A fourth‐order finite‐difference method for solving the system of two‐dimensional Burgers' equations

Liao, Wenyuan

doi:10.1002/fld.2163

Cited by 46 publications

(39 citation statements)

References 26 publications

(38 reference statements)

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“…The noncompact stencil not only complicates the formulations near the boundaries but also increases the bandwidth of the resulting coefficient matrix. Motivated by these problems, various higher-order compact finite difference discretization techniques for different equations have been developed (see, e.g., [1,14,18,19,20,21,36,39]). …”

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confidence: 99%

Fourth-Order Compact Finite Difference Methods and Monotone Iterative Algorithms for Quasi-Linear Elliptic Boundary Value Problems

Wang¹

2015

SIAM J. Numer. Anal.

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This paper is concerned with numerical methods for a class of two-dimensional quasilinear elliptic boundary value problems. A compact finite difference method with a nonisotropic mesh is proposed for the problems. The existence of a maximal and a minimal compact difference solution is proved by the method of upper and lower solutions, and two sufficient conditions for the uniqueness of the solution are also given. The optimal error estimate in the discrete L ∞ norm is obtained under certain conditions. The error estimate shows the fourth-order accuracy of the proposed method when two spatial mesh sizes are proportional. By using an upper solution or a lower solution as the initial iteration, an "almost optimal" Picard type of monotone iterative algorithm is presented for solving the resulting nonlinear discrete system efficiently. Applications using two model problems give numerical results that confirm our theoretical analysis.Key words. quasi-linear elliptic boundary value problem, compact finite difference method, error estimate, fourth-order accuracy, monotone iterative algorithm 1. Introduction. Finite difference methods have long been used to approximate the solution of ordinary or partial differential equations. Central differencing is one of the most popular finite difference methods. However, it has only second-order accuracy in general. A simple technique for improving the accuracy is to include additional mesh points into the difference approximations of the derivatives [6, 7, 30]. The higher-order methods derived in this manner are noncompact in the sense that the required stencils always utilize mesh points located beyond those directly adjacent to the node about which the differences are taken. The noncompact stencil not only complicates the formulations near the boundaries but also increases the bandwidth of the resulting coefficient matrix. Motivated by these problems, various higher-order compact finite difference discretization techniques for different equations have been developed (see, e.g., [1,14,18,19,20,21,36,39]).In recent years, there has been growing interest in developing fourth-order compact finite difference methods for elliptic differential equations. Such a method uses a similar compact stencil as the standard second-order central finite difference method, but it still achieves higher-order accuracy. The existing fourth-order compact finite difference methods are mostly derived for linear elliptic differential equations (see, e.g., [4,5,11,12,16,34,35,38,41]). The main idea of these methods is to increase the accuracy of the standard central finite difference approximation from the second-

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confidence: 99%

Fourth-Order Compact Finite Difference Methods and Monotone Iterative Algorithms for Quasi-Linear Elliptic Boundary Value Problems

Wang¹

2015

SIAM J. Numer. Anal.

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“…However, higher-order methods derived in this manner not only complicate the formulations near the boundaries but also increase the bandwidth of the resulting coefficient matrix. Motivated by these problems, various higher-order compact finite difference discretization techniques for different equations have been developed (see, e.g., [4][5][6][7][8][9][10][11]). In recent years, there has been growing interest in developing fourth-order compact finite difference methods for elliptic differential equations.…”

Section: Introductionmentioning

confidence: 99%

Fourth-order compact finite difference methods and monotone iterative algorithms for semilinear elliptic boundary value problems

Wang

Guo

2014

Computers & Mathematics with Applications

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“…Mittal and Jiwari (2012) proposed differential quadrature methods to obtain numerical solutions of 2D Burgers equation. A Lattice Boltzmann method has been proposed by Zhang and Yan (2008) while a fourth-order compact finite difference method has been proposed by Liao (2010).…”

Section: Introductionmentioning

confidence: 99%

“…In terms of various applications, all of these functions have their own advantages and disadvantages. In some earlier studies (Dehghan and Lakestani, 2008;Lakestani and Dehghan, 2009, 2010, 2013Dehghan et al, 2012Dehghan et al, , 2014, authors used collocation method and B-spline functions to obtain numerical solutions of various partial differential equations. However, modified B-spline basis functions draw the attention among these functions because of their ease of implementation, low computational efforts and cost.…”

Section: Introductionmentioning

confidence: 99%

Numerical solutions of two-dimensional Burgers’ equations using modified Bi-cubic B-spline finite elements

Mittal

Tripathi

2015

Engineering Computations

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Article information:To cite this document: R C Mittal Amit Tripathi , (2015),"Numerical solutions of two-dimensional Burgers' equations using modified Bi-cubic B-spline finite elements", Engineering Computations, Vol. 32 Iss 5 pp. 1275 -1306 Permanent link to this document: http://dx. AbstractPurpose -The purpose of this paper is to develop an efficient numerical scheme for non-linear twodimensional (2D) parabolic partial differential equations using modified bi-cubic B-spline functions. As a test case, method has been applied successfully to 2D Burgers equations. Design/methodology/approach -The scheme is based on collocation of modified bi-cubic B-Spline functions. The authors used these functions for space variable and for its derivatives. Collocation form of the partial differential equation results into system of first-order ordinary differential equations (ODEs). The obtained system of ODEs has been solved by strong stability preserving Runge-Kutta method. The computational complexity of the method is O(p log(p)), where p denotes total number of mesh points. Findings -Obtained numerical solutions are better than those available in literature. Ease of implementation and very small size of computational work are two major advantages of the present method. Moreover, this method provides approximate solutions not only at the grid points but also at any point in the solution domain. Originality/value -First time, modified bi-cubic B-spline functions have been applied to non-linear 2D parabolic partial differential equations. Efficiency of the proposed method has been confirmed with numerical experiments. The authors conclude that the method provides convergent approximations and handles the equations very well in different cases.

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A fourth‐order finite‐difference method for solving the system of two‐dimensional Burgers' equations

Cited by 46 publications

References 26 publications

Fourth-Order Compact Finite Difference Methods and Monotone Iterative Algorithms for Quasi-Linear Elliptic Boundary Value Problems

Fourth-Order Compact Finite Difference Methods and Monotone Iterative Algorithms for Quasi-Linear Elliptic Boundary Value Problems

Fourth-order compact finite difference methods and monotone iterative algorithms for semilinear elliptic boundary value problems

Numerical solutions of two-dimensional Burgers’ equations using modified Bi-cubic B-spline finite elements

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