We consider the long-time instability issue associated with finite difference simulation of seismic acoustic wave equations on discontinuous grids. This issue is exhibited by a prototype algebraic problem abstracted from practical application settings. Analysis of this algebraic problem leads to better understanding of the cause of the instability and provides guidance for its treatment. Specifically, we use the concept of discrete energy to derive the proper solution transfer operators and design an effective way to damp the unstable solution modes. Our investigation shows that the interpolation operators need to be matched with their companion restriction operators in order to properly couple the coarse and fine grids. Moreover, to provide effective damping, specially designed diffusive terms are introduced to the equations at designated locations and discretized with specially designed schemes. These techniques are applied to simulations in practical settings and are shown to lead to superior results in terms of both stability and accuracy.
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