equiv - 2 ( G-H ) H + . . . + ( G-H ) H n-1 Every term has...

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Equivalence Theorem (Lax-Richtmyer) The Fundamental Theorem of Numerical Analysis .F o r 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 .W ew i l la s s um et h ep r o b l em sa r e close, and prove that the solutions are close. Consistency implies f n f and L n u Lu as n →∞ t 0, with n Δ t = t ±xed. 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 + ... +
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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 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/ t terms in Eq. ( ). Therefore as n , | G n-H n | t t e Kt C t p +1 = O ( 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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This note was uploaded on 03/11/2012 for the course MAT 421 taught by Professor Staff during the Fall '11 term at ASU.

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equiv - 2 ( G-H ) H + . . . + ( G-H ) H n-1 Every term has...

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