Higher order time integration formula with application on Burgers’ equation

Abstract

We derive fifth-order convergent method in time for the initial value problem When applied to test equation y′=−λ y, λ>0, it gives y_n+1=Ψ(z)y_n, where Ψ(z) does not satisfy the condition for A-stability but Ψ(z)→0 as z→∞. To develop this method we use a higher order average approximation which is based on osculatory cubic polynomial interpolation coupled with fourth-order backward Taylor's series approximation. We also test this method on Burgers’ equation. Computed solutions are quite encouraging and does not give oscillations when inconsistencies are present in terms of initial and boundary conditions.

Keywords:

• Hopf–Cole transformation
• Burgers’ equation
• Boole's rule
• stability
• Mathematica 7.0
• Origin 7.5

2010 AMS Subject Classification::

• 35K60

Acknowledgements

We are thankful to the reviewers and editors for their expert comments. This helped us a lot and also helped to improve the quality of the paper.