A computational approach using modified trigonometric cubic B-spline for numerical solution of Burgers’ equation in one and two dimensions

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.…”

The local discontinuous Galerkin method with generalized alternating flux for solving Burger's equation

“…From these tables, we can observe the different convergence with different θ. For θ = 1 2 , the suboptimal accuracy of (k + 1 2 )-th order for odd k and optimal accuracy of (k + 1)-th order with for even k are obtained. For θ = 1 2 , the suboptimal accuracy of k-th order for odd k and optimal (k + 1)-th order with for even k can be obtained.…”

“…Shao et al have presented LDG method for a nonlinear Burger's equation with Dirichlet boundary conditions [24]. They have proved that the LDG method has the (k + 1)-th order of convergence rate, while the numerical experiments show only (k + 1 2 )-th order convergence which is inconsistent with the theoretical results. In [27,28], we have applied the direct DG method to Burger's, modified Burger's and coupled Burger's equation by LDG method.…”

“…In this paper we will apply LDG method based on the generalized alternating numerical flux. By virtue of the generalized Gauss-Radau projection, we can prove that the proposed method can get sub-optimal rate of convergence in L 2 -norm of O(h k+ 1 2 ) with polynomial of degree k for nonlinear Burger's equation.…”

“…Many researchers have developed various numerical methods for the solution of Burger's equation. The available numerical techniques include finite element method [7,22], Local Crank-Nicolson method [15], the finite difference methods [4,19], lattice Boltzmann method [13], Chebyshev spectral collocation method [18], cubic B-spline differential quadrature method [1,2], boundary element method [5] and differential quadrature method [17]. More recently, Mukundan and Awasthi [21] developed the finite difference method coupled with an low-dispersion and low-dissipation implicit Runge-Kutta method for (1.1); Tamsir et al proposed an algorithm based on exponential modified cubic B-spline differential quadrature method [25]; Jiwari developed the hybrid numerical scheme based on Euler implicit method, quasilinearization and uniform Haar wavelets [16].…”