Summary: The solution of the Riemann problem was a building block for general Cauchy problems in conservation laws. A Cauchy problem is approximated by a series of Riemann problems in many numerical schemes. But, since the structure of the Riemann solution holds locally in time only, and, furthermore, a Riemann solution is not piecewise constant in general, there are several fundamental issues in this approach such as the stability and the complexity of computation.
We introduce a new approach which is based on piecewise self-similar solutions. The scheme proposed in this article solves the problem without the time marching process. The approximation error enters in the step for the initial discretization only, which is given as a similarity summation of base functions. The complexity of the scheme is linear. Convergence to the entropy solution and the error estimate are shown. The mechanism of the scheme is introduced in detail together with several interesting properties of the scheme.