Efficient implementation of weighted ENO schemes. (English) Zbl 0877.65065

The weighted essentially non-oscillatory (WENO) finite difference schemes of X.-D. Liu, S. Osher and T. Chan [J. Comput. Phys. 115, No. 1, 200-212 (1994; Zbl 0811.65076)] for the approximation of hyperbolic conservation laws (HCLs) of the type \[ u_t+\text{div } f(u)=0 \] are analyzed, tested and improved. A new way of measuring the smoothness of a numerical solution that is based upon minimizing the \(L^2\) norm of the derivatives of the reconstruction polynomials, emulating the idea of minimizing the total variation of the approximation is proposed. It is proved that for HCLs with smooth solutions, all WENO schemes are convergent.
Many numerical tests are presented to demonstrate the remarkable capability of the WENO schemes, especially the WENO scheme using the new smoothness measurement in resolving complicated shock and flow structures. Also H. Yang’s artificial compression method [J. Comput. Phys. 89, No. 1, 125-160 (1990; Zbl 0705.65062)] to the WENO schemes to sharpen contact discontinuities is applied.
Reviewer: K.Zlateva (Russe)


65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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