reference to the Darcy law, allowing for the chemorheological properties of the reacting resin. Thermal profile calculations have been extended to a three phase domain, namely the mold, the dry preform and the filled preform. The mold has been included in order to evaluate the thermal inertial effects. The energy balance equation includes the reaction term together with the conductive and convective terms, and particular attention has been devoted to setting the thermal boundary condition at the flow front surface. The moving boundary condition has been derived by a jump equation. The simulation performance has been tested by comparing the predicted temperature profiles with experimental data from literature. Further numerical analysis assessed the relevance of using the jump equation at the flow front position for both filling time and thermal profile determination. [source] Qualitative model of concrete acidification due to cathodic protection,MATERIALS AND CORROSION/WERKSTOFFE UND KORROSION, Issue 2 2008W. H. A. Peelen In this paper a mathematical description and numerical implementation for ion transport in concrete due to current passage is developed, in which the heterogeneous equilibrium between Ca2+, OH, and the solid Ca(OH)2 is incorporated. The description is based on the Nernst,Planck equation for ion transport, and reaction terms for the dissolution/precipitation of Ca(OH)2. This description was implemented in the finite element package Comsol Multiphysics. In this way Ca(OH)2 depletion in a zone at a CP anode adjacent to a bulk of concrete with Ca(OH)2 could be modelled in one calculation. Drawback of this model is that the kinetic parameters in the reaction terms are not known, and must be chosen high to ensure the dissolution of Ca(OH)2 to be in equilibrium. This proved numerically challenging and sometimes caused long calculation times. The growth rate of the zone without solid depends on the current density applied, concrete cover, the pore liquid composition and the diffusion constants of Ca2+ and OH,. This rate must be evaluated numerically. This qualitative model of anode acidification shows no participation of Na+; therefore transport properties of this ion do not affect the acidification rate of concrete. The same would hold for any other ion included in the model, which is not involved in electrochemical or chemical reactions. [source] On the subdomain-Galerkin/least squares method for 2- and 3-D mixed elliptic problems with reaction termsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2002Suh-Yuh Yang Abstract In this article we apply the subdomain-Galerkin/least squares method, which is first proposed by Chang and Gunzburger for first-order elliptic systems without reaction terms in the plane, to solve second-order non-selfadjoint elliptic problems in two- and three-dimensional bounded domains with triangular or tetrahedral regular triangulations. This method can be viewed as a combination of a direct cell vertex finite volume discretization step and an algebraic least-squares minimization step in which the pressure is approximated by piecewise linear elements and the flux by the lowest order Raviart-Thomas space. This combined approach has the advantages of both finite volume and least-squares methods. Among other things, the combined method is not subject to the Ladyzhenskaya-Babus,ka-Brezzi condition, and the resulting linear system is symmetric and positive definite. An optimal error estimate in the H1(,) × H(div; ,) norm is derived. An equivalent residual-type a posteriori error estimator is also given. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 738,751, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10030. [source] An efficient high-order algorithm for solving systems of reaction-diffusion equationsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2002Wenyuan Liao Abstract 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 [source] | |||||||||||||