The nonlinear modified Burgers equation is solved numerically by method of Galerkin using quintic B-splines as both shape and weight functions over the finite intervals. The same method is also applied to the time-split modified Burgers equation. Numerical comparison of results of both algorithms and some other published papers is done by studying standard test problem.

Purpose – The purpose of this paper is to investigate the numerical solutions of the Burgers' and modified Burgers' equation using sextic B‐spline collocation method.Design/methodology/approach – Crank‐Nicolson central differencing scheme has been used for the time integration and sextic B‐spline functions have been used for the space integration to the modified and time splitted modified Burgers' equation.Findings – It has been found that the proposed method is unconditionally stable and obtained results are consistent with some earlier published studies.Originality/value – Sextic B‐spline collocation method for the Burgers' and modified Burgers' equation is given.

Numerical Solution of the Modified Burgers Equation by the Quintic B-spline Galerkin Finite Element Method

Sextic B‐spline collocation method for the modified Burgers' equation

In this study, the Quintic B-spline Differential Quadrature method (QBDQM) is applied to find the numerical solution of the modified Burgers’ equation (MBE). The efficiency and accuracy of the method are measured by calculating the maximum error norm L∞ and the discrete root mean square error L2. The obtained numerical results are compared with published numerical results and the comparison shows that the method is an effective numerical scheme to solve the MBE. A rate of convergence analysis is also given.

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B-spline Differential Quadrature Method for the Modified Burgers' Equation

Purpose




This paper aims to investigate the features of three vectorized iterative numerical schemes used to simulate the behavior of modified Burgers equation (MBE).




Design/methodology/approach




Two of the schemes comprise differential quadrature and finite difference methods, while the third scheme consists of only differential quadrature for the derivative approximations. Proposed schemes are simulated for well-posed problems of MBE having known the analytic solution. The computational complexity of the schemes is examined through monitoring the time taken to complete the simulation. The results are compared with the analytic solution with the help of discrete error norms. Also, the accuracy of the proposed schemes is compared with that of the existing schemes in the literature. Vectorized MATLAB programs of the schemes are used for all investigations.




Findings




It is observed that all the three schemes succeeded in producing a good replication of the exact solution. The results are closer to the analytical solution than the results in the literature. Among the three schemes, the scheme labeled as FDTDQS is found highly accurate and computationally cheaper using fewer grid points. From the vectorized MATLAB programs provided, it is evident that the implementation of the schemes is simple.




Originality/value




This study gives an idea about three numerical schemes for a highly nonlinear problem. This mathematical framework can be adopted to any one-dimensional partial differential equation as well, and the provided program will be helpful to generate more fast and accurate vectorized code in MATLAB.

Numerical Solution of the Modified Burgers Equation by the Quintic B-spline Galerkin Finite Element Method

Citations

Sextic B‐spline collocation method for the modified Burgers' equation

B-spline Differential Quadrature Method for the Modified Burgers' Equation

Iterative differential quadrature algorithms for modified Burgers equation

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