High-resolution scheme

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: 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.

To avoid spurious or non-physical oscillations where shocks are present, schemes that exhibit a Total Variation Diminishing (TVD) characteristic are especially attractive.

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.

The method of holistic discretisation 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).