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

Abstract

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.

Effective numerical technique applied for Burgers' equation of (1 + 1)‐, (2 + 1)‐dimensional, and coupled forms

“…Table 6 illustrates absolute errors comparison between numerical solutions of Liu et al, 57 and the proposed scheme with Δ t = 0.001, Re = 20, and grid size = 16 × 16 at some specific points for 2‐D Burgers' model. Also, the proposed scheme L 2 and L ∞ error norms are compared to other authors' works, 57–59 and the comparison is tabulated in Table 7. Moreover, Table 8 presents L 2 and L ∞ error norms with Δ t = 0.0005 and grid size of 16 × 16 at different time and Re values.…”

An efficient cubic B‐spline and bicubic B‐spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations

“…Meanwhile, spline basis functions have attracted many researchers because of their ease of implementation and low computing. In recent years, numerous numerical methods on the basis of various types of spline functions have been proposed to solve boundary value problems, partial differential equations, integral equations, and integro‐differential equations such as, Abd Hamid et al used cubic Beta‐spline to linear two‐point boundary value problems, Ebrahimi and Rashidinia presented spline collocation for linear and nonlinear Fredholm and Volterra integral equations, Fredholm and Volterra integro‐differential equations, and system of Fredholm and Volterra integro‐differential equations, numerical solutions of two‐dimensional Burgers' equations and two‐dimensional unsteady convection‐diffusion problems using modified bi‐cubic B‐spline finite elements offered by Mittal and Tripathi . Also, they applied collocation of cubic B‐splines finite element method to solve symmetric regularized long wave equations .…”

Numerical Solution of High-Dimensional Shockwave Equations by Bivariate Multi-Quadric Quasi-Interpolation

“…In recent years, the meshless collocation method has drawn considerable attention for solving PDEs [11][12][13]. Mittal and Tripathi [14] proposed a collocation method based on Modified bi-cubic B-Spline functions. However, it requires to solve the inverse of a large-scale matrix on each time step, which costs computational time and might be ill-conditioned.…”