Basing on the flux-vector splitting

a space 3-order accuracy total variation diminishing(TVD) scheme by applying the Taylorseries theory and the control of the limiter was constructed.The scheme was 1-order in time.In the region of the shock wave the scheme changed into 1-order upwind scheme

Then the simple linear differential equation was spread nonlinear equation.It was applied in solving typical numerical experiments.The present results were compared with those results computed by the 2 nd order TVD scheme.It testifys that the scheme has high order accuracy and high resolution for shocks waves

therefore they are non-oscillatory