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

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Equivalence Theorem (Lax-Richtmyer) The Fundamental Theorem of Numerical Analysis . For consistent numerical approximations, stability and convergence are equivalent. A numerical method is 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 ff in floating point arithmetic are not exponentially amplified. A numerical method is convergent if || u ( t n ) - u n || 0 in the limit Δ t 0 and n → ∞ , with t n = n Δ t held fixed. A-stability and L-stability A time integration method for 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 } < 0) is L-stable if it is A-stable and lim Δ t →∞ || G
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Unformatted text preview: || G || = 0 . TR is A-stable, but not L-stable. Backward Euler (³rst- 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 Δ t ). The one-step discretized version of the initial value problem ( ± ) can be written as u n +1 = u n + Δ t Φ( u n , u n +1 , Δ t ) . We de³ne the local truncation error (LTE = Δ t τ ) for the initial value prob-lem ( ± ) by u (( n + 1) Δ t ) = u ( n Δ t ) + Δ t Φ( u ( n Δ t ) , u (( n + 1) Δ t ) , Δ t ) + Δ t τ where u is the exact solution of ( ± ). A one-step method is p th order accurate if the (global) error τ ∼ Δ t p ....
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