In this paper, we consider linear stability issues for one-dimensional hyperbolic conservation laws using a class of conservative high order upwind-biased finite difference schemes, which is a prototype for the weighted essentially non-oscillatory (WENO) schemes, for initial-boundary value problems (IBVP). The inflow boundary is treated by the so-called inverse Lax-Wendroff (ILW) or simplified inverse Lax-Wendroff (SILW) procedure, and the outflow boundary is treated by the classical high order extrapolation. A third order total variation diminishing (TVD) Runge-Kutta time discretization is used in the fully discrete case. Both GKS (Gustafsson, Kreiss and Sundström) and eigenvalue analysis are performed for both semi-discrete and fully discrete schemes. The two different analysis techniques yield consistent results. Numerical tests are performed to demonstrate the stability results predicted by the analysis.

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Summary In this paper, a lattice Boltzmann model for simulating linear elastic Lame equation is proposed. Differently from the classic lattice Boltzmann models, this lattice Boltzmann model is based on displacement distribution function in lattice Boltzmann equation. By using the technique of the higher‐order moments of equilibrium distribution functions and a series of partial differential equations in different time scales, we obtain the Lame equation with fourth‐order truncation errors. Based on this model, some problems with small deflection are simulated. The comparisons between the numerical results and the analytical solutions are given in detail. The numerical examples show that the lattice Boltzmann model can be used to solve problems of the linear elastic displacement field with small deflection. Copyright © 2015 John Wiley & Sons, Ltd.

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