Study of quasi-linear spectral analysis method of high-order weighted nonlinear schemes

The quasi-linear spectral analysis method based on an approximate dispersion relation (ADR) can give spectral properties of nonlinear space-discrete schemes more accurately. It has become to be a very important tool for the assessment of nonlinear schemes up to date. However, some factors, such as time-discrete schemes and number of computation grid points, may affect the prediction accuracy seriously and make the ADR formula hard to be employed. In some cases, the spectral curve may even jump at some points under some conditions. In order to develop the formula independent of the influences of these factors, time-discrete independent ADR formula is proposed firstly through theoretical analysis. The meanings of each term in the ADR formula are explained, and the influence of time step on the time dependent ADR is investigated. Secondly, the reason resulting in the curve jumps at some wave numbers is analyzed using this new time-independent ADR formula. It is shown that the jumps may happen in case the number of grid points is improperly chosen. The proper method to selected grid points number is introduced, the proposed formula may eliminate the influence of number of grid points and initial phase angle effectively when the number of grid points is larger than this supposed number. Based on these works, the spectral properties of two 3rd-order WCNS schemes are investigated, and some typical cases are simulated using these two schemes. The results show that the quasi-linear spectral curve calculated by the ADR formula evaluates the properties of nonlinear schemes qualitatively. However, in respect of quantity, the errors cannot be ignored.