Dispersion-relation-preserving finite difference schemes for computational acoustics
Journal of Computational Physics
Weighted essentially non-oscillatory schemes
Journal of Computational Physics
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
Optimized weighted essentially nonoscillatory schemes for linear waves with discontinuity: 381
Journal of Computational Physics
Conservative hybrid compact-WENO schemes for shock-turbulence interaction
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Analysis of central and upwind compact schemes
Journal of Computational Physics
Journal of Computational Physics
Stability criteria for hybrid difference methods
Journal of Computational Physics
Convergence of Godunov-Type Schemes for Scalar Conservation Laws under Large Time Steps
SIAM Journal on Numerical Analysis
A systematic methodology for constructing high-order energy stable WENO schemes
Journal of Computational Physics
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In the present study, the stability condition for semi-discrete difference schemes of hyperbolic conservation laws obtained from Fourier analysis is simplified. This stability condition can be applied only to linear difference schemes with constant coefficients implemented with periodic boundary treatment. It could often give useful results for other cases, such as schemes with variable coefficients, schemes for nonperiodic problem and nonlinear problem. However, this condition usually leads to a trigonometric inequality, which makes it not convenient to use. For explicit difference schemes on uniform grids, this trigonometric inequality can be converted to polynomial form. Furthermore, if the scheme is a high-order one, the polynomial can be factorized into a simple form. Thus, it is much easier to solve than the inequality obtained directly from Fourier analysis. For compact difference schemes and conservative schemes, similar results are obtained. Some applications of this new stability criterion are shown, including judging the stability of two schemes, proving the upstream central schemes to be stable, constructing a stable upwind dissipation relation preserving (DRP) scheme and constructing an optimized weighted essentially non-oscillatory (WENO) scheme. Since WENO schemes are nonlinear schemes, the stability analysis in the present study is performed on their underlying linear schemes. According to the numerical tests, the underlying linear scheme should be stable, otherwise the corresponding WENO scheme may display instability. These applications demonstrate that this criterion is convenient and efficient for judging the linear stability of semi-discrete difference schemes and constructing stable upwind difference schemes.