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High-resolution schemes are used in the numerical solution of partial differential equations where high accuracy is required in the presence of shocks or discontinuities. They have the following properties: *Second or higher order spatial accuracy is obtained in smooth parts of the solution. *Solutions are free from spurious oscillations or wiggles. *High accuracy is obtained around shocks and discontinuities. *The number of mesh points containing the wave is small compared with a first-order scheme with similar accuracy. General methods are often not adequate for accurate resolution of steep gradient phenomena; they usually introduce non-physical effects such as ''smearing'' of the solution or ''spurious oscillations''. Since publication of ''Godunov's order barrier theorem'', which proved that linear methods cannot provide non-oscillatory solutions higher than first order (Godunov-1954, Godunov-1959), these difficulties have attracted a lot of attention and a number of techniques have been developed that largely overcome these problems. To avoid spurious or non-physical oscillations where shocks are present, schemes that exhibit a Total Variation Diminishing (TVD) characteristic are especially attractive. Two techniques that are proving to be particularly effective are MUSCL (''Monotone Upstream-Centered Schemes for Conservation Laws'') a flux/slope limiter method (van Leer-1979, Hirsch-1990, Tannehill-1997, Laney-1998, Toro-1999) and the WENO (''Weighted Essentially Non-Oscillatory'') method (Shu-1998, Shu-2009). Both methods are usually referred to as ''high resolution schemes'' (see diagram). MUSCL methods are generally second-order accurate in smooth regions (although they can be formulated for higher orders) and provide good resolution, monotonic solutions around discontinuities. They are straight-forward to implement and are computationally efficient. For problems comprising both shocks and complex smooth solution structure, WENO schemes can provide higher accuracy than second-order schemes along with good resolution around discontinuities. Most applications tend to use a fifth order accurate WENO scheme, whilst higher order schemes can be used where the problem demands improved accuracy in smooth regions. ==See also== *Godunov's theorem *Sergei K. Godunov *Total variation diminishing *Shock capturing methods 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「High-resolution scheme」の詳細全文を読む スポンサード リンク
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