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A numerical study of the convergence properties of ENO schemes. (English) Zbl 0732.65086
Essentially nonoscillatory (ENO) finite difference schemes have been developed to treat hyperbolic conservation laws, see A. Harten, S. Osher [SIAM J. Numer. Anal. 24, 279-309 (1987; Zbl 0627.65102)] or A. Harten, B. Engquist, S. Osher and S. Chakravarthy [J. Comput. Phys. 71, 231-303 (1987; Zbl 0652.65067)]. Essentially nonoscillatory schemes, like any nonoscillatory high-order schemes for shock calculations, are nonlinear even for linear problems. Traditional stability analysis is not applicable. Convergence for higher order essentially nonoscillatory schemes is still an open theoretical problem.
For some special initial conditions the surprising observation is made that the calculations do not converge at the expected rate, and that sometimes the error in fact increases as the discretization is refined. It can be concluded that the difficulties with the convergence must be due to the nonlinear nature of the scheme, which in turn results from the use of adaptive stencils. Concepts to overcome these difficulties have been given by C.-W. Shu [ibid. 5, No.2, 127-149 (1990; Zbl 0732.65085)].

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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[1] Harten, A., and Osher, S. (1987). Uniformly high-order accurate nonoscillatory schemes,SIAM J. Numer. Anal. 24, 279. · Zbl 0627.65102
[2] Harten, A., Engquist, B., Osher, S., and Chakravarthy, S. (1987). Uniformly high-order accurate essentially nonoscillatory schemes,J. Comput. Phys. 71, 231. · Zbl 0652.65067
[3] Shu, C.-W. (1990). Numerical experiments on the accuracy of ENO and modified ENO schemes,J. Sci. Comput. 5, 2. · Zbl 0732.65085
[4] Shu, C.-W., and Osher, S. (1988). Efficient implementation of essentially nonoscillatory shock-capturing schemes,J. Comput. Phys. 77, 439. · Zbl 0653.65072
[5] Shu, C.-W., and Osher, S. (1989). Efficient implementations of essentially nonoscillatory shock-capturing schemes,J. Comput. Phys. 83, 32. · Zbl 0674.65061
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