In this paper we consider scalar hyperbolic equations in one space dimension of the type u t(x,t)+ d dx f(u;x)=h(u;x) , u(x,0)=u 0(x), x∈ Rt>0 , where ƒ ϵ C 1 and h continuous w.r.t. u, x. The initial condition is assumed to be piecewise continuous. We present a new method for constructing the entropy solution of (1) at a fixed time t = T > 0 in one time step based on transporting the initial values along characteristics. If the solution of (1) is smooth, we get the exact solution; in case of shocks the multivalued graph of the initial data is corrected by a geometrical averaging technique via the conservation principle. The method is also applicable to a scalar equation in which there is a mild coupling between the physical dimensions in the problem, for example, u t(x,y)+ d dx f(u;x,y)+ d dy f(u;x,y)=h(u;x,y) . By a change of variables, (2) can be reduced to a quasi one-dimensional problem. We conjecture that the advantage of computing the entropy solution at a fixed time in one time step cannot easily be carried over to systems. But we have some hints that this might be possible in case of scalar equations in two space dimensions with arbitrary fluxes ƒ 1, ƒ 2 . The CPU time depends only on the total number of shocks which occur in the entropy solution up to time T; the accuracy of the computed shock position is of order at least 10 −2. Since our method is not based on a time discretisation, questions (and problems) concerning stability and convergence do not arise.