Question

Question is as follows:

NOTE:

3. Consider the 2-step Implicit Nyström Method (aka Simpson's Method): Un+2 – Un = (F(UM) + 4f(Un+1) + f(Un+2)]. (a) Determine the two characteristic polynomials, p() and o(5), for this method. Use these to show that this method is consistent. (b) Prove that this method is zero-stable.

• Subject: Numerical Methods for Ordinary Differential Equations: Initial Value Problems.

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Useful Information

Implicit Nystrom Method: (aka Milne-Simpson)

Explicit Nystrom Method: (for reference)

Adams-Bashforth Method (AB): (for reference)

Zero-Stability:

Root Condition: (for reference)

Answer 1

solution:

Page 1 Solutions given 2. step Implicit Nystrom method part @ - v n comparing from k seun) 4+ f (uht") + f(untry] unt2 + a, unt! + a un = k ( be f (untry + by flunu) #bo few - a -o ao=-1, b2 = 1, bi = 4, boat Now characterstic polynomials ple), och are defined as ele- & ou + a 4 a 64-bout but by ² substituting values of as, a, bob, br. PREI= 4-1) 16(e) = 1 + 1 2 + 1 2 3 a linear multistep method is consistent ich and only if. & ai = -1 and & bi = 2 + & iai 2-1 2-1 - it=0 -1 +0=-1 (satisfies) , ses), + 2 2 2 + 1 2 = 2 + Oxact 1 a I 4 I to - 2 = 2 toto satisfies) - Satisfies root condition given method Q is consistent. part 6: A linear multistep method is zero-stable if it e at all zeros of P(G) satisfies lelse and is lel=I, P(G) & 0 (ho repeated roots on the unit ceros of ela 4-1 = 0 = 11 satisfies tels also (CE) does not have repeated roots on unit circle ie e(t)=2570 This Ple) satisfies root condition hence Method Coin circle) zero stable