The novel approximation system introduced by Liu is an accurate approximation to systems of hyperbolic conservation laws. We develop a class of global and local alternating evolution (AE) schemes for one- and two-dimensional hyperbolic conservation law and one-dimensional Hamilton-Jacobi equations, where we take advantage of the high accuracy of the AE approximation. The nature of solutions having singularities, which is generic to these equations in handled using the AE methodology. The numerical scheme is constructed from the AE system by sampling over alternating computational grid points. Higher order accuracy is achieved by a combination of high-order polynomial reconstruction and a stable Runge-Kutta discretization in time. Local AE schemes are made possible by letting the scale parameter [epsilon] reflect the local distribution of nonlinear waves. The AE schemes have the advantage of easier formulation and implementation, and efficient computation of the solution. Theoretical numerical stability is proved mainly for the first and second order schemes of hyperbolic conservation law and Hamilton-Jacobi equations. In the case of hyperbolic conservation law, we have also shown that the numerical solutions converge to the weak solution. The designed methods have the advantage of being Riemann solver free, and the performs comparably to the finite volume/difference methods currently used. A series of numerical tests illustrates the capacity and accuracy of our method in describing the solutions.