Central Schemes and Systems of Balance Laws

The development of shock capturing schemes for the numerical approximation of the solution of conservation laws has been a very active field of research in the last two decades. There are several motivations for this effort. First, the challenge is a mathematical one. Solutions of conservation laws may develop jump discontinuities in finite time. To understand how to obtain numerical approximations that converge to the (discontinuous) solution has been a non trivial task. The mathematical theory of quasilinear hyperbolic systems of conservation laws has been used as a guideline in the development of modern shock capturing schemes. The concept of entropy condition and total variation diminishing are common to mathematical analysts and numerical analysts who deal with the problem. Another motivation for the development of shock capturing schemes is provided by the large number of applications. Complex flows in gas dynamics require the use of efficient and accurate schemes, which are able to deal with complex geometries. Unstructured mesh or adaptive mesh refinement become necessary so solve realistic problems. The schemes that are used for practical problems are usually different from the schemes for which theoretical results can be proven. For example, very little is known about the convergence property of high order schemes.

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