In this paper, a high-order finite-volume scheme is presented for the onedimensional scalar and inviscid Euler conservation laws. The Simpson,s quadrature rule is used to achieve high-order accuracy in time. To get the point value of the Simpson,s quadrature, the characteristic theory is used to obtain the positions of the grid points at each sub-time stage along the characteristic curves, and the third-order and fifthorder central weighted essentially non-oscillatory (CWENO) reconstruction is adopted to estimate the cell point values. Several standard one-dimensional examples are used to verify the high-order accuracy, convergence and capability of capturing shock.