Part CCSIR NET December 2023rk2-order-conditions

Rk2 order conditions

For the IVP dy/dx $= f (x, y), y (x_{0}) = y_{0}$ , with step size h, let the first iterate of a second-order scheme be $y_{1} = y_{0} +$ Pk $_{1} +$ Qk $_{2}$ , where $k_{1} = h f (x_{0}, y_{0}), k_{2} = h f (x_{0} + α_{0} h, y_{0} + β_{0} k_{1})$ and $P, Q, α_{0}, β_{0} \in R$ . Which of the following statements are correct?

A.If $α_{0} = 2$ , then $β_{0} = 2, P = 3/4, Q = 1/4$
B.If $β_{0} = 3$ , then $α_{0} = 3, P = 5/6, Q = 1/6$
C.If $α_{0} = 2$ , then $β_{0} = 2, P = 1/4, Q = 3/4$
D.If $β_{0} = 3$ , then $α_{0} = 3, P = 1/6, Q = 5/6$

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The trap it tests

Numerical convergence

A scheme assumed to converge, or to converge at its advertised rate.

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Related counterexample: A convergent method is stable for any step size

Rk2 order conditions

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