Midpoint method: Difference between revisions

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the implicit midpoint method by

:<math> y_{n+1} = y_n + hf\left(t_n+\frac{h}{2},\frac12 (y_n+y_{n+1})\right), \qquad\qquad (1i)</math>

for <math>n=0, 1, 2, \dots</math> Here, <math>h</math> is the ''step size'' &mdash; a small positive number, <math>t_n=t_0 + n h,</math> and <math>y_n</math> is the computed approximate value of <math>y(t_n).</math> The explicit midpoint method is sometimes also known as the '''modified Euler method''',<ref>{{harvnb|Süli|Mayers|2003|p=328}}</ref>, the implicit method is the most simple [[collocation method]], and, applied to Hamiltonian dynamics, a [[symplectic integrator]]. Note that the '''modified Euler method''' can refer to [[Heun's method]],<ref>{{harvnb|Burden|Faires|2011|p=286}}</ref>, for further clarity see [[List of Runge–Kutta methods]].

The name of the method comes from the fact that in the formula above, the function <math>f</math> giving the slope of the solution is evaluated at <math>t=t_n+h/2= \tfrac{t_n+t_{n+1}}{2},</math> the midpoint between <math>t_n</math> at which the value of <math>y(t)</math> is known and <math>t_{n+1}</math> at which the value of <math>y(t)</math> needs to be found.