Compute the coefficients for the most accurate (in the sense of satisfying as ma

Question

Compute the coefficients for the most accurate (in the sense of satisfying as many exactness constraints as possible explicit linear multi-step formulas for which alpha_0 = 1, alpha_1 = -1, and alpha_3 = ... = alpha_k = 0. Compute the coefficients for k = 1, 2, 3, 4. What power of delta t is LTE proportional to for k = 1, 2, 3, 4? Compute the coefficients for the most accurate (in the sense of satisfying as many exactness constraints as possible) implicit linear multi-step formulas for which alpha_0 = 1, alpha_1 = -1, and alpha_3 = ... = alpha_k = 0. Compute the coefficients for k = 1, 2, 3, 4. What power of delta t is LTE proportional to for k = 1, 2, 3, 4? Compute the coefficients for the most accurate (in the sense of satisfying as many exactness constraints as possible) implicit linear multi-step formulas for which alpha_0 = 1, and beta_1 = ... = beta_k = 0. Compute the coefficients for k = 1, 2, 3, 4. What power of delta t is LTE proportional to for k = 1, 2, 3, 4?

Explanation / Answer

A)

explicit linear multistep:

In the established notation we have k = 1, 0 = 1, 1 = 1. The terms similarly come from the coefficients on the right hand side

Now

Here k = 1

so that 1yn+1 + 0yn = h(1fn+1 + 0fn)

substituting the values in for the coefficients gives

yn+1 yn = h.

B)

Implicit linear multistep:

k = 1, 0 = 1, 1 = 1.

Here k = 1

so that 1yn+1 + 0yn = h(1fn+1 + 0fn)

substituting the values in for the coefficients gives

yn+1 yn = h.

C)

Implicit linear multistep:

k = 1, 0 = 1,0 = 1 and k = 0

Here k = 1

we have 1yn+1 + 0yn = h(1fn+1 + 0fn).

substituting the values in for the coefficients gives

yn = hfn.

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