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.
Harten, A. (1983), High Resolution Schemes for Hyperbolic Conservation Laws. J. Comput. Phys., 49:357-293.
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.