Solving ODEs with Heun's Method: A Question

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Heun's method for solving ordinary differential equations (ODEs) involves calculating new values using the formula y(new) = y(old) + 0.5(k1 + k2)Δh, where k1 is f(y(old)) and k2 is f(y(old) + q11k1Δh). In the example of the differential equation du/dt = au, the calculations lead to confusion regarding the multiplication of the k2 term by 'a'. The user initially struggled to understand this aspect, noting discrepancies in their notes and textbooks. Ultimately, the user resolved their confusion and indicated that the thread could be deleted.
yugeci
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Hi there,

in my notes for Heun's method for solving an ODE, I have

y(new) = y(old) + 0.5(k1 + k2)Δh

And k1 is supposed to be f(y(old)) while k2 is f(y(old) + q11k1Δh) and q11 is 1

So if for example I have a simple differential equation like du/dt = au

It would be du/dt = 0.5(k1 + k2)

du/dt = 0.5(au(told) + a((u(told)+au(told)Δt))

The first term on the right of the equal is k1, and the term after is k2. What I don't understand is why the entire k2 term is multiplied by a? It's not in any formula so what mistake have I made here. I couldn't really find this in any of my books either, especially since the notation is so different.
 
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I figured it out. This thread can be deleted.
 

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