Let $u(t) = (u_1(t), u_2(t), u_3(t))$ be a solution of the ODE $$\frac{d\mathbf{u}}{dt}=\mathbf{a}\times\mathbf{u}.$$ where $\times$ denotes the cross product and $\mathbf{a} = (a_1, a_2, a_3) \neq 0$. Consider a forward difference scheme in time $$\mathbf{u}^{n+1}=\mathbf{u}^n+\Delta t(\mathbf{a}\times\mathbf{u}^n)$$
Show that the scheme is always unstable. $$$$ My intuition for this problem is that you cannot just use the definition to expand the cross product, analyzing all three terms at once will be too difficult. Therefore I believe there must be some other way of dealing with this problem which probably uses the intrinsic property of cross product and stability. Are there any ways of linking the two terms together? Also, are there any ways to improve the stability of this unstable scheme? Thanks!
You know that the cross product is orthogonal to its factors. So employ Pythagoras $$ \|u_{n+1}\|^2=\|u_n\|^2+(Δt)^2\|a\times u_n\|^2 $$ Can there be length conservation?