LTE - || G || = 0 . TR is A-stable, but not L-stable....

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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. Anumer ica lmethodis consistent if the (global) error is proportional to Δ t p , p> 0. For initial value problems, u n = G n u 0 ,where G is the growth factor of the numerical method. The method is stable if || G || ≤ 1, or equivalently, if numerical errors introduced by roundo± in ²oating point arithmetic are not exponentially ampli³ed. Anumer ica lmethodis convergent if || u ( t n ) - u n || → 0 in the limit Δ t 0and n →∞ ,w ith t n = n Δ t held ³xed. A-stability and L-stability At imeintegrat ionmethodfor du/dt = au (Re { a } < 0) is A-stable if || G || ≤ 1 for all Δ t> 0. A time integration method for du/dt = au (Re { a
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Unformatted text preview: || G || = 0 . TR is A-stable, but not L-stable. Backward Euler (rst- and second-order) and TRBDF2 are L-stable. Local Truncation Error ( ) du dt = f ( u ) , u ( t = 0) = u The discrete approximation is u n u ( n t ). The one-step discretized version of the initial value problem ( ) can be written as u n +1 = u n + t ( u n , u n +1 , t ) . We dene the local truncation error (LTE = t ) for the initial value prob-lem ( ) by u (( n + 1) t ) = u ( n t ) + t ( u ( n t ) , u (( n + 1) t ) , t ) + t where u is the exact solution of ( ). A one-step method is p th order accurate if the (global) error t p ....
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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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