Stability LeapFrog method

Question

I have a system of ODE's $x''=Ax$ being $A$ a matrix. Denoting y=x', I obtain the first order system $$ x'=y\\ y'=Ax. $$ If we denote $x^n\approx x(t_n)$ and $y^n\approx y(t_n)$, and we apply the LeapFrog method $$ w=x^n+\Delta ty^n/2\\ y^{n+1}=y^n+\Delta t(Aw)\\ x^{n+1}=w+\Delta t/2 y^{n+1} $$ what can I say about its stability? Thanks

Answer 1

If $z= \pmatrix{x\cr y\cr}$, we have $$ z^{n+1} = \pmatrix{I + \frac{(\Delta t)^2}{2} A & \Delta t I + \frac{(\Delta t)^3}{4} A\cr \Delta t A & I + \frac{(\Delta t)^2}{2} A\cr} z^n $$ The eigenvalues of the matrix are $$1 + \frac{\lambda (\Delta t)^2}{2} \pm \sqrt{\lambda}\Delta t \sqrt{1 + \frac{A (\Delta t)^2}{4}} $$ where $\lambda$ is an eigenvalue of $A$. If any of these has absolute value $> 1$, the method is unstable; if all have absolute value $< 1$, it is stable.