Although the 6-point scheme proposed by Komatsu et al. is the 1st order scheme, computation results are obtained with high accuracy. The 6-point scheme is able to give numerical solutions with few clipping errors. However, there is a room for improvement for the reproduction of the semi-ellipse profile. We propose the 3rd order and 4th order 6-point scheme using the idea for the derivation of the 6-point scheme. Theses schemes are called the characteristics type high-order 6-point scheme. The high-order 6-point scheme obtained in this study is modified into the conservative form in order to incorporate the universal limiter. Theses schemes are called the conservative high-order 6-point scheme. The discriminator, which is an algorithm to distinguish between numerical oscillations and true physical maxima, is also introduced. The discriminator proposed by Asai et al. is modified in this study. A high-accuracy and high-resolution numerical method is proposed by introducing the universal limiter and the modified discriminator.