Higher order time integration formula with application on Burgers’ equation

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 yn+1=Ψ(z)yn, 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.

[1]  M. M. Chawla,et al.  New L-stable modified trapezoidal formulas for the numerical integration of Y1=f(x, y) , 1997, Int. J. Comput. Math..

[2]  E. Süli,et al.  Numerical Solution of Partial Differential Equations , 2014 .

[3]  M. K. Kadalbajoo,et al.  A numerical method based on Crank-Nicolson scheme for Burgers' equation , 2006, Appl. Math. Comput..

[4]  A. R. Gourlay,et al.  The Extrapolation of First Order Methods for Parabolic Partial Differential Equations, II , 1978 .

[5]  E. Hopf.  The partial differential equation ut + uux = μxx , 1950 .

[6]  G. Dahlquist.  A special stability problem for linear multistep methods , 1963 .

[7]  Mustafa Inç.  New L-stable method for numerical solutions of ordinary differential equations , 2007, Appl. Math. Comput..

[8]  Amit K. Verma,et al.  L-stable Simpson's 3/8 rule and Burgers' equation , 2011, Appl. Math. Comput..

[9]  W. Ramirez.  Chapter 8 – SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS , 1997 .

[10]  J. Cole.  On a quasi-linear parabolic equation occurring in aerodynamics , 1951 .

[11]  M. M. Chawla,et al.  Stabilized fourth order extended methods for the numerical solution of odes , 1994 .

[12]  J. Crank,et al.  A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type , 1947, Mathematical Proceedings of the Cambridge Philosophical Society.

[13]  Tien D. Bui.  Some A-Stable and L-Stable Methods for the Numerical Integration of Stiff Ordinary Differential Equations , 1979, JACM.

[14]  R. D. Richtmyer,et al.  Difference methods for initial-value problems , 1959 .

[15]  J. Lambert.  Numerical Methods for Ordinary Differential Systems: The Initial Value Problem , 1991 .

[16]  David J. Evans,et al.  A new L-stable Simpson-type rule for the diffusion equation , 2005, Int. J. Comput. Math..