A non-standard finite difference scheme for convection in a porous cavity

In this paper, a non-standard finite difference scheme is proposed for solving a steady finite Rayleigh number convection in a porous cavity with an inclined magnetic field and non-uniform internal heating. Numerical results are compared with the classical finite difference scheme.     

Peculiarities of error accumulation in solving problems for simple equations of mathematical physics by finite difference methods

A mixed problem for the one-dimensional heat conduction equation with several versions of initial and boundary conditions is considered. To solve this problem, explicit and implicit schemes are used. Sweep and iterative methods are used for the implicit scheme when solving the system of equations. Numerical filtering of a finite sequence of results obtained for various grids with an increasing number of node points is used to analyze the method and rounding errors. To investigate rounding errors, the results obtained for various machine word mantissa lengths are compared. The numerical solution of a mixed problem for the wave equation is studied by similar methods. Some deterministic dependencies of the numerical method and rounding errors on the spatial coordinates, time, and the number of nodes are found. Some models of sources to describe the behavior of the errors in time are constructed. They are based on the results of computational experiments under various conditions. According to these models, which have been experimentally verified, the errors increase, decrease, or stabilize in time under the conditions, similarly to energy or mass.     

求解一维定常对流扩散反应方程的混合型紧致差分格式

借助显式紧致格式和隐式紧致格式的思想,基于截断误差余项修正,并结合原方程本身,构造出了一种求解一维定常对流扩散反应方程的高精度混合型紧致差分格式.格式仅用到三个点上的未知函数值及一阶导数值,而一阶导数值利用四阶Pade格式进行计算,格式整体具有四阶精度.数值实验结果验证了格式的精确性和可靠性.     

On some finite difference schemes for solution of hyperbolic heat conduction problems

We consider the accuracy of two finite difference schemes proposed recently in [Roy S., Vasudeva Murthy A.S., Kudenatti R.B., A numerical method for the hyperbolic-heat conduction equation based on multiple scale technique, Appl. Numer. Math., 2009, 59(6), 1419–1430], and [Mickens R.E., Jordan P.M., A positivity-preserving nonstandard finite difference scheme for the damped wave equation, Numer. Methods Partial Differential Equations, 2004, 20(5), 639–649] to solve an initial-boundary value problem for hyperbolic heat transfer equation. New stability and approximation error estimates are proved and it is noted that some statements given in the above papers should be modified and improved. Finally, two robust finite difference schemes are proposed, that can be used for both, the hyperbolic and parabolic heat transfer equations. Results of numerical experiments are presented.     

Wavelet optimized finite difference method using interpolating wavelets for self-adjoint singularly perturbed problems

We design a wavelet optimized finite difference (WOFD) scheme for solving self-adjoint singularly perturbed boundary value problems. The method is based on an interpolating wavelet transform using polynomial interpolation on dyadic grids. Small dissipation of the solution is captured significantly using an adaptive grid. The adaptive feature is performed automatically by thresholding the wavelet coefficients. Numerical examples have been solved and compared with non-standard finite difference schemes in [J.M.S. Lubuma, K.C. Patidar, Uniformly convergent non-standard finite difference methods for self-adjoint singular perturbation problems, J. Comput. Appl. Math. 191 (2006) 228–238]. The proposed method outperforms the non-standard finite difference for studying singular perturbation problems for small dissipations (very small ) and effective grid generation. Therefore, the proposed method is better for studying the more challenging cases of singularly perturbed problems.     

On non-standard finite difference models of reaction–diffusion equations

Reaction–diffusion equations arise in many fields of science and engineering. Often, their solutions enjoy a number of physical properties. We design, in a systematic way, new non-standard finite difference schemes, which replicate three of these properties. The first property is the stability/instability of the fixed points of the associated space independent equation. This property is preserved by non-standard one- and two-stage theta methods, presented in the general setting of stiff or non-stiff systems of differential equations. Schemes, which preserve the principle of conservation of energy for the corresponding stationary equation (second property) are constructed by non-local approximation of nonlinear reactions. Assembling of theta-methods in the time variable with energy-preserving schemes in the space variable yields non-standard schemes which, under suitable functional relation between step sizes, display the boundedness and positivity of the solution (third property). A spectral method in the space variable coupled with a suitable non-standard scheme in the time variable is also presented. Numerical experiments are provided.     

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.     

New higher-order compact finite difference schemes for 1D heat conduction equations

In this paper, we present two higher-order compact finite difference schemes for solving one-dimensional (1D) heat conduction equations with Dirichlet and Neumann boundary conditions, respectively. In particular, we delicately adjust the location of the interior grid point that is next to the boundary so that the Dirichlet or Neumann boundary condition can be applied directly without discretization, and at the same time, the fifth or sixth-order compact finite difference approximations at the grid point can be obtained. On the other hand, an eighth-order compact finite difference approximation is employed for the spatial derivative at other interior grid points. Combined with the Crank–Nicholson finite difference method and Richardson extrapolation, the overall scheme can be unconditionally stable and provides much more accurate numerical solutions. Numerical errors and convergence rates of these two schemes are tested by two examples.     

A Nonstandard Finite Difference Scheme for Nonlinear Heat Transfer in a Thin Finite Rod

A nonstandard finite difference scheme is constructed to solve an initial-boundary value problem involving a quartic nonlinearity that arises in heat transfer involving conduction with thermal radiation. It is noted that the positivity condition is equivalent to the usual linear stability criteria and it is shown that the representation of the nonlinear term in the finite difference scheme, in addition to the magnitudes of the equation parameters, has a direct bearing on the scheme's stability. Finally, solution profiles are plotted and avenues of further inquiry are discussed.     

On convergence of finite volume schemes for one-dimensional two-phase flow in porous media

This paper deals with development and analysis of finite volume schemes for a one-dimensional nonlinear, degenerate, convection-diffusion equation having application in petroleum reservoir and groundwater aquifer simulation. The main difficulty is that the solution typically lacks regularity due to the degenerate nonlinear diffusion term. We analyze and compare three families of numerical schemes corresponding to explicit, semi-implicit, and implicit discretization of the diffusion term and a Godunov scheme for the advection term. L^∞ stability under appropriate CFL conditions and BV estimates are obtained. It is shown that the schemes satisfy a discrete maximum principle. Then we prove convergence of the approximate solution to the weak solution of the problem. Results of numerical experiments using the present approach are reported.     

A positivity‐preserving nonstandard finite difference scheme for the damped wave equation

A positivity‐preserving nonstandard finite difference scheme is constructed to solve an initial‐boundary value problem involving heat transfer described by the Maxwell‐Cattaneo thermal conduction law, i.e., a modified form of the classical Fourier flux relation. The resulting heat transport equation is the damped wave equation, a PDE of hyperbolic type. In addition, exact analytical solutions are given, special cases are mentioned, and it is noted that the positivity condition is equivalent to the usual linear stability criteria. Finally, solution profiles are plotted and possible extensions to a delayed diffusion equation and nonlinear reaction‐diffusion systems are discussed. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

An inverse problem of finding a source parameter in a semilinear parabolic equation

An inverse problem concerning diffusion equation with source control parameter is considered. Several finite-difference schemes are presented for identifying the control parameter. These schemes are based on the classical forward time centred space (FTCS) explicit formula, and the 5-point FTCS explicit method and the classical backward time centred space (BTCS) implicit scheme, and the Crank–Nicolson implicit method. The classical FTCS explicit formula and the 5-point FTCS explicit technique are economical to use, are second-order accurate, but have bounded range of stability. The classical BTCS implicit scheme and the Crank–Nicolson implicit method are unconditionally stable, but these schemes use more central processor (CPU) times than the explicit finite difference mehods. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. The results of a numerical experiment are presented, and the accuracy and CPU time needed for this inverse problem are discussed.     

The Conservation and Convergence of Two Finite Difference Schemes for KdV Equations with Initial and Boundary Value Conditions

Korteweg-de Vries equation is a nonlinear evolutionary partial differential equation that is of third order in space. For the approximation to this equation with the initial and boundary value conditions using the finite difference method, the difficulty is how to construct matched finite difference schemes at all the inner grid points. In this paper, two finite difference schemes are constructed for the problem. The accuracy is second-order in time and first-order in space. The first scheme is a two-level nonlinear implicit finite difference scheme and the second one is a three-level linearized finite difference scheme. The Browder fixed point theorem is used to prove the existence of the nonlinear implicit finite difference scheme. The conservation, boundedness, stability, convergence of these schemes are discussed and analyzed by the energy method together with other techniques. The two-level nonlinear finite difference scheme is proved to be unconditionally convergent and the three-level linearized one is proved to be conditionally convergent. Some numerical examples illustrate the efficiency of the proposed finite difference schemes.     

关于U_t=aU_(xx)差分格式的进一步研究(英文)

对于热传导方程构造了两个高阶精度的差分格式，一个是三层七点显格式，另一个是三层九点隐格式．证明了差分格式的收敛性和稳定性，最后给出数值计算结果．     

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 heat and moisture transport with a finite difference method

In this article, we study a system of nonlinear parabolic partial differential equations arising from the heat and moisture transport through textile materials with phase change. A splitting finite difference method with semi‐implicit Euler scheme in time direction is proposed for solving the system of equations. We prove the existence and uniqueness of a classical positive solution to the parabolic system as well as the existence and uniqueness of a positive solution to the splitting finite difference system. We provide optimal error estimates for the splitting finite difference system under the condition that the mesh size and time step size are smaller than a positive constant which solely depends upon the physical parameters involved. Numerical results are presented to confirm our theoretical analysis. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Implicit Solution of a Two-Dimensional Parabolic Inverse Problem with Temperature Overspecification

Three different implicit finite difference schemes for solving the two-dimensional parabolic inverse problem with temperature overspecification are considered. These schemes are developed for indentifying the control parameter which produces, at any given time, a desired temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the second-order (5,1) Backward Time Centered Space (BTCS) implicit formula, and the second-order (5,5) Crank-Nicolson implicit finite difference formula and the fourth-order (9,9) implicit scheme. These finite difference schemes are unconditionally stable. The (9,9) implicit formula takes a huge amount of CPU time, but its fourth-order accuracy is significant. The results of a numerical experiment are presented, and the accuracy and central processor (CPU) times needed for each of the methods are discussed and compared. The implicit finite difference schemes use more central processor times than the explicit finite difference techniques, but they are stable for every diffusion number.     

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.     

A new combined finite element‐upwind finite volume method for convection‐dominated diffusion problems

In this article, we develop a combined finite element‐weighted upwind finite volume method for convection‐dominated diffusion problems in two dimensions, which discretizes the diffusion term with the standard finite element scheme, and the convection and source terms with the weighted upwind finite volume scheme. The developed method leads to a totally new scheme for convection‐dominated problems, which overcomes numerical oscillation, avoids numerical dispersion, and has high‐order accuracy. Stability analyses of the scheme are given for the problems with constant coefficients. Numerical experiments are presented to illustrate the stability and optimal convergence of our proposed method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 799–818, 2016     

求解一维非定常对流扩散反应方程的高精度紧致差分格式

提出了数值求解一维非定常对流扩散反应方程的一种高精度紧致隐式差分格式,其截断误差为O(τ~4+τ~2h~2+h~4),即格式整体具有四阶精度.差分方程在每一时间层上只用到了三个网格节点,所形成的代数方程组为三对角型,可采用追赶法进行求解,最后通过数值算例验证了格式的精确性和可靠性.