Problem 2.23. Heun's method (or improved Euler) is given by xm+1 = xm + / -...

  

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Problem 2.23. Heun's method (or improved Euler) is given by xm+1 = xm + / - (ƒ (tm, xm) + f(tm+1, Ym)), Um = m + f(tm, Em)h. Show that using this method the error during one step is of O(h³) (provided ƒ=C²): h o(to + h) = xo + (ƒ(to, xo) + f(t₁, yo)) + O(h³). Note that this is not the only possible scheme with this error order since h o(to + h) = xo +(f(t₁, xo) + f(to. yo)) + 0(h³) as well. Problem 2.23. Heun's method (or improved Euler) is given by xm+1 = xm + / - (ƒ (tm, xm) + f(tm+1, Ym)), Um = m + f(tm, Em)h. Show that using this method the error during one step is of O(h³) (provided ƒ=C²): h o(to + h) = xo + (ƒ(to, xo) + f(t₁, yo)) + O(h³). Note that this is not the only possible scheme with this error order since h o(to + h) = xo +(f(t₁, xo) + f(to. yo)) + 0(h³) as well.

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We want to prove the first equation in the given second picture the other equation is for the reade... View the full answer