Past exam of the numerical analysis course of the University of Oxford / 2023 / 3 / d

Applying the implicit midpoint rule to the simple harmonic oscillator system givesFactor the change in the quadratic form $y^2+z^2$ and substitute these equations:Thus every step preserves the first integralFor completeness, the implicit equations are solvable for every real $h$: putting $s=h/2$, their coefficient matrix has determinant $1+s^2>0$ and the update isThis is an orthogonal matrix, as follows from $(1-s^2{)}^2+4s^2=(1+s^2{)}^2$. It therefore preserves the Euclidean norm, explaining geometrically the exact energy conservation. This is a concrete instance of Runge-Kutta conservation of quadratic invariants. Preservation of the circle does not assert that the numerical and exact solutions have the same phase at a fixed nonzero step size.