 Solution

Substitute the exact smooth solution into the leapfrog advection scheme and expand the centered differences. With $\tau =\Delta t$ and $h=\Delta x$, the normalized local truncation error isBecause $u_t=u_x$, repeated differentiation gives $u_{ttt}=u_{xxx}$. HenceEquivalently the unnormalized one-update residual is $\tau ({\tau }^2-h^2)u_{xxx}/3+O({\tau }^5+\tau h^4)$. Thus the scheme is second order in space and time. When $\tau =h$, the exact translating solution $u(x,t)=F(x+t)$ satisfies the grid recurrence exactly; initialization still matters for the independent computational branch.