This paper is concerned with the numerical solution of hyperbolic equations. Initially, it concentrates on the scalar conservation law \(u_ t+f(u)_ x=0\), with special reference given to the behavior of the solution near sonic points (\(u\approx \bar u\), \(f'(\bar u)=0\)). It is well known that details of the numerical flux calculation under these conditions are very critical for the convergence of an algorithm. `` Sonic flux formulae'' are recently developed to guarantee such convergence [e.g. \textit{S. Osher}, SIAM J. Numer. Anal. 21, 217-235 (1984; Zbl 0592.65069)]. Even when convergence does take place, it may be slow. Also, even though the local errors may be small, they may tend to accumulate some kind of local pathology. In this paper a more accurate treatment of sonic points is presented that involves modifying the sonic flux and both its neighbors, and some numerical experiments are shown. A generalization to systems of equations is discussed in the paper. As a test the one-dimensional Euler equations are investigated, and the numerical results are reported.