Summary: New five-point total variation diminishing (TVD) schemes with third- and fourth-order accuracy in space and second-order accuracy in time are constructed by applying the Taylor series theory and the TVD sufficient conditions. Comparative results for solving a linear hyperbolic equation are presented using the present schemes, the second-order TVD schemes and the traditional TVD schemes. It demonstrates that the presented high-order TVD schemes, especially the fourth-order accuracy scheme, have good numerical features that the peak values attenuate more slowly and discontinuities are steeper than the new second-order TVD schemes and the traditional TVD schemes. However, the computational time with the present schemes is almost the same as that with the traditional TVD schemes.