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 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.

See also

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.
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