Before revision:  Improved numerical results are obtained for one-dimensional linear advection cases, especially over long output times . Numerical solutions of the 1D Euler equations with the Sod's shock tube problem, the Shu-Osher shock-entropy wave interaction problem and the Woodward-Colella blastwave problem are compared with results in references. Quite satisfied performances using the new method are demonstrated in the simulation of two severe problems as the Titarev-Toro shock-entropy wave interaction problem and the 123 problem .
After revision:  Improved numerical results are obtained for one-dimensional linear advection cases, especially over long output times .