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# equiv - 2 G-H H G-H H n-1 Every term has a factor G-H and |...

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Equivalence Theorem (Lax-Richtmyer) The Fundamental Theorem of Numerical Analysis . For consistent numerical approximations, stability and convergence are equivalent. Lax proved for IVPs. Applies as well to BVPs, approximations to func- tions and integrals, and PDEs. Approximate Lu = f by L n u n = f n . We will assume the problems are close, and prove that the solutions are close. Consistency implies f n f and L n u Lu as n → ∞ , Δ t 0, with n Δ t = t fixed. Stability implies L - 1 n remains uniformly bounded. Then as n → ∞ , u n converges to u : u - u n L - 1 n ( L n u - Lu ) + L - 1 n ( f - f n ) 0 Let’s look at the details for the IVP du/dt = au , u ( t 0 ) = u 0 (proof due to Strang). The exact solution is u ( t ) = e at u 0 H n u 0 , where H = exp { a Δ t } is the exact growth factor. The approximate solution is u n = G n u 0 . Stability implies | G n | e Kn Δ t = e Kt where K is a positive constant independent of n . Consistency implies | G - H | C Δ t p +1 , p > 0 Then we’ll prove convergence : | G n u 0 - H n u 0 | 0 as n → ∞ . We use a telescoping identity ( ) G n - H n = G n - G n - 1 H + G n - 1 H - G n - 2 H 2 + . . . + GH n - 1 - H

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Unformatted text preview: 2 ( G-H ) H + . . . + ( G-H ) H n-1 Every term has a factor G-H , and | G-H | ≤ C Δ t p +1 by consistency. Every term has a power of G (possibly G ) which is bounded by stability. Every term has a power of H which is bounded since the continuous problem is well-posed. There are n = t/ Δ t terms in Eq. ( ± ). Therefore as n → ∞ , | G n-H n | ≤ t Δ t e Kt C Δ t p +1 = O (Δ t p ) → The telescoping series ( ± ) is exactly how error accumulates in a di f erence equation. ( ± ) G n-H n = n ± j =1 G n-j ( G-H ) H j-1 H j-1 propagates the exact solution to timelevel j-1; ( G-H ) is the local truncation error going From timelevel j-1 to j ; and G n-j propagates this error Forward with the diference method to timelevel n . 2...
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