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ODEsPDEs2

# ODEsPDEs2 - Numerical Methods for ODEs PDEs Carl Gardner...

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Numerical Methods for ODEs & PDEs Carl Gardner Arizona State University

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Equivalence Theorem (Lax-Richtmyer) For consistent numerical approximations, stability & convergence ( || u ( t n ) - u n || → 0) are equivalent. Consistency u n + 1 - u n Δ t = du dt + Δ t 2 d 2 u dt 2 + ··· Instability
A-Stability & L-Stability 2 4 6 8 y 0.5 1 1.5 2 C implant TR TRBDF after one timestep

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Derivatives & Notation Second-order accurate central difference approx. to Frst derivative ± df dx ² i f i + 1 - f i - 1 2 Δ x Second-order accurate central difference approx. to second derivative ± d 2 f dx 2 ² i f i + 1 - 2 f i + f i - 1 Δ x 2 ±irst-order accurate one-sided difference approx. to Frst derivative du dt u n + 1 - u n Δ t
Taylor Series To verify formulas, use Taylor series f i ± 1 = f ( x i ± Δ x )= f i ± Δ xf ² i + Δ x 2 2 ! f ² i ± Δ x 3 3 ! f ² i + ··· Then f i + 1 - f i - 1 2 Δ x = f ² i + Δ x 2 6 f ² i +

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ODE (IVP) Methods on One Page! For du / dt = f ( u ) : du / dt ( u n + 1 - u n ) / Δ t = f ( u ) u n + 1 = u n tf ( u n )( Forward Euler ) FE is explicit, stable if Δ t is small, & ±rst-order accurate.
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ODEsPDEs2 - Numerical Methods for ODEs PDEs Carl Gardner...

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