High-resolution schemes are used in the numerical solution of
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
where high accuracy is required in the presence of shocks or discontinuities. They have the following properties:
*Second- or higher-
order
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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 much 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 In numerical methods, total variation diminishing (TVD) is a property of certain discretization schemes used to solve hyperbolic partial differential equations. The most notable application of this method is in computational fluid dynamics. The conc ...
(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 straightforward 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.
The method of
holistic discretisation
Holism () is the idea that various systems (e.g. physical, biological, social) should be viewed as wholes, not merely as a collection of parts. The term "holism" was coined by Jan Smuts in his 1926 book ''Holism and Evolution''."holism, n." OED Onl ...
systematically analyses subgrid scale dynamics to algebraically construct closures for numerical discretisations that are both accurate to any specified order of error in smooth regions, and automatically adapt to cater for rapid grid variations through the algebraic learning of subgrid structures (Roberts 2003).
A web service analyses any PDE in a class that may be submitted
See also
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Godunov's theorem In numerical analysis and computational fluid dynamics, Godunov's theorem — also known as Godunov's order barrier theorem — is a mathematical theorem important in the development of the theory of high resolution schemes for the numerical solutio ...
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Sergei K. Godunov
Sergei Konstantinovich Godunov (russian: Серге́й Константи́нович Годуно́в; born July 17, 1929) is a Soviet and Russian professor at the Sobolev Institute of Mathematics of the Russian Academy of Sciences in Novosibirs ...
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Total variation diminishing In numerical methods, total variation diminishing (TVD) is a property of certain discretization schemes used to solve hyperbolic partial differential equations. The most notable application of this method is in computational fluid dynamics. The conc ...
*
Shock capturing method In computational fluid dynamics, shock-capturing methods are a class of techniques for computing inviscid flows with shock waves. The computation of flow containing shock waves is an extremely difficult task because such flows result in sharp, disco ...
References
*Godunov, Sergei K. (1954), ''Ph.D. Dissertation: Different Methods for Shock Waves'', Moscow State University.
* translated US Joint Publ. Res. Service, JPRS 7226, 1969.
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* Toro, E. F. (1999), Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag.
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{{DEFAULTSORT:High-Resolution Scheme
Numerical differential equations
Computational fluid dynamics