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One-dimensional Problems (one-dimensional + problem)
Selected AbstractsA variational multiscale model for the advection,diffusion,reaction equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2009Guillaume Houzeaux Abstract The variational multiscale (VMS) method sets a general framework for stabilization methods. By splitting the exact solution into coarse (grid) and fine (subgrid) scales, one can obtain a system of two equations for these unknowns. The grid scale equation is solved using the Galerkin method and contains an additional term involving the subgrid scale. At this stage, several options are usually considered to deal with the subgrid scale equation: this includes the choice of the space where the subgrid scale would be defined as well as the simplifications leading to compute the subgrid scale analytically or numerically. The present study proposes to develop a two-scale variational method for the advection,diffusion,reaction equation. On the one hand, a family of weak forms are obtained by integrating by parts a fraction of the advection term. On the other hand, the solution of the subgrid scale equation is found using the following. First, a two-scale variational method is applied to the one-dimensional problem. Then, a series of approximations are assumed to solve the subgrid space equation analytically. This allows to devise expressions for the ,stabilization parameter' ,, in the context of VMS (two-scale) method. The proposed method is equivalent to the traditional Green's method used in the literature to solve residual-free bubbles, although it offers another point of view, as the strong form of the subgrid scale equation is solved explicitly. In addition, the authors apply the methodology to high-order elements, namely quadratic and cubic elements. The proposed model consists in assuming that the subgrid scale vanishes also on interior nodes of the element and applying the strategy used for linear element in the segment between these interior nodes. The proposed scheme is compared with existing ones through the solution of a one-dimensional numerical example for linear, quadratic and cubic elements. In addition, the mesh convergence is checked for high-order elements through the solution of an exact solution in two dimensions. Copyright © 2008 John Wiley & Sons, Ltd. [source] An accurate integral-based scheme for advection,diffusion equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 10 2001Tung-Lin Tsai Abstract This paper proposes an accurate integral-based scheme for solving the advection,diffusion equation. In the proposed scheme the advection,diffusion equation is integrated over a computational element using the quadratic polynomial interpolation function. Then elements are connected by the continuity of first derivative at boundary points of adjacent elements. The proposed scheme is unconditionally stable and results in a tridiagonal system of equations which can be solved efficiently by the Thomas algorithm. Using the method of fractional steps, the proposed scheme can be extended straightforwardly from one-dimensional to multi-dimensional problems without much difficulty and complication. To investigate the computational performances of the proposed scheme five numerical examples are considered: (i) dispersion of Gaussian concentration distribution in one-dimensional uniform flow; (ii) one-dimensional viscous Burgers equation; (iii) pure advection of Gaussian concentration distribution in two-dimensional uniform flow; (iv) pure advection of Gaussian concentration distribution in two-dimensional rigid-body rotating flow; and (v) three-dimensional diffusion in a shear flow. In comparison not only with the QUICKEST scheme, the fully time-centred implicit QUICK scheme and the fully time-centred implicit TCSD scheme for one-dimensional problem but also with the ADI-QUICK scheme, the ADI-TCSD scheme and the MOSQUITO scheme for two-dimensional problems, the proposed scheme shows convincing computational performances. Copyright © 2001 John Wiley & Sons, Ltd. [source] Development of a class of multiple time-stepping schemes for convection,diffusion equations in two dimensionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2006R. K. Lin Abstract In this paper we present a class of semi-discretization finite difference schemes for solving the transient convection,diffusion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection,diffusion (CD) equation to the inhomogeneous steady convection,diffusion-reaction (CDR) equation after using different time-stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one-dimensional framework. For the sake of increasing accuracy, the exact solution for the one-dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one-dimensional problem. Development of the proposed time-stepping schemes is rooted in the Taylor series expansion. All higher-order time derivatives are replaced with spatial derivatives through use of the model differential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection,diffusion-reaction differential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions. Copyright © 2006 John Wiley & Sons, Ltd. [source] Analysis of algebraic systems arising from fourth-order compact discretizations of convection-diffusion equationsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2002Ashvin Gopaul Abstract 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 [source] Chemically induced deformation of a porous layer coupled with advective,dispersive transport.INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 8 2001Analytical solutions Abstract In this paper a chemically induced deformation of porous material taking place during advective,dispersive transport of a chemical is considered. Linearized governing equations are derived and analytical solutions of 2 one-dimensional problems for a homogeneous layer with drained boundaries are developed. Numerical results for a particular clayey material and a chemical migrating through the layer showing distributions of concentration of chemical, changes in porosity of the material and pore fluid pressure, and evolution of settlement of the layer as functions of time are discussed. Copyright © 2001 John Wiley & Sons, Ltd. [source] On discontinuous Galerkin methodsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2003O. C. Zienkiewicz Abstract Discontinuous Galerkin methods have received considerable attention in recent years for problems in which advection and diffusion terms are present. Several alternatives for treating the diffusion and advective fluxes have been introduced. This report summarizes some of the methods that have been proposed. Several numerical examples are included in the paper. These present discontinuous Galerkin solutions of one-dimensional problems with a scalar variable. Results are presented for diffusion,reaction problems and advection,diffusion problems. We discuss the performance of various formulations with respect to accuracy as well as stability of the method. Copyright © 2003 John Wiley & Sons, Ltd. [source] Simple efficient algorithm (SEA) for shallow flows with shock wave on dry and irregular bedsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2008Alireza Zia Abstract An explicit Godunov-type solution algorithm called SEA (simple efficient algorithm) has been introduced for the shallow water equations. The algorithm is based on finite volume conservative discretisation method. It can deal with wet/dry and irregular beds. Second-order accuracy, in both time and space, is achieved using prediction and correction steps. A very simple and efficient flux limiting technique is used to equip the algorithm with total variation dimensioning property for shock capturing purposes. In order to make sure about the balance between the flux gradient and the bed slope, treatment of the source term has been done using a new procedure inspired mainly by the physical rather than mathematical consideration. SEA has been applied to one-dimensional problems, although it can equally be applied to multi-dimensional problems. In order to assess the capability of proposed algorithm in dealing with practical applications, several test cases have been examined. Copyright © 2007 John Wiley & Sons, Ltd. [source] Simulating spatially and physiologically structured populationsJOURNAL OF ANIMAL ECOLOGY, Issue 6 2001William S. C. Gurney Summary 1,Population dynamics are frequently the product of a subtle interplay between development and dispersal in an inhomogeneous environment. Simulations of spatially inhomogeneous populations with physiologically distinguishable individuals are a known source of numerical difficulty. 2,This paper reports a new and highly efficient algorithm for this problem, whose accuracy we demonstrate by comparison with conventional numerical solutions of one-dimensional problems. 3,As an illustration, we construct a two (space)-dimensional model of a copepod (Calanus finmarchicus) in the NE Atlantic, and demonstrate that its predictions correspond closely with those of an equivalent Lagrangian ensemble. [source] Heat transfer in composite materials with Stefan,Boltzmann interface conditionsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2008Yang Gufan Abstract In this paper, we discuss nonstationary heat transfer problems in composite materials. This problem can be formulated as the parabolic equation with Stefan,Boltzmann interface conditions. It is proved that there exists a unique global classical solution to one-dimensional problems. Moreover, we propose a numerical algorithm by the finite difference method for this nonlinear transmission problem. Copyright © 2007 John Wiley & Sons, Ltd. [source] Smoothed Particle Magnetohydrodynamics , III.MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 2 2005Multidimensional tests, ·B= 0 constraint ABSTRACT In two previous papers (Papers I and II), we have described an algorithm for solving the equations of Magnetohydrodynamics (MHD) using the Smoothed Particle Hydrodynamics (SPH) method. The algorithm uses dissipative terms in order to capture shocks and has been tested on a wide range of one-dimensional problems in both adiabatic and isothermal MHD. In this paper, we investigate multidimensional aspects of the algorithm, refining many of the aspects considered in Papers I and II and paying particular attention to the code's ability to maintain the ,·B= 0 constraint associated with the magnetic field. In particular, we implement a hyperbolic divergence cleaning method recently proposed by Dedner et al. in combination with the consistent formulation of the MHD equations in the presence of non-zero magnetic divergence derived in Papers I and II. Various projection methods for maintaining the divergence-free condition are also examined. Finally, the algorithm is tested against a wide range of multidimensional problems used to test recent grid-based MHD codes. A particular finding of these tests is that in Smoothed Particle Magnetohydrodynamics (SPMHD), the magnitude of the divergence error is dependent on the number of neighbours used to calculate a particle's properties and only weakly dependent on the total number of particles. Whilst many improvements could still be made to the algorithm, our results suggest that the method is ripe for application to problems of current theoretical interest, such as that of star formation. [source] ON AXISYMMETRIC TRAVELING WAVES AND RADIAL SOLUTIONS OF SEMI-LINEAR ELLIPTIC EQUATIONSNATURAL RESOURCE MODELING, Issue 3 2000THOMAS P. WITELSKI ABSTRACT. Combining analytical techniques from perturbation methods and dynamical systems theory, we present an elementaryapproach to the detailed construction of axisymmetric diffusive interfaces in semi-linear elliptic equations. Solutions of the resulting non-autonomous radial differential equations can be expressed in terms of a slowlyvarying phase plane system. Special analytical results for the phase plane system are used to produce closed-form solutions for the asymptotic forms of the curved front solutions. These axisym-metric solutions are fundamental examples of more general curved fronts that arise in a wide variety of scientific fields, and we extensivelydiscuss a number of them, with a particular emphasis on connections to geometric models for the motion of interfaces. Related classical results for traveling waves in one-dimensional problems are also reviewed briefly. Manyof the results contained in this article are known, and in presenting known results, it is intended that this article be expositoryin nature, providing elementarydemonstrations of some of the central dynamical phenomena and mathematical techniques. It is hoped that the article serves as one possible avenue of entree to the literature on radiallysymmetric solutions of semilinear elliptic problems, especiallyto those articles in which more advanced mathematical theoryis developed. [source] | ||||||||||||||||