WENOCLAW: A higher order wave propagation method. Many important physical phenomena are governed by hyperbolic systems of conservation laws 𝐪 t +f(𝐪) x =0,(1) for which a wide range of numerical methods have been developed. In this paper we present a numerical method for solution of (1) that is also applicable to general hyperbolic systems of the form 𝐪 t +A(𝐪,x,t)𝐪 x =0·(2) In the nonlinear nonconservative case, the method may be applied if the structure of the Riemann solution is understood. Examples of (1–2) include acoustics and elasticity in heterogeneous media. The method described in this work combines the notions of wave propagation and the method of lines, and can in principle be extended to arbitrarily high order accuracy by the use of high order accurate spatial reconstruction and a high order accurate ordinary differential equation solver. In this work, we use Runge-Kutta methods.