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Numerical Methods

Quick questions on Numerical solution of differential equations explained: H2 Further Mathematics

6short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is euler's method?
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Given dydx=f(x,y)\dfrac{\mathrm{d}y}{\mathrm{d}x} = \mathrm{f}(x, y) and a starting point (x0,y0)(x_0, y_0), Euler's method steps forward by following the tangent:
What is the improved Euler (midpoint) method?
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Euler's method is crude because it uses only the gradient at the start of each step. The improved method corrects this by also using the gradient at the predicted end (or midpoint) and averaging:
What is wrong step count?
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To reach a target xx, the number of steps is xtarget−x0h\dfrac{x_{\text{target}} - x_0}{h}; miscounting stops short or overshoots.
What is q1?
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Write Euler's method for dydx=f(x,y)\dfrac{\mathrm{d}y}{\mathrm{d}x} = \mathrm{f}(x, y). [1 mark]
What is q2?
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How does reducing the step size hh affect the accuracy? [1 mark]
What is q3?
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Why is the improved Euler method more accurate than basic Euler? [2 marks]

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