Showing papers in "Numerical Methods for Partial Differential Equations in 2006"
TL;DR: In this paper, a theoretical framework for the Galerkin finite element approximation to the steady state fractional advection dispersion equation is presented, and appropriate fractional derivative spaces are defined and shown to be equivalent to the usual fractional dimension Sobolev spaces Hs.
Abstract: In this article a theoretical framework for the Galerkin finite element approximation to the steady state fractional advection dispersion equation is presented. Appropriate fractional derivative spaces are defined and shown to be equivalent to the usual fractional dimension Sobolev spaces Hs. Existence and uniqueness results are proven, and error estimates for the Galerkin approximation derived. Numerical results are included that confirm the theoretical estimates. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005
707 citations
TL;DR: In this article, the numerical solution of the one-dimensional parabolic equation subject to the specification of mass, which have been considered in the literature, is reported. And the performance of the proposed algorithm, considering a test problem, is investigated.
Abstract: Certain problems arising in engineering are modeled by nonstandard parabolic initial-boundary value problems in one space variable, which involve an integral term over the spatial domain of a function of the desired solution. Hence, in the past few years interest has substantially increased in the solutions of these problems. As a result numerous research papers have also been devoted to the subject. Although considerable amount of work has been done in the past, there is still a lack of a completely satisfactory computational scheme. Also, there are some cases that have not been studied numerically yet. In the current article several approaches for the numerical solution of the one-dimensional parabolic equation subject to the specification of mass, which have been considered in the literature, are reported. Finite difference methods have been proposed for the numerical solution of the new nonclassic boundary value problem. To investigate the performance of the proposed algorithm, we consider solving a test problem. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006
154 citations
TL;DR: In this paper, a convergence analysis of the multi point flux approximation control volume method, MPFA, in two space dimensions is presented, where the discretization is based on local mappings onto a reference square.
Abstract: This paper presents a convergence analysis of the multi point flux approximation control volume method, MPFA, in two space dimensions. The MPFA version discussed here is the so–called O–method on general quadrilateral grids. The discretization is based on local mappings onto a reference square. The key ingredient in the analysis is an equivalence between the MPFA method and a mixed finite element method, using a specific numerical quadrature, such that the analysis of the MPFA method can be done in a finite element setting. c © ??? John Wiley & Sons, Inc.
100 citations
TL;DR: In this article, the authors derived stencils for differential operators for which the discretization error becomes isotropic in the lowest order, i.e., difference schemes.
Abstract: We derive stencils, i.e., difference schemes, for differential operators for which the discretization error becomes isotropic in the lowest order. We treat the Laplacian, Bilaplacian (= biharmonic operator), and the gradient of the Laplacian both in two and three dimensions. For three dimensions [MATHEMATICAL SCRIPT CAPITAL O](h2) results are given while for two dimensions both [MATHEMATICAL SCRIPT CAPITAL O](h2) and [MATHEMATICAL SCRIPT CAPITAL O](h4) results are presented. The results are also available in electronic form as a Mathematica file. It is shown that the extra computational cost of an isotropic stencil usually is less than 20%. Keywords: difference schemes;Laplacian;isotropic discretization
70 citations
TL;DR: In this article, a subgrid eddy viscosity method for solving the steady-state incompressible flow problem is proposed, which does not act on the large flow structures.
Abstract: We formulate a subgrid eddy viscosity method for solving the steady-state incompressible flow problem. The eddy viscosity does not act on the large flow structures. Optimal error estimates are obtained for velocity and pressure. The numerical illustrations agree completely with the theoretical results. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005
69 citations
TL;DR: The Perfectly Matched Layer (PML) is improved by using an automatic hp‐adaptive discretization to recover the property of the PML of having a zero reflection coefficient for all angles of incidence and all frequencies on the continuum level.
Abstract: We improve the performance of the Perfectly Matched Layer (PML) by using an automatic hp-adaptive discretization. By means of hp-adaptivity, we obtain a sequence of discrete solutions that converges exponentially to the continuum solution. Asymptotically, we thus recover the property of the PML of having a zero reflection coefficient for all angles of incidence and all frequencies on the continuum level. This allows us to minimize the reflections from the discrete PML to an arbitrary level of accuracy while retaining optimal computational efficiency. Since our hp-adaptive scheme is automatic, no interaction with the user is required. This renders tedious parameter tuning of the PML obsolete. We demonstrate the improvement of the PML performance by hp-adaptivity through numerical results for acoustic, elastodynamic, and electromagnetic wave-propagation problems in the frequency domain and in different systems of coordinates. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 832-858, 2007
68 citations
TL;DR: Banks, V A Bokil and N L Gibson as discussed by the authors analyzed stability and dispersion in a Finite Element Method for Debye and Lorentz Media, 25(4), pp 885-917, July 2009.
Abstract: This is the pre-peer reviewed version of the following article: H T Banks, V A Bokil and N L Gibson, Analysis of Stability and Dispersion in a Finite Element Method for Debye and Lorentz Media, Numerical Methods for Partial Differential Equations, 25(4), pp 885-917, July 2009, which has been
published in final form at http://www3intersciencewileycom/journal/122341241/issue
61 citations
TL;DR: The spectral smoothed boundary method (SSBM) as discussed by the authors is a recently proposed numerical method to approximate the solution of partial differential equations in irregular domains with no-flux boundary conditions by means of Fourier spectral methods.
Abstract: The spectral smoothed boundary method (SSBM) is a recently proposed numerical method to approximate the solution of partial differential equations in irregular domains with no-flux boundary conditions by means of Fourier spectral methods. In this article we explore the robustness and accuracy of the scheme under variations of the artificial boundary conditions that must be imposed on the boundary of the enlarged domain in which the problem is solved. As a test model, we present quantitative numerical results based on a problem of propagation of waves of electrical activity in cardiac tissue for which the method is relevant. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006
53 citations
51 citations
TL;DR: In this paper, a relaxation scheme is developed which reliably captures δ-shocks and vacuum states, and the stability conditions for relaxation methods for the pressureless case are refined, useful properties for the time stepping procedure are established and two-dimensional numerical results are presented.
Abstract: In the present work, we consider the numerical approximation of pressureless gas dynamics in one and two spatial dimensions. Two particular phenomena are of special interest for us, namely δ-shocks and vacuum states. A relaxation scheme is developed which reliably captures these phenomena. In one space dimension, we prove the validity of several stability criteria, i.e., we show that a maximum principle as well as the TVD property for the discrete velocity component and the validity of discrete entropy inequalities hold. Several numerical tests considering not only the developed first-order scheme but also a classical MUSCL-type second-order extension confirm the reliability and robustness of the relaxation approach. The paper extends previous results on the topic: the stability conditions for relaxation methods for the pressureless case are refined, useful properties for the time stepping procedure are established and two-dimensional numerical results are presented. c © ??? John Wiley & Sons, Inc.
46 citations
TL;DR: A high‐order compact alternating direction implicit (ADI) method for solving three‐dimentional unsteady convection‐diffusion problems and it is shown through a discrete Fourier analysis that the method is unconditionally stable in the diffusion case.
Abstract: We derive a high-order compact alternating direction implicit (ADI) method for solving three-dimentional unsteady convection-diffusion problems. The method is fourth-order in space and second-order in time. It permits multiple uses of the one-dimensional tridiagonal algorithm with a considerable saving in computing time and results in a very efficient solver. It is shown through a discrete Fourier analysis that the method is unconditionally stable in the diffusion case. Numerical experiments are conducted to test its high order and to compare it with the standard second-order Douglas-Gunn ADI method and the spatial fourth-order compact scheme by Karaa. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006
TL;DR: The main topic of as mentioned in this paper is the study of a nonlinear partial differential equation, the Camassa-Holm (CH) equation: ut - utxx + 3uux - 2uxuxx -uuxxx = 0.
Abstract: The main topic of this thesis is the study of a nonlinear partial differential equation, the Camassa-Holm (CH) equation: ut - utxx + 3uux - 2uxuxx -uuxxx = 0.
TL;DR: A local defect correction technique for time-dependent problems suitable for solving partial differential equations characterized by a high activity, which is mainly located, at each time, in a small part of the physical domain.
Abstract: In this article a local defect correction technique for time-dependent problems is presented. The method is suitable for solving partial differential equations characterized by a high activity, which is mainly located, at each time, in a small part of the physical domain. The problem is solved at each time step by means of a global uniform coarse grid and a local uniform fine grid. Local and global approximation are improved iteratively. Results of numerical experiments illustrate the accuracy, the efficiency, and the robustness of the method.
TL;DR: In this paper, the authors proposed and studied different mixed variational methods in order to approximate the Signorini problem with friction using finite elements, and the discretized normal and tangential constraints at the contact interface were expressed by using either continuous piecewise linear or piecewise constant Lagrange multipliers in the saddle−point formulation.
Abstract: In this article, we propose and study different mixed variational methods in order to approximate the Signorini problem with friction using finite elements. The discretized normal and tangential constraints at the contact interface are expressed by using either continuous piecewise linear or piecewise constant Lagrange multipliers in the saddle−point formulation. A priori error estimates are established and several numerical examples corresponding to the different choices of the discretized normal and tangential constraints are carried out. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006
TL;DR: In this paper, a defect correction method for the approximation of viscoelastic fluid flow is proposed, where the constitutive equation is computed with an artificially reduced Weissenberg parameter for stability, and the resulting residual is corrected in the correction step.
Abstract: We study a defect correction method for the approximation of viscoelastic fluid flow. In the defect step, the constitutive equation is computed with an artificially reduced Weissenberg parameter for stability, and the resulting residual is corrected in the correction step. We prove the convergence of the defect correction method and derive an error estimate for the Oseen-viscoelastic model problem. The derived theoretical results are supported by numerical tests for both the Oseen-viscoelastic problem and the Johnson-Segalman model problem. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 21: 000‐000, 2005
TL;DR: In this paper, a pseudospectral Legendre method is proposed for the numerical solution of the parabolic equation subject to a given initial condition and non-local boundary specifications.
Abstract: In this research, the problem of solving the two-dimensional parabolic equation subject to a given initial condition and nonlocal boundary specifications is considered. A technique based on the pseudospectral Legendre method is proposed for the numerical solution of the studied problem. Several examples are given and the numerical results are shown to demonstrate the efficiently of the newly proposed method. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006
TL;DR: In this paper, the authors apply the univariate multiquadric (MQ) quasi-interpolation to solve the hyperbolic conservation laws and obtain the numerical schemes to solve partial differential equations.
Abstract: In this article, we apply the univariate multiquadric (MQ) quasi-interpolation to solve the hyperbolic conservation laws. At first we construct the MQ quasi-interpolation corresponding to periodic and inflow-outflow boundary conditions respectively. Next we obtain the numerical schemes to solve the partial differential equations, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the differential equation and a low-order explicit difference to approximate the temporal derivative of the differential equation. Then we verify our scheme for the one-dimensional Burgers' equation (without viscosity). We can see that the numerical results are very close to the exact solution and the computational accuracy of the scheme is (τ), where τ is the temporal step. We can improve the accuracy by using the high-order quasi-interpolation. Moreover the methods can be generalized to the other equations. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006
TL;DR: In this paper, the authors investigated the application of the method of fundamental solutions (MFS) to the Dirichlet problem for Laplace's equation in an annular domain, and developed an efficient matrix decomposition algorithm using fast Fourier transforms (FFTs) for the computation of the MFS approximation.
Abstract: In this study, we investigate the application of the method of fundamental solutions (MFS) to the Dirichlet problem for Laplace's equation in an annular domain. We examine the properties of the resulting coefficient matrix and its eigenvalues. The convergence of the method is proved for analytic boundary data. An efficient matrix decomposition algorithm using fast Fourier transforms (FFTs) is developed for the computation of the MFS approximation. We also tested the algorithm numerically on several problems confirming the theoretical predictions. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005
TL;DR: It is proved that optimal error bounds can be obtained for a fourth‐order elliptic problem and a fourth-order parabolic problem solved by mixed finite element methods on quasi‐uniform rectangular meshes.
Abstract: By using a special interpolation operator developed by Girault and Raviart (finite element methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986), we prove that optimal error bounds can be obtained for a fourth-order elliptic problem and a fourth-order parabolic problem solved by mixed finite element methods on quasi-uniform rectangular meshes. Optimal convergence is proved for all continuous tensor product elements of order k ≥ 1. A numerical example is provided for solving the fourth-order elliptic problem using the bilinear element. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006
TL;DR: In this article, the authors present a new relaxation method for the numerical approximation of the two-dimensional Riemann problems in gas dynamics, which does not require either a RiemANN solver or a characteristics decomposition.
Abstract: We present a new relaxation method for the numerical approximation of the two-dimensional Riemann problems in gas dynamics. The novel feature of the technique proposed here is that it does not require either a Riemann solver or a characteristics decomposition. The high resolution of the method is achieved by using a third-order reconstruction for the space discretization and a third-order TVD Runge-Kutta scheme for the time integration. Numerical experiments, using several configurations of Riemann problems in gas dynamics, are included to confirm the high resolution of the new relaxation scheme. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006
TL;DR: In this paper, the Adomian decomposition method was used to approximate the solution of the Boussinesq equation and the results obtained were compared to the theoretical solution for single soliton wave.
Abstract: We will consider the application of the Adomian decomposition method to approximate the solution of the Boussinesq equation. Both the well-posed and the ill-posed cases will be considered. The results obtained will be compared to the theoretical solution for single soliton wave. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006
TL;DR: This article presents a posteriori error estimates for the mixed discontinuous Galerkin approximation of the stationary Stokes problem with anisotropic finite element discretizations, i.e., elements with very large aspect ratio.
Abstract: The paper presents a posteriori error estimates for the mixed discontinuous Galerkin approximation of the stationary Stokes problem. We consider anisotropic finite element discretizations, i.e. elements with very large aspect ratio. Our analysis covers two- and three-dimensional domains. Lower and upper error bounds are proved with minimal assumptions on the meshes. The lower error bound is uniform with respect to the mesh anisotropy. The upper error bound depends on a proper alignment of the anisotropy of the mesh which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimator.
TL;DR: This work discretizes in space the equations obtained at each time step when discretizing in time a Navier‐Stokes system modelling the two‐dimensional flow in a horizontal pipe of two immiscible fluids with comparable densities, but very different viscosities.
Abstract: We discretize in space the equations obtained at each time step when discretizing in time a Navier-Stokes system modelling the two-dimensional flow in a horizontal pipe of two immiscible fluids with comparable densities, but very different viscosities. At each time step the system reduces to a generalized Stokes problem with nonstandard conditions at the boundary and at the interface between the two fluids. We discretize this system with the “mini-element” and establish error estimates. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005
TL;DR: In this paper, the generalized finite element method (GFEM) is used to approximate a harmonic function u ∈ H 1−k(Ω), k > 0, on a relatively compact subset A of Ω, using the GFEM.
Abstract: We study the approximation properties of a harmonic function u ∈ H1−k(Ω), k > 0, on a relatively compact subset A of Ω, using the generalized finite element method (GFEM). If Ω = , for a smooth, bounded domain , we obtain that the GFEM-approximation uS ∈ S of u satisfies ‖u − uS‖ ≤ Chγ‖u‖, where h is the typical size of the “elements” defining the GFEM-space S and γ ≥ 0 is such that the local approximation spaces contain all polynomials of degree k + γ. The main technical ingredient is an extension of the classical super-approximation results of Nitsche and Schatz (Applicable Analysis 2 (1972), 161–168; Math Comput 28 (1974), 937–958). In addition to the usual “energy” Sobolev spaces H1(), we need also the duals of the Sobolev spaces Hm(), m ∈ ℤ+. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006
TL;DR: In this paper, a high-order implicit difference scheme presented by Mohanty, Jain, and George for the nonlinear hyperbolic equation utt = A(x, t)uxx + F(n, t, u, ut, ux) with Dirichlet boundary conditions is considered.
Abstract: This article is concerned with a high-order implicit difference scheme presented by Mohanty, Jain, and George for the nonlinear hyperbolic equation utt = A(x, t)uxx + F(x, t, u, ut, ux) with Dirichlet boundary conditions. Some prior estimates of the difference solution are obtained by the energy methods. The solvability of the difference scheme is proved by the energy method and Brower's fixed point theorem. Similarly, the uniqueness, the convergence in L∞-norm and the stability of the difference solution are obtained. A numerical example is provided to demonstrate the validity of the theoretical results. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 484–498, 2007
TL;DR: In this paper, the Gauss Legendre Quadrature method was used for numerical integration over the standard tetrahedron in Cartesian three-dimensional (x, y, z) space.
Abstract: In this article we consider the Gauss Legendre Quadrature method for numerical integration over the standard tetrahedron: {(x, y, z)|0 ≤ x, y, z ≤ 1, x + y + z ≤ 1} in the Cartesian three-dimensional (x, y, z) space. The mathematical transformation from the (x, y, z) space to (ξ, η, ζ) space is described to map the standard tetrahedron in (x, y, z) space to a standard 2-cube: {(ξ, η, ζ)| − 1 ≤ ζ, η, ζ ≤ 1} in the (ξ, η, ζ) space. This overcomes the difficulties associated with the derivation of new weight coefficients and sampling points. The effectiveness of the formulas is demonstrated by applying them to the integration of three nonpolynomial, three polynomial functions and to the evaluation of integrals for element stiffness matrices in linear three-dimensional elasticity. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006
TL;DR: An alternative approach to develop an Eulerian‐Lagrangian control‐volume method (ELCVM) for transient advection‐diffusion equations is adopted, which is locally conservative and maintains the accuracy of characteristics methods even if a very simple tracking is used, while retaining the advantages of characteristic methods in general.
Abstract: Characteristic methods generally generate accurate numerical solutions and greatly reduce grid orientation effects for transient advection-diffusion equations. Nevertheless, they raise additional numerical difficulties. For instance, the accuracy of the numerical solutions and the property of local mass balance of these methods depend heavily on the accuracy of characteristics tracking and the evaluation of integrals of piecewise polynomials on some deformed elements generally with curved boundaries, which turns out to be numerically difficult to handle. In this article we adopt an alternative approach to develop an Eulerian-Lagrangian control-volume method (ELCVM) for transient advection-diffusion equations. The ELCVM is locally conservative and maintains the accuracy of characteristic methods even if a very simple tracking is used, while retaining the advantages of characteristic methods in general. Numerical experiments show that the ELCVM is favorably comparable with well-regarded Eulerian-Lagrangian methods, which were previously shown to be very competitive with many well-perceived methods. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005
TL;DR: The semi-analytical integration of an 8-node plane strain finite element stiffness matrix is presented in this article, where the element is assumed to be super-parametric, having straight sides.
Abstract: The semi-analytical integration of an 8-node plane strain finite element stiffness matrix is presented in this work. The element is assumed to be super-parametric, having straight sides. Before carrying out the integration, the integral expressions are classified into several groups, thus avoiding duplication of calculations. Symbolic manipulation and integration is used to obtain the basic formulae to evaluate the stiffness matrix. Then, the resulting expressions are postprocessed, optimized, and simplified in order to reduce the computation time. Maple symbolic-manipulation software was used to generate the closed expressions and to develop the corresponding Fortran code. Comparisons between semi-analytical integration and numerical integration were made. It was demonstrated that semi-analytical integration required less CPU time than conventional numerical integration (using Gaussian-Legendre quadrature) to obtain the stiffness matrix. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006