In this paper, we present an alternating evolution (AE) approximation [Formula: see text] to systems of hyperbolic conservation laws [Formula: see text] in arbitrary spatial dimension. We prove the convergence of the approximate solutions towards an entropy solution of scalar multi-D conservation laws, and the L1 contraction property for the approximate solution is established as well. It is also shown that such an approximation is extremely accurate in the sense that if initial data is prepared such that u0 = v0 = U0, then no method error is induced as time evolves, and the exact entropy solution is precisely captured. Furthermore, in the approximation system time evolution of one variable is associated with spatial redistribution in another variable. These features render such an approximation ideal to be used for construction of high resolution numerical schemes to solve hyperbolic conservation laws. The usual obstacles caused by jumps crossing computational cell interfaces are not felt when both u and v are sampled alternatively, and reconstructed independently. Herewith we discuss the designing principle for constructing AE schemes, with illustration of two preliminary schemes for systems of conservation laws in one dimension. Both l∞ monotonicity and the TVD (Total Variational Diminishing) property are established for these schemes when applied to the scalar laws.
Well-posedness of the global entropy solution to the Cauchy problem of a hyperbolic conservation laws with relaxation