5. Consider the general formula for all explicit two-step methods, yn+1 =a0yn +a1yn−1 +h[b0f(tn,yn)+ b1f(tn−1,yn−1)], n≥1. (a) Consider finding all such two-step methods that are of order 2. Show that the coefficients must satisfy the equations a0 +a1 =1, −a1 +b0 +b1 =1, a1 −2b1 =1. Solve for {a1, b0, b1} in terms of a0. (b) Find a formula for the leading term in the truncation error, written as in (7.15). It will depend on a0. (c) What are the restrictions on a0 for this two-step method to be stable? To be convergent?
5. Consider the general formula for all explicit two-step methods, yn+1 =a0yn +a1yn−1 +h[b0f(tn,yn)+ b1f(tn−1,yn−1)], n≥1. (a) Consider finding all such two-step methods that are of order 2. Show that the coefficients must satisfy the equations a0 +a1 =1, −a1 +b0 +b1 =1, a1 −2b1 =1. Solve for {a1, b0, b1} in terms of a0. (b) Find a formula for the leading term in the truncation error, written as in (7.15). It will depend on a0. (c) What are the restrictions on a0 for this two-step method to be stable? To be convergent?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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5. Consider the general formula for all explicit two-step methods,
yn+1 =a0yn +a1yn−1 +h[b0f(tn,yn)+ b1f(tn−1,yn−1)], n≥1.
(a) Consider finding all such two-step methods that are of order 2. Show that
the coefficients must satisfy the equations
a0 +a1 =1, −a1 +b0 +b1 =1, a1 −2b1 =1. Solve for {a1, b0, b1} in terms of a0.
(b) Find a formula for the leading term in the truncation error, written as in (7.15). It will depend on a0.
(c) What are the restrictions on a0 for this two-step method to be stable? To be convergent?
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