New finite-difference formulations are given for the free surface of an elastic solid. It is argued that the new methods are better than earlier formulations in terms of both accuracy and stability. This conclusion is based on some new algebraic results and on a thorough analysis of the stability problems that can occur. It is backed up by extensive numerical testing. The explicit grid mode that makes the earlier “composed approximation” unstable is constructed. The construction generalizes to give general tests for instabilities. A criterion for using formulations that have relatively minor stability problems (the “useful life”) is stated and applied.

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