Uniformly high order accurate essentially non-oscillatory schemes, 111
Journal of Computational Physics
Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Total-variation-diminishing time discretizations
SIAM Journal on Scientific and Statistical Computing
Efficient implementation of essentially non-oscillatory shock-capturing schemes,II
Journal of Computational Physics
Weighted essentially non-oscillatory schemes
Journal of Computational Physics
Convergence to steady state solutions of the Euler equations on unstructured grids with limiters
Journal of Computational Physics
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
A high-order WENO finite difference scheme for the equations of ideal magnetohydrodynamics
Journal of Computational Physics
Resolution of high order WENO schemes for complicated flow structures
Journal of Computational Physics
Power ENO methods: a fifth-order accurate weighted power ENO method
Journal of Computational Physics
Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points
Journal of Computational Physics
Development of nonlinear weighted compact schemes with increasingly higher order accuracy
Journal of Computational Physics
Approximate solution of hyperbolic conservation laws by discrete mollification
Applied Numerical Mathematics
A Speed-Up Strategy for Finite Volume WENO Schemes for Hyperbolic Conservation Laws
Journal of Scientific Computing
Improvement of Convergence to Steady State Solutions of Euler Equations with the WENO Schemes
Journal of Scientific Computing
A Robust Reconstruction for Unstructured WENO Schemes
Journal of Scientific Computing
Journal of Computational Physics
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The convergence to steady state solutions of the Euler equations for the fifth-order weighted essentially non-oscillatory (WENO) finite difference scheme with the Lax---Friedrichs flux splitting [7, (1996) J. Comput. Phys. 126, 202---228.] is studied through systematic numerical tests. Numerical evidence indicates that this type of WENO scheme suffers from slight post-shock oscillations. Even though these oscillations are small in magnitude and do not affect the "essentially non-oscillatory" property of WENO schemes, they are indeed responsible for the numerical residue to hang at the truncation error level of the scheme instead of settling down to machine zero. We propose a new smoothness indicator for the WENO schemes in steady state calculations, which performs better near the steady shock region than the original smoothness indicator in [7, (1996) J. Comput. Phys. 126, 202---228.]. With our new smoothness indicator, the slight post-shock oscillations are either removed or significantly reduced and convergence is improved significantly. Numerical experiments show that the residue for the WENO scheme with this new smoothness indicator can converge to machine zero for one and two dimensional (2D) steady problems with strong shock waves when there are no shocks passing through the domain boundaries.