On Error Estimates of the Projection Methods for the Navier-Stokes Equations: Second-order Schemes

We present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible Navier-Stokes equations. The error analysis is accomplished by interpreting the respective projection schemes as second-order time discretizations of a perturbed system which approximates the Navier-Stokes equations. Numerical results in agreement with the error analysis are also presented.

     

Finite element methods of an operator splitting applied to population balance equations

In population balance equations, the distribution of the entities depends not only on space and time but also on their own properties referred to as internal coordinates. The operator splitting method is used to transform the whole time-dependent problem into two unsteady subproblems of a smaller complexity. The first subproblem is a time-dependent convection–diffusion problem while the second one is a transient transport problem with pure advection. We use the backward Euler method to discretize the subproblems in time. Since the first problem is convection-dominated, the local projection method is applied as stabilization in space. The transport problem in the one-dimensional internal coordinate is solved by a discontinuous Galerkin method. The unconditional stability of the method will be presented. Optimal error estimates are given. Numerical tests confirm the theoretical results.     

A second order characteristics finite element scheme for natural convection problems

In this paper a second order characteristics finite element scheme is applied to the numerical solution of natural convection problems. Firstly, after recalling the mathematical model, a second order time discretization of the material time derivative is introduced. Next, fully discretized schemes are proposed by using finite element methods. Numerical results for the two-dimensional problem of buoyancy-driven flow in a square cavity with differentially heated side walls are given and compared with a reference solution.     

二阶椭圆问题新的混合元格式

本文基于二阶椭圆问题一种新的混合变分形式,给出同时满足强椭圆性和B-B条件的任意次的求解格式.理论分析表明这些单元论证简单而且用了较少的自由度达到最优误差估计.同时我们还给出了它们在各向异性网格下的误差估计.     

A fractional step θ-method for convection-diffusion problems

In this article, we analyze the fractional step θ-method for the time-dependent convection-diffusion equation. In our implementation, we completely separate the convection operator from the diffusion operator, and stabilize the convective problem using a Streamline Upwinded Petrov-Galerkin (SUPG) method. We establish a priori error estimates and show that the optimal value of θ yields a scheme that is second-order in time. Numerical computations are presented which demonstrate the method and support the theoretical results.     

Convergence of Extrapolated BDF2 Finite Element Schemes for Unsteady Penetrative Convection Model

Extrapolated two-step backward difference (BDF2) in time and finite element in space discretization for the unsteady penetrative convection model is analyzed. Penetrative convection model employs a nonlinear equation of state making the problem more nonlinear. Optimal order error estimates are derived for the semi-discrete finite element spatial discretization. Two time discretization schemes based on linear extrapolation are proposed and analyzed, namely a coupled and a decoupled scheme. In particular, we show that although both schemes are unconditionally nonlinearly stable, the decoupled scheme converges unconditionally whereas coupled scheme requires that the time step be sufficiently small for convergence. These time discretization schemes can be implemented efficiently in practice, saving computational memory. Numerical computations and numerical convergence checks are presented to demonstrate the efficiency and the accuracy of the schemes.     

Alternating triangular schemes for convection–diffusion problems

Explicit–implicit approximations are used to approximate nonstationary convection–diffusion equations in time. In unconditionally stable two-level schemes, diffusion is taken from the upper time level, while convection, from the lower layer. In the case of three time levels, the resulting explicit–implicit schemes are second-order accurate in time. Explicit alternating triangular (asymmetric) schemes are used for parabolic problems with a self-adjoint elliptic operator. These schemes are unconditionally stable, but conditionally convergent. Three-level modifications of alternating triangular schemes with better approximating properties were proposed earlier. In this work, two- and three-level alternating triangular schemes for solving boundary value problems for nonstationary convection–diffusion equations are constructed. Numerical results are presented for a two-dimensional test problem on triangular meshes, such as Delaunay triangulations and Voronoi diagrams.     

Analysis of a decoupled time‐stepping algorithm for reduced MHD system modeling magneto‐convection

In this article, we analyze the stability and error estimate of a decoupled algorithm for a magneto‐convection problem. Magneto‐convection is assumed to be modeled by a coupled system of reduced magneto‐hydrodynamic (RMHD) equations and convection‐diffusion equation. The proposed algorithm applies the second‐order backward difference formula in time and finite element in space. To obtain a noniterative decouple algorithm from the fully discrete nonlinear system, we use a second‐order extrapolation in time to the nonlinear terms such that their skew symmetry properties are preserved. We prove the stability of the algorithm and derive error estimates without assuming any stability conditions. The algorithm is unconditionally stable and requires the solution of one RMHD problem and one convection‐diffusion equation per time step. Numerical test is presented that illustrates the accuracy and efficiency of the algorithm.     

Difference Schemes on Nonuniform Grids for the Two-Dimensional Convection–Diffusion Equation

New second-order accurate monotone difference schemes on nonuniform spatial grids for two-dimensional stationary and nonstationary convection–diffusion equations are proposed. The monotonicity and stability of the solutions of the computational methods with respect to the boundary conditions, the initial condition, and the right-hand side are proved. Two-sided and corresponding a priori estimates are obtained in the grid norm of C. The convergence of the proposed algorithms to the solution of the original differential problem with the second order is proved.     

Finite-element method for time-dependent euler equation

This paper presents finite element methods to approximate inviscid incompressible flow problems. First we emphasize the conservation properties of these problems, and we show that finite element methods appear as a very natural way to find conservative schemes such as Arakawa's scheme. We give convergence theorems and an error analysis of finite element discretization schemes. We turn then to the time differencing problem. We derive stability and convergence results for a second-order semi-implicit scheme and for the leap-frog scheme.     

A stabilized implicit fractional-step method for the time-dependent Navier–Stokes equations using equal-order pairs

A stabilized implicit fractional-step method for numerical solutions of the time-dependent Navier–Stokes equations is presented in this paper. The time advancement is decomposed into a sequence of two steps: the first step has the structure of the linear elliptic problem; the second step can be seen as the generalized Stokes problem. The two problems satisfy the full homogeneous Dirichlet boundary conditions on the velocity. On the other hand, a locally stabilized term is added in the second step of the schemes. It allows one to enhance the numerical stability and efficiency by using the equal-order pairs. Convergence analysis and error estimates for the velocity and pressure of the schemes are established via the energy method. Some numerical experiments are also used to demonstrate the efficiency of this new method.     

A higher order uniformly convergent method for singularly perturbed parabolic turning point problems

In this article, we study numerical approximation for a class of singularly perturbed parabolic (SPP) convection-diffusion turning point problems. The considered SPP problem exhibits a parabolic boundary layer in the neighborhood of one of the sides of the domain. Some a priori bounds are given on the exact solution and its derivatives, which are necessary for the error analysis. A numerical scheme comprising of implicit finite difference method for time discretization on a uniform mesh and a hybrid scheme for spatial discretization on a generalized Shishkin mesh is proposed. Then Richardson extrapolation method is applied to increase the order of convergence in time direction. The resulting scheme has second-order convergence up to a logarithmic factor in space and second-order convergence in time. Numerical experiments are conducted to demonstrate the theoretical results and the comparative study is done with the existing schemes in literature to show better accuracy of the proposed schemes.     

对流扩散方程一类改进的特征线修正有限元方法

１引言在地下水污染,地下渗流驱动,核污染,半导体等问题的数值模拟中,均涉及抛物型对流扩散方程（或方程组）的数值求解问题．这些对流扩散型偏微分方程（或方程组）具有共同的特点：对流的影响远大于扩散的影响,即对流占优性,对流占优性给问题的数值求解带来许多困难,因此对流占优问题的有效数值解法一直是计算数学中重要的研究内容．用通常的差分法或有限元法进行数值求解将出现数值振荡．为了克服数值振荡,提出各种迎风方法和修正的特征方法并在这些问题上得到成功的实际应用、８０年代,Ｄｏｕｇｌａｓ和Ｒｕｓｓｅｌｌ［２］等…     

Second-order characteristic schemes in time and age for a nonlinear age-structured population model

In this paper, we analyze two new second-order characteristic schemes in time and age for an age-structured population model with nonlinear diffusion and reaction. By using the characteristic difference to approximate the transport term and the average along the characteristics to treat the nonlinear spatial diffusion and reaction terms, an implicit second-order characteristic scheme is proposed. To compute the nonlinear approximation system, an explicit second-order characteristic scheme in time and age is further proposed by using the extrapolation technique. The global existence and uniqueness of the solution of the nonlinear approximation scheme are established by using the theory of variation methods, Schauder’s fixed point theorem, and the technique of prior estimates. The optimal error estimates of second order in time and age are strictly proved for both the implicit and the explicit characteristic schemes. Numerical examples are given to illustrate the performance of the methods.     

Numerical analysis of a method for high Peclet number transport in porous media

A variationally consistent eddy viscosity discretization is presented in [W.J. Layton, A connection between subgrid scale eddy viscosity and mixed methods, Appl. Math. Comput. 133 (2002) 147-157] for the stationary convection diffusion problem. This discretization is extended to the evolutionary problem in [N. Heitmann, Subgridscale stabilization of time-dependent convection dominated diffusive transport, J. Math. Anal. Appl. 331 (2007) 38-50] with a near optimal error bound. In the following, we couple this discretization with the porous media problem. We present a comprehensive analysis of stability and error for the velocity field derived from the porous media problem. Next, using a backward Euler approximation for the time derivative we follow the inherited error in velocity through the coupling with the convection diffusion problem. The method is shown to be stable and the error near optimal and independent of the diffusion coefficient, ?.     

High-order finite difference schemes for the solution of second-order BVPs

We introduce new methods in the class of boundary value methods (BVMs) to solve boundary value problems (BVPs) for a second-order ODE. These formulae correspond to the high-order generalizations of classical finite difference schemes for the first and second derivatives. In this research, we carry out the analysis of the conditioning and of the time-reversal symmetry of the discrete solution for a linear convection–diffusion ODE problem. We present numerical examples emphasizing the good convergence behavior of the new schemes. Finally, we show how these methods can be applied in several space dimensions on a uniform mesh.     

Two-level finite difference scheme of improved accuracy order for time-dependent problems of mathematical physics

In the theory of finite difference schemes, the most complete results concerning the accuracy of approximate solutions are obtained for two- and three-level finite difference schemes that converge with the first and second order with respect to time. When the Cauchy problem is numerically solved for a system of ordinary differential equations, higher order methods are often used. Using a model problem for a parabolic equation as an example, general requirements for the selection of the finite difference approximation with respect to time are discussed. In addition to the unconditional stability requirements, extra performance criteria for finite difference schemes are presented and the concept of SM stability is introduced. Issues concerning the computational implementation of schemes having higher approximation orders are discussed. From the general point of view, various classes of finite difference schemes for time-dependent problems of mathematical physics are analyzed.     

Adiabatic Integrators for Highly Oscillatory Second-Order Linear Differential Equations with Time-Varying Eigendecomposition

Numerical integrators for second-order differential equations with time-dependent high frequencies are proposed and analysed. We derive two such methods, called the adiabatic midpoint rule and the adiabatic Magnus method. The integrators are based on a transformation of the problem to adiabatic variables and an expansion technique for the oscillatory integrals. They can be used with far larger step sizes than those required by traditional schemes, as is illustrated by numerical experiments. We prove second-order error bounds with step sizes significantly larger than the almost-period of the fastest oscillations.AMS subject classification (2000) 65L05, 65L70.Received February 2004. Accepted February 2005. Communicated by Syvert Nørsett.     

Finite element methods of an operator splitting applied to population balance equations

In population balance equations, the distribution of the entities depends not only on space and time but also on their own properties referred to as internal coordinates. The operator splitting method is used to transform the whole time-dependent problem into two unsteady subproblems of a smaller complexity. The first subproblem is a time-dependent convection-diffusion problem while the second one is a transient transport problem with pure advection. We use the backward Euler method to discretize the subproblems in time. Since the first problem is convection-dominated, the local projection method is applied as stabilization in space. The transport problem in the one-dimensional internal coordinate is solved by a discontinuous Galerkin method. The unconditional stability of the method will be presented. Optimal error estimates are given. Numerical tests confirm the theoretical results.     

一类半线性对流扩散问题特征-有限体积法H1模误差估计

1引言有限体积法是由Baliga和Patankar提出的一种数值求解偏微分方程,特别是物理学中保持守恒律方程的有效方法．由于其运用原方程的体积积分公式和有限控制体积来离散方程．使方程在控制体积上保持守恒律这一重要的物理特性,自出现以来,有了很大的发展([2-4],[10])．特征线方法([1],[8],[9])则是一种非常适合求解对流占优扩散方程的数值