Consider the implicit leapfrog scheme $$ \frac{u_{m}^{n+1}-u_{m}^{n-1}}{2 k}+a\left(1+\frac{h^{2}}{6} \delta^{2}\right)^{-1} \delta_{0} u_{m}^{n}=f_{m}^{n} $$ for the one-way wave equation $$ u_{t}+a u_{x}=f . $$ Here $\delta^{2}$ is the central second difference operator, and $\delta_{0}$ is the central first difference operator.
(1) show that the scheme is of order $(2,4)$.
(2) show that the scheme is stable if and only if $\left|\frac{a k}{h}\right|<\frac{1}{\sqrt{3}}$.$$$$ This is a compact fourth order difference scheme. Since there's a $\left(1+\frac{h^{2}}{6} \delta^{2}\right)^{-1}$ term here, to deal with this, my idea is to take the operator $\left(1+\frac{h^{2}}{6} \delta^{2}\right)$ at both sides to cancel it. This is a very direct method, but it requires a lot of calculation, the scheme will become very long after the expansion. My question is, is there any other way to deal with such an $\left(1+\frac{h^{2}}{6} \delta^{2}\right)^{-1}$ operator which needs not that much computation? Thanks!