Quasi-linear analysis of dispersion relation preservation for nonlinear schemes

by   Fengyuan Xu, et al.

In numerical simulations of complex flows with discontinuities, it is necessary to use nonlinear schemes. The spectrum of the scheme used have a significant impact on the resolution and stability of the computation. Based on the approximate dispersion relation method, we combine the corresponding spectral property with the dispersion relation preservation proposed by De and Eswaran (J. Comput. Phys. 218 (2006) 398-416) and propose a quasi-linear dispersion relation preservation (QL-DRP) analysis method, through which the group velocity of the nonlinear scheme can be determined. In particular, we derive the group velocity property when a high-order Runge-Kutta scheme is used and compare the performance of different time schemes with QL-DRP. The rationality of the QL-DRP method is verified by a numerical simulation and the discrete Fourier transform method. To further evaluate the performance of a nonlinear scheme in finding the group velocity, new hyperbolic equations are designed. The validity of QL-DRP and the group velocity preservation of several schemes are investigated using two examples of the equation for one-dimensional wave propagation and the new hyperbolic equations. The results show that the QL-DRP method integrated with high-order time schemes can determine the group velocity for nonlinear schemes and evaluate their performance reasonably and efficiently.



There are no comments yet.


page 9

page 10

page 12

page 14


Numerical scheme based on the spectral method for calculating nonlinear hyperbolic evolution equations

High-precision numerical scheme for nonlinear hyperbolic evolution equat...

A stability property for a mono-dimensional three velocities scheme with relative velocity

In this contribution, we study a stability notion for a fundamental line...

A quasi-conservative DG-ALE method for multi-component flows using the non-oscillatory kinetic flux

A high-order quasi-conservative discontinuous Galerkin (DG) method is pr...

Implicit-explicit-compact methods for advection diffusion reaction equations

We provide a preliminary comparison of the dispersion properties, specif...

Preventing pressure oscillations does not fix local linear stability issues of entropy-based split-form high-order schemes

Recently, it was discovered that the entropy-conserving/dissipative high...

Nonlinear Model Order Reduction using Diffeomorphic Transformations of a Space-Time Domain

In many applications, for instance when describing dynamics of fluids or...

Very-high-order TENO schemes with adaptive accuracy order and adaptive dissipation control

In this paper, a new family of very-high-order TENO schemes with adaptiv...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.