What is the reason behind the multiplication of the function's derivative with the step size (and the subsequent addition) in the numerical Euler method? y n + 1 = y n + h f ( t n , y n )I can't figure out why exactly this would work for generating a new value. How can scaling the output of the function f and subsequently adding it to the prior value y n approximate a new value? Is there some formal or graphical explanation of this? The formula seems somewhat counterintuitive.

Question

What is the reason behind the multiplication of the function&#039;s derivative with the step size (and the subsequent addition) in the numerical Euler method?      y          n      +      1        =      y    n    +  h  f  (      t    n    ,      y    n    )I can&#039;t figure out why exactly this would work for generating a new value. How can scaling the output of the function   f and subsequently adding it to the prior value       y    n   approximate a new value? Is there some formal or graphical explanation of this? The formula seems somewhat counterintuitive.

Laura Fletcher · Accepted Answer

I would say Euler&#039;s method is actually the most intuitive of the numerical methods!We have a function   f that tells us the derivative of a solution to the ODE passing through our current point   (      t    n    ,      y    n    ). What is the derivative? In a small enough interval around       y    n  , it&#039;s approximately the &quot;rise over run&quot;, so if   h  =      t          n      +      1        −      t    n   is &#039;small&#039; then we can use this approximation:            y      (              t                  n          +          1                    )      −              y        n                            t                  n          +          1                    −              t        n              ≃  f  (      t    n    ,      y    n    )    ⟹    y  (      t          n      +      1        )  ≃      y    n    +  h  ⋅  f  (      t    n    ,      y    n    )

kemecryncqe9 · Accepted Answer

It helped a lot)

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