High-resolution scheme

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.

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

References

• Godunov, Sergei K. (1954), Ph.D. Dissertation: Different Methods for Shock Waves, Moscow State University.
• Godunov, Sergei K. (1959), A Difference Scheme for Numerical Solution of Discontinuous Solution of Hydrodynamic Equations, Math. Sbornik, 47, 271-306, translated US Joint Publ. Res. Service, JPRS 7226, 1969.
• 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.
• Shu, C-W. (1998), Essentially Non-oscillatory and Weighted Essential Non-oscillatory Schemes for Hyperbolic Conservation Laws. In: Cockburn, B., Johnson, C., Shu, C-W., Tadmor, E. (Eds.), Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, vol 1697. Springer, 325-432.
• Shu, C-W. (2009), High Order Weighted Essentially Non-oscillatory Schemes for Convection Dominated Problems, SIAM Review, 51, No. 1, 82-126.
• Tannehill, John C., et al. (1997), Computational Fluid mechanics and Heat Transfer, 2nd Ed., Taylor and Francis.
• Toro, E. F. (1999), Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag.
• Van Leer, B. (1979), Towards the ultimate conservative difference scheme V. A second order sequel to Godunov's method. Comp. Phys. 32, 101–136.