Numerical Simulation and Dynamics of Burgers’ Equation Using the Modified Cubic B-Spline Differential Quadrature Method

Abstract

In the present work, a numerical approach using the Crank–Nicolson scheme along with the modified cubic B-spline differential quadrature (CN-MCDQ) method is proposed to find the numerical approximations to Burgers’ equation. After applying the well-known Crank–Nicolson technique, Burgers’ equation is solved in this study by using the differential quadrature approach to approximate the derivatives that lead to a system of equations to be solved. When compared to other methods for obtaining numerical solutions, the proposed method is shown to be efficient and easy to implement while still providing accurate results. The obtained results are in agreement with the earlier available approaches and are even better in comparison in terms of less domain partition. Three test problems were used to evaluate the methodology, and the results are tabulated and graphically shown below.

A New Efficient Hybrid Method Based on FEM and FDM for Solving Burgers’ Equation with Forcing Term

“…The advantages of the hybrid methods are seen especially from the steep behavior of the produced results. We show that our methods stabilize the solutions much earlier than the methods suggested by some literature [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]26]. The present works of literature are aimed at obtaining numerical solutions accurately.…”

“…The numerical techniques are valuable equipment for understanding the process of the physical model, and for this reason, many numerical studies have been developed in the literature, and we mention some of them in this study, for example, the implicit finite difference method [7], the implicit fourth-order compact finite difference scheme [8], the seventh-order weighted essentially nonoscillatory (WENO) schemes [9], a nonlinear Hopf-Cole transformation and backward differentiation formula method [10], the finite element method based on the method of discretization in time [11], a simple finite element approximation to the Burgers' equation diminished by Hopf-Cole transformation [12], and a weak finite element method [13]. Recently, spline functions with some numerical schemes have been used in acquiring numerical solutions of the Burgers' equation such as cubic and quadratic B-spline collocation method [14], modified cubic B-spline collocation method [15], B-spline Galerkin method and B-spline collocation method [16], collocation method based on Hermite formula and cubic B-splines [17], a cubic B-spline Galerkin method with higher order splitting approaches [18], cubic B-spline and fourth-order compact finite difference method [19], and cubic B-spline and differential quadrature method [20]. Also, implicit fractional step θ-scheme and conforming finite element method [21], radial basis functions (RBF) meshless method [22], nonstandard finite difference method [23], and a sixth-order compact finite difference scheme for space integration and Crank-Nicolson scheme for time discretization were used in [24].…”