The steady-state Burgers equation with high Reynolds number is a singularly perturbed boundary value problem. In order to depress the singularity we consider a coordinate transformation from the z-domain to the t-domain. Then we construct a very effective Lie-group shooting method to search a missing initial condition of slope through a weighting factor r ∈ (0,1). Furthermore, a closed-form formula is derived to calculate the unknown slope in terms of r in a more refined range identified. Numerical examples were examined to show that the new approach has high efficiency and high accuracy.
lie-group shooting method burgers equation singularly perturbed boundary value problem one-step group preserving scheme
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KADALBAJOO, M. K.; PATIDAR, K. C. A survey of numerical techniques for solving singularly perturbed ordinary differential equations [J]. Applied Mathematics and Computation, 2002, 130: 457–510.MathSciNetCrossRefGoogle Scholar
LIU, C.-S. Cone of non-linear dynamical system and group preserving schemes [J]. International Journal of Non-Linear Mechanics, 2001, 36: 1047–1068.MathSciNetCrossRefGoogle Scholar
LIU, C.-S. Nonstandard group-preserving schemes for very stiff ordinary differential equations [J]. CMES: Computer Modeling in Engineering & Sciences, 2005, 9: 255–272.MathSciNetzbMATHGoogle Scholar
O’MALLEY, R. E. Singular Perturbation Methods for Ordinary Differential Equations [M]. New York: Springer-Verlag, 1991.CrossRefGoogle Scholar