High-resolution scheme

File:MUSCL HiRes.jpg

Typical high-resolution scheme based on MUSCL reconstruction.

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.

High-resolution schemes often use flux/slope limiters to limit the gradient around shocks or discontinuities. A particularly successful high-resolution scheme is the MUSCL scheme which uses state extrapolation and limiters to achieve good accuracy – see diagram below.

See also

• Godunov's theorem
• Sergei K. Godunov
• Total variation diminishing
• Shock capturing methods

Further reading

• Harten, A. (1983), High Resolution Schemes for Hyperbolic Conservation Laws. J. Comput. Phys., 49:357–393.
• Hirsch, C. (1990), Numerical Computation of Internal and External Flows, vol 2, Wiley.
• Laney, Culbert B. (1998), Computational Gas Dynamics, Cambridge University Press.
• Toro, E. F. (1999), Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag.