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Lax-Richtmyer equivalence theorem

The Lax-Richtmyer equivalence theorem states that ``a consistent finite-difference scheme for a partial differential equation for which the initial-value problem is well posed is convergent if and only if it is stable.'' For a proof, see [481, Ch. 10].

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Passivity of a Finite-Difference Scheme
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Stability of a Finite-Difference Scheme