I am new to numerical methods, in my understanding, the central differentiation method is used to calculate derivatives such as $$\frac{dy}{dx}=\frac{y(x+h)-y(x-h)}{2h}$$ How could we use this to discretize equations?
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What equations? Note that the central Euler method is only weakly stable. See also the midpoint method, explicit as well as implicit. – Lutz Lehmann Nov 17 '22 at 21:52
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Hi! Thank you for your reply! For example how could we use central method to discretize: d/dx * x * dy/dx = 0 ? – jjk Nov 17 '22 at 21:59
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Then $f(x)=xy'(x)$ so that the discretization becomes $\dfrac{x_{k+1}·(y_{k+2}-y_k)-x_{k-1}·(y_k-y_{k-2})}{4h^2}$. Or condense it to $\dfrac{x_{k+1/2}·(y_{k+1}-y_k)-x_{k-1/2}·(y_k-y_{k-1})}{h^2}$. How useful either of the variants is has to be tested against other methods. – Lutz Lehmann Nov 17 '22 at 22:04