References
- T.D. Bui, Some A-stable and L-stable methods for the numerical integration of stiff ordinary differential equations, J. Assoc. Comput. Mach. 26(3) (1979), pp. 483–493. doi: 10.1145/322139.322147
- M.M. Chawla and M.A. Al-Zanaidi, New L-stable modified trapezoidal formulas for the numerical integration of y′=f (x, y), Int. J. Comput. Math. 63 (1997), pp. 279–288. doi: 10.1080/00207169708804567
- M.M. Chawla and D.J. Evans, A new L-stable Simpson type rule for the diffusion equation, Int. J. Comput. Math. 82 (2005), pp. 601–607. doi: 10.1080/00207160512331331138
- M.M. Chawla, M.A. Al-Zanaidi, and M.S. Al-Sahhar, Stabilized fourth order extended methods for the numerical solution of ODEs, Int. J. Comput. Math. 52 (1994), pp. 99–107. doi: 10.1080/00207169408804294
- J.D. Cole, On a quasilinear parabolic equation occurring in aerodynamics, Q. Appl. Math. 9 (1951), pp. 225–236.
- J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Proc. Cambridge Philos. Soc. 43 (1947), pp. 50–67. doi: 10.1017/S0305004100023197
- G. Dahlquist, A special stability problem for linear multistep methods, BIT 3 (1963), pp. 27–43. doi: 10.1007/BF01963532
- E. Hopf, The partial differential equation , Commun. Pure Appl. Math. 3 (1950), pp. 201–230. doi: 10.1002/cpa.3160030302
- M. Inc, New L-stable method for numerical solutions of ordinary differential equations, Appl. Math. Comput. 188 (2007), pp. 779–785. doi: 10.1016/j.amc.2006.10.055
- M.K. Kadalbajoo and A. Awasthi, A numerical method based on Crank–Nicolson scheme for Burgers’ equation, Appl. Math. Comput. 182 (2006), pp. 1430–1442. doi: 10.1016/j.amc.2006.05.030
- J.D. Lambert, Numerical Methods for Ordinary Differential Systems, John Wiley, New York, 1991.
- J.D. Lawson and J.L. Morris, The extrapolation of first order methods for parabolic partial differential equations I, SIAM J. Numer. Anal. 15 (1978), pp. 1212–1224. doi: 10.1137/0715082
- R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, Philadelphia, 2007.
- K. Pandey, L. Verma, and A.K. Verma, L-stable Simpson's 3/8 rule and Burgers’ equation, Appl. Math. Comput. 218 (2011), pp. 1342–1352. doi: 10.1016/j.amc.2011.06.017
- R.D. Richtmyer and K.W. Morton, Difference Methods for Initial-Value Problems, Wiley-Interscience, New York, 1967.
- G.D. Smith, Numerical Solution of Partial Differential Equations, Oxford University Press, Oxford, 1978.
- L. Verma, L-stable derivative-free error-corrected trapezoidal rule for Burgers’ equation with inconsistent initial and boundary conditions, Int. J. Math. Math. Sci. 2012 (2012), Article ID 821907, 13 pages.