Propagation of nonlinear acoustic waves radiated from a spherically concave source into an ideal gas is numerically studied in the case where the dissipation effect is negligible everywhere except for the shock front. In order to obtain the precise wave profile including shock waves, a high-resolution upwind finite difference scheme is employed to solve the Euler equations of gasdynamics. The linear wave equation and the Zabolotskaya--Khokhlov equation are also solved by the high-resolution upwind scheme, and the results are compared with those obtained from the Euler equations. It is shown that moderately strong shocks cause strong streaming in the focal region. The maximum streaming velocity amounts to the particle velocity of the gas at the sound source.