Weighted essentially non-oscillatory schemes. (English) Zbl 0811.65076

Liu, Xu-Dong; Osher, Stanley; Chan, Tony

J. Comput. Phys. 115, No. 1, 200-212 (1994).

The weighted essentially non-oscillatory (ENO) schemes are based on cell averages and a total variation diminishing Runge-Kutta time discretization. The main idea is instead of choosing the “smoothest” stencil to pick one interpolating polynomial for the ENO reconstruction. To this end the authors use a convex combination of all candidates to achieve the essentially non-oscillatory property, while additionally obtaining one order of improvement in accuracy.

Reviewer: K.Zlateva (Russe)

MSC:

65M20	Method of lines for initial value and initial-boundary value problems involving PDEs
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65L06	Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations

Keywords:

essentially non-oscillatory schemes; total variation diminishing Runge- Kutta time discretization

References:

[1]	Abgrall, R.: On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. J. comput. Phys. 114, 45 (1994) · Zbl 0822.65062
[2]	Test Cases for Inviscid Flow Field Methods
[3]	Friedrich, O.: A new method for generating inner points of triangulations in two dimensions. Comput. methods appl. Mech. engrg. 104, 77 (1993) · Zbl 0771.76056
[4]	Harten, A.; Chakravarthy, S. R.: Multi-dimensional ENO schemes for general geometries. (1991)
[5]	Hietel, D.; Meister, A.; Sonar, Th.: On the comparision of four different implementations of an implicit third-order ENO scheme of box type for the computation of unsteady compressible flow. Numer. algorithms 13, 77 (1996) · Zbl 0865.76052
[6]	Jiang, G. -S.; Shu, C. -W.: Efficient implementation of weighted ENO schemes. J. comput. Phys. 126, 202 (1996) · Zbl 0877.65065
[7]	Liu, X. -D.; Osher, S.; Chan, T.: Weighted essentially non-oscillatory schemes. J. comput. Phys. 115, 200 (1994) · Zbl 0811.65076
[8]	Osher, S.; Solomon, F.: Upwind difference schemes for hyperbolic conservation laws. Math. comp. 38, 339 (1982) · Zbl 0483.65055
[9]	Shu, C. -W.; Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. comput. Phys. 77, 439 (1988) · Zbl 0653.65072
[10]	Sonar, Th.: On the construction of essentially non-oscillatory finite volume approximations to hyperbolic conservation laws on general triangulations: polynomial recovery, accuracy and stencil selection. Comput. methods appl. Mech. engrg. 140, 157 (1997) · Zbl 0898.76086
[11]	Woodward, P.; Colella, Ph.: The numerical simulation of two-dimensional fluid flows with strong shocks. J. comput. Phys. 54, 115 (1984) · Zbl 0573.76057