Numerical simulation of the coupled viscous Burgers equation using the Hermite formula and cubic B-spline basis functions

Abstract

A numerical procedure dependent on the cubic B-spline and the Hermite formula is developed for the coupled viscous Burgers’ equation (CVBE). The method uses a combination of the Hermite formula and the cubic B-spline for discretization of the space dimension while the time dimension is approximated using the typical finite differences. A piecewise continuous sufficiently smooth function is obtained as a solution which allows to approximate solution at any location in the domain of interest. The scheme is tested for stability analysis and is proved to be unconditionally stable. Numerical experiments and comparison of outcomes reveal that the suggested scheme comes up with better accuracy and is extremely productive.

An Innovative Numerical Method Utilizing Novel Cubic B-Spline Approximations to Solve Burgers’ Equation

“…The recently derived estimation for the second derivative of (v n j ) zz is presented in [31] as follows:…”

“…B-spline methods excel in handling complex geometries and smooth solutions, while finite element methods offer a well-established and versatile approach that is widely adopted in the field. In order to explore further research on Burgers' equation and the related nonlinear phenomenon, please consult [28][29][30][31][32][33][34][35][36][37][38] and the sources cited within.…”

Fast algorithms for coupled Burgers' equations by sharing characteristic curves within BSLM

“…Over the last 10 years, various numerical methods have been developed to solve the viscous coupled Burgers' equations. For example, Kapoor and Joshi developed a differential quadrature method with uniform algebraic trigonometric tension B-spline [11], Başhan applied a mixed method with the finite difference and differential quadrature method based on third order modified cubic B-spline functions [4], Hussein and Kashkool used a weak Galerkin finite element method [9], Zhang et al introduced an improved backward substitution method [19], Abdullah et al developed a numerical procedure based on the cubic B-spline and the Hermite formula [1], Mohanty and Sharma presented a high accuracy two-level implicit method based on cubic spline approximation [16], Bhatt and Khaliq modified exponential time-differencing Runge-Kutta method based on Padé approximation [6], Kumar and Pandit proposed a composite scheme based on finite difference and Haar wavelets [12], Jiwari and Alshomrani developed a new collocation method based on modified cubic trigonometric B-spline function [10], Mohanty et al proposed a two-level implicit compact operator method [13]. Among the abovementioned methods, there is a backward semi-Lagrangian (BSLM) to solve the model problem.…”

“…In Sec. 2, we review and extend CBSM2 for solving (1) and the Cauchy problem solver with third-order accuracy, which individually determines all characteristic curves of each particle. In Sec.…”

Numerical solution of the coupled Burgers equation by trigonometric B‐spline collocation method

“…Siraj‐ul‐Islam et al 16 have formulated a simple classical radial basis functions (RBFs) collocation (Kansa) method for the numerical solution of the Korteweg‐de Vries equations, coupled Burgers equations, and quasi non‐linear hyperbolic equations. Abdullah et al 17 have developed a numerical procedure dependent on the cubic B‐spline and the Hermite formula for the coupled viscous Burgers equation. Mittal and Jiwari 18 have solved the coupled viscous Burgers equations by using the differential quadrature method.…”