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In this paper,I first give a generalized Lax-Friedrichs scheme（GLxF scheme） for a convective-diffusion equation ut+aux=εuxx,α∈R,ε＞0,and discuss the consistency,stability and convergence.For the discretization of some special initial data under stable conditions,local oscillations in numerical solutions are observed.Then I show three strategies about how to control those oscillations via some numerical examples.Finally,I focus on investigating oscillations by the discrete Fourier analysis and the modified equation analysis,respectively,in order to clarify the reason of oscillations and the interrelation among oscillations, numerical dissipations,numerical damping and relative phase errors.
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