The development of numerical methods for hyperbolic conservation laws has been a rapidly growing area for the last ten years. Many of the fundamental concepts and state-of-the-art developments can only be found in meeting proceedings or internal reports. This review paper attempts to give an overview and a unified formulation of a class of shock-capturing methods. Special emphasis will be on the construction of the basic nonlinear scalar second-order schemes and the methods of extending these nonlinear scalar schemes to nonlinear systems via the exact Riemann solver, approximate Riemann solvers, and flux-vector splitting approaches. Generalization of these methods to efficiently include real gases and large systems of nonequilibrium flows will be discussed. The performance of some of these schemes is illustrated by numerical examples for one-, two- and three-dimensional gas-dynamics problems.
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