The upwind finite difference fractional steps method for combinatorial system of dynamics of fluids in porous media and its application

For combinatorial system of multilayer dynamics of fluids in porous media, the second order and first order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward and two-dimensional and three-dimensional schemes are used to form a complete set. Some techniques, such as implicit-explicit difference scheme, calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates, are adopted. Optimal order estimates in L ² norm are derived to determine the error in the second order approximate solution. This method has already been applied to the numerical simulation of migration-accumulation of oil resources. Keywords: combinatorial system, multilayer dynamics of fluids in porous media, two-class upwind finite difference fractional steps method, convergence, numerical simulation of energy sources.     

Error analysis of staggered finite difference finite volume schemes on unstructured meshes

This work combines the consistency in lower‐order differential operators with external approximations of functional spaces to obtain error estimates for finite difference finite volume schemes on unstructured nonuniform meshes. This combined approach is first applied to a one‐dimensional elliptic boundary value problem on nonuniform meshes, and a first‐order convergence rate is obtained, which agrees with the results previously reported. The approach is also applied to the staggered Marker‐and‐Cell scheme for the two‐dimensional incompressible Stokes problem on unstructured meshes. A first‐order convergence rate is obtained, which improves over a previously reported result in that it also holds on unstructured meshes. For both problems considered in this work, the convergence rate is one order higher on meshes satisfying special requirements. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1159–1182, 2017     

High order schemes for the tempered fractional diffusion equations

Lévy flight models whose jumps have infinite moments are mathematically used to describe the superdiffusion in complex systems. Exponentially tempering Lévy measure of Lévy flights leads to the tempered stable Lévy processes which combine both the α-stable and Gaussian trends; and the very large jumps are unlikely and all their moments exist. The probability density functions of the tempered stable Lévy processes solve the tempered fractional diffusion equation. This paper focuses on designing the high order difference schemes for the tempered fractional diffusion equation on bounded domain. The high order difference approximations, called the tempered and weighted and shifted Grünwald difference (tempered-WSGD) operators, in space are obtained by using the properties of the tempered fractional calculus and weighting and shifting their first order Grünwald type difference approximations. And the Crank-Nicolson discretization is used in the time direction. The stability and convergence of the presented numerical schemes are established; and the numerical experiments are performed to confirm the theoretical results and testify the effectiveness of the schemes.     

FINITE DIFFERENCE FRACTIONAL STEP METHODS FOR THE TRANSIENT BEHAVIOR OF A SEMICONDUCTOR DEVICE

Characteristic finite difference fractional step schemes are put forward. The electric potential equation is described by a seven-point finite difference scheme, and the electron and hole concentration equations are treated by a kind of characteristic finite difference fractional step methods. The temperature equation is described by a fractional step method. Thick and thin grids are made use of to form a complete set. Piecewise threefold quadratic interpolation, symmetrical extension, calculus of variations, commutativity of operator product, decomposition of high order difference operators and prior estimates are also made use of. Optimal order estimates in l2 norm are derived to determine the error of the approximate solution. The well-known problem is thorongley and completely solred.     

非线性多层渗流系统的数值方法及其应用

对多层非线性渗流系统提出适合并行计算的迎风分数步差分格式，利用变分形式、能量方法、差分算子乘积交换性、高阶差分算子的分解、先验估计的理论和技巧，得到收敛性的最佳阶的误差估计．该方法已成功的应用到油资源渗流力学运移聚集数值模拟的生产实际中．     

The finite difference method for the three-dimensional nonlinear coupled system of dynamics of fluids in porous media

For the three-dimensional coupled system of multilayer dynamics of fluids in porous media, the second-order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward. Some techniques, such as calculus of variations, energy method, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates are adopted. Optimal order estimates in l2 norm are derived to determine the error in the second-order approximate solution. These methods have already been applied to the numerical simulation of migration-accumulation of oil resources.     

非线性渗流耦合系统动边值问题二阶迎风分数步差分方法

对多层非线性渗流方程耦合系统三维动边值问题, 提出适合并行计算的一类二阶迎风分数步差分格式, 利用区域变换、变分形式、能量方法、隐显格式的相互结合、差分算子乘积交换性、高阶差分算子的分解、先验估计的理论和技巧, 得到收敛性的最佳阶l² 误差估计. 该方法已成功地应用到多层油资源运移聚集的资源评估生产实际中, 得到了很好的数值模拟结果.     

The characteristic finite difference fractional steps methods for compressible two-phase displacement problem

For compressible two-phase displacement problem, a kind of characteristic finite difference fractional steps schemes is put forward and thick and thin grids are used to form a complete set. Some techniques, such as piecewise biquadratic interpolation, of calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates are adopted. Optimal order estimates inL ² norm are derived to determine the error in the approximate solution. Project supported by the National Scaling Program, the National Tackling Key Problems Program and the Doctorate Foundation of the State Education Commission of China.     

多层非线性渗流耦合系统的特征分数步差分方法

对多层非线性渗流耦合系统提出适合并行计算的特征分数步差分格式, 利用变分形式、能量方法、粗细网格配套、分片双二次插值、差分算子乘积交换性、高阶差分算子的分解、先验估计的理论和技巧, 得到收敛性的最佳阶的l₂误差估计. 该方法已成功的应用到多层油资源评估的生产实际中.     

The upwind finite difference fractional steps method for nonlinear coupled systems

For nonlinear coupled system of multilayer dynamics of fluids in porous media, a second‐order upwind finite‐difference fractional‐steps scheme applicable to parallel arithmetic are put forward, and two‐ and three‐dimensional schemes are used to form a complete set. Some techniques, such as calculus of variations, multiplicative commutation rule of difference operators, decomposition of high‐order difference operators, and prior estimates are adopted. Optimal order estimates in l² norm are derived to determine the error in the approximate solution. This method has already been applied to the numerical simulation of migration‐accumulation of oil resources. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

非线性渗流耦合系统的数值方法及其应用

对非线性二相渗流耦合系统提出修正迎风分数步差分格式,利用变分形式、归纳法假定、高阶差分算子的分解、微分方程的先验估计理论和技巧,得到收敛性的最佳阶误差估计．该方法已成功应用于防治海水入侵主要工程后效预测与调控模式的生产实践中．     

Higher order finite difference schemes for the magnetic induction equations

We describe high order accurate and stable finite difference schemes for the initial-boundary value problem associated with the magnetic induction equations. These equations model the evolution of a magnetic field due to a given velocity field. The finite difference schemes are based on Summation by Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation Term (SAT) technique for imposing boundary conditions. We present various numerical experiments that demonstrate both the stability as well as high order of accuracy of the schemes.      

A second-order accurate numerical method for a fractional wave equation

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k ² + h ², where k and h are the parameters for the time and space meshes, respectively.     

Convergence analysis of some high–order time accurate schemes for a finite volume method for second order hyperbolic equations on general nonconforming multidimensional spatial meshes

The present work is an extension of our previous work (Bradji, Numer Methods Partial Differ Equations, to appear) which dealt with error analysis of a finite volume scheme of a first convergence order (both in time and space) for second‐order hyperbolic equations on general nonconforming multidimensional spatial meshes introduced recently in (Eymard et al. IMAJ Numer Anal 30(2010), 1009–1043). We aim in this article to get some higher‐order time accurate schemes for a finite volume method for second‐order hyperbolic equations using the same class of spatial generic meshes stated above. We derive a family of finite volume schemes approximating the wave equation, as a model for second‐order hyperbolic equations, in which the discretization in time is performed using a one‐parameter scheme of the Newmark's method. We prove that the error estimate of these finite volume schemes is of order two (or four) in time and it is of optimal order in space. These error estimates are analyzed in several norms which allow us to derive approximations for the exact solution and its first derivatives whose the convergence order is two (or four) in time and it is optimal in space. We prove in particular, when the discrete flux is calculated using a stabilized discrete gradient, that the convergence order is \begin{align*}k^2+h_\mathcal{D}\end{align*} or \begin{align*}k^4+h_\mathcal{D}\end{align*}, where \begin{align*}h_\mathcal{D}\end{align*} (resp. k) is the mesh size of the spatial (resp. time) discretization. These estimates are valid under the regularity assumption \begin{align*}u\in C^4(\lbrack 0,T\rbrack;C^2(\overline{\Omega}))\end{align*}, when the schemes are second‐order accurate in time, and \begin{align*}u\in C^6(\lbrack 0,T\rbrack;C^2(\overline{\Omega}))\end{align*}, when the schemes are four‐order accurate in time for the exact solution u. The proof of these error estimates is based essentially on a comparison between the finite volume approximate solution and an auxiliary finite volume approximation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

MIXED FINITE ELEMENT METHODS FOR FRACTIONAL NAVIER-STOKES EQUATIONS

This paper gives the detailed numerical analysis of mixed finite element method for fractional Navier-Stokes equations.The proposed method is based on the mixed finite element method in space and a finite difference scheme in time.The stability analyses of semi-discretization scheme and fully discrete scheme are discussed in detail.Furthermore,We give the convergence analysis for both semidiscrete and flly discrete schemes and then prove that the numerical solution converges the exact one with order O(h2+k),where h and k:respectively denote the space step size and the time step size.Finally,numerical examples are presented to demonstrate the effectiveness of our numerical methods.     

A series of high‐order quasi‐compact schemes for space fractional diffusion equations based on the superconvergent approximations for fractional derivatives

Based on the superconvergent approximation at some point (depending on the fractional order α, but not belonging to the mesh points) for Grünwald discretization to fractional derivative, we develop a series of high‐order quasi‐compact schemes for space fractional diffusion equations. Because of the quasi‐compactness of the derived schemes, no points beyond the domain are used for all the high‐order schemes including second‐order, third‐order, fourth‐order, and even higher‐order schemes; moreover, the algebraic equations for all the high‐order schemes have the completely same matrix structure. The stability and convergence analysis for some typical schemes are made; the techniques of treating the fractional derivatives with nonhomogeneous boundaries are introduced; and extensive numerical experiments are performed to confirm the theoretical analysis or verify the convergence orders. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1345–1381, 2015     

The modified characteristic finite difference fractional steps method for the coupled system of fluid dynamics in porous media and its analysis

For the coupled system of multilayer fluid dynamics in porous media, the modified characteristic finite difference fractional steps method applicable to parallel arithmetic is put forward and two‐dimensional and three‐dimensional schemes are used to form a complete set. Some techniques, such as calculus of variations, energy method, piecewise biquadratic interpolation, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates are adopted. Optimal order estimates in L² norm are derived to determine the error in the approximate solution. This method has already been applied to the numerical simulation of multilayer fluid dynamics in porous media. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 665–681, 2003.     

HIGH ORDER FINITE DIFFERENCE/SPECTRAL METHODS TO A WATER WAVE MODEL WITH NONLOCAL VISCOSITY

In this paper, efficient numerical schemes are proposed for solving the water wave model with nonlocal viscous term that describe the propagation of surface water wave. By using the Caputo fractional derivative definition to approximate the nonlocal fractional operator, finite difference method in time and spectral method in space are constructed for the considered model. The proposed method employs known 5/2 order scheme for fractional derivative and a mixed linearization for the nonlinear term. The analysis shows that the proposed numerical scheme is unconditionally stable and error estimates are provided to predict that the second order backward differentiation plus 5/2 order scheme converges with order 2 in time, and spectral accuracy in space. Several numerical results are provided to verify the efficiency and accuracy of our theoretical claims. Finally, the decay rate of solutions is investigated.     

A Survey of High Order Schemes for the Shallow Water Equations

In this paper, we survey our recent work on designing high order positivity-preserving well-balanced finite difference and finite volume WENO (weighted essentially non-oscillatory) schemes, and discontinuous Galerkin finite element schemes for solving the shallow water equations with a non-flat bottom topography. These schemes are genuinely high order accurate in smooth regions for general solutions, are essentially non-oscillatory for general solutions with discontinuities, and at the same time they preserve exactly the water at rest or the more general moving water steady state solutions. A simple positivity-preserving limiter, valid under suitable CFL condition, has been introduced in one dimension and reformulated to two dimensions with triangular meshes, and we prove that the resulting schemes guarantee the positivity of the water depth.