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In this paper we develop a novel two-stage fourth order time-accurate discretization for time-dependent flow problems, particularly for hyperbolic conservation laws. Different from the classical Runge--Kutta (R--K) temporal discretization for first order Riemann solvers as building blocks, the current approach is solely associated with Lax--Wendroff (L--W) type schemes as the building blocks. As a result, a two-stage procedure can be constructed to achieve a fourth order temporal accuracy, rather than using the well-developed four-stage procedure for R--K methods. The generalized Riemann problem (GRP) solver is taken as a representative of L--W type schemes for the construction of a two-stage fourth order scheme.

A Two-Stage Fourth Order Time-Accurate Discretization for Lax--Wendroff Type Flow Solvers I. Hyperbolic Conservation Laws