## Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points.(English)Zbl 1072.65114

Summary: A new fifth-order weighted essentially non-oscillatory scheme is developed. Necessary and sufficient conditions on the weights for fifth-order convergence are derived; one more condition than previously published is found. A detailed analysis reveals that the version of this scheme implemented by G.-S. Jiang and C.-W. Shu [ibid. 126, No. 1, 202–228 (1996; Zbl 0877.65065)] is, in general, only third-order accurate at critical points. This result is verified in a simple example. The magnitude of $$\epsilon$$, a parameter which keeps the weights bounded, and the level of grid resolution are shown to determine the order of the scheme in a nontrivial way. A simple modification of the original scheme is found to be sufficient to give optimal order convergence even near critical points. This is demonstrated using the one-dimensional linear advection equation. Also, four examples utilizing the compressible Euler equations are used to demonstrate the scheme’s improved behavior for practical shock capturing problems.

### MSC:

 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35L45 Initial value problems for first-order hyperbolic systems 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35L65 Hyperbolic conservation laws

Zbl 0877.65065
Full Text:

### References:

 [1] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. comput. phys., 126, 202-228, (1996) · Zbl 0877.65065 [2] Harten, A., High resolution schemes for hyperbolic conservation laws, J. comput. phys., 49, 357-393, (1983) · Zbl 0565.65050 [3] Van Leer, B., Towards the ultimate conservative difference scheme II, monotonicity and conservation combined in a second order scheme, J. comput. phys., 14, 361-470, (1974) · Zbl 0276.65055 [4] Van Leer, B., Towards the ultimate conservative difference scheme V, a second order sequel to godunov’s method, J. comput. phys., 32, 101-136, (1979) · Zbl 1364.65223 [5] Harten, A.; Enquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order accurate essentially non-oscillatory schemes III, J. comput. phys., 71, 231-303, (1987) · Zbl 0652.65067 [6] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. comput. phys., 77, 439-471, (1988) · Zbl 0653.65072 [7] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes II, J. comput. phys., 83, 32-78, (1989) · Zbl 0674.65061 [8] Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. comput. phys., 115, 200-212, (1994) · Zbl 0811.65076 [9] Balsara, D.S.; Shu, C.-W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. comput. phys., 160, 405-452, (2000) · Zbl 0961.65078 [10] Fedkiw, R., Simplified discretization of systems of hyperbolic conservation laws containing advection equations, J. comput. phys., 157, 302-326, (2000) · Zbl 0959.76059 [11] S. Osher, R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, vol. 152, Springer, New York, 2003, p. 154 · Zbl 1026.76001 [12] Sod, G.A., Survey of several finite-difference methods for systems of non-linear hyperbolic conservation laws, J. comput. phys., 27, 1-31, (1978) · Zbl 0387.76063 [13] IEEE Standards Board, IEEE Standard for Binary Floating-Point Arithmetic, IEEE Standard 754-1985, 1985 [14] Xu, S.; Aslam, T.; Stewart, D.S., High resolution numerical simulation of ideal and non-ideal compressible reacting flows with embedded internal boundaries, Combust. theory model., 1, 1, 113-142, (1997) · Zbl 1046.80505 [15] Liu, X.-D.; Osher, S., Convex ENO high order multi-dimensional schemes without field-by-field decomposition or staggered grids, J. comput. phys., 142, 304-330, (1998) · Zbl 0941.65082 [16] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. comput. phys., 54, 115-173, (1984) · Zbl 0573.76057 [17] Lax, P.D., Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. pure appl. math., 7, 159-193, (1954) · Zbl 0055.19404 [18] Aslam, T.D., A level set algorithm for tracking discontinuities in hyperbolic conservation laws I: scalar equations, J. comput. phys., 167, 413-438, (2001) · Zbl 0989.65089 [19] Donat, R.; Osher, S., Propagation of error into regions of smoothness for non-linear approximations to hyperbolic equations, Comput. meth. appl. mech. eng., 80, 59-64, (1990) · Zbl 0728.65082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.