Uniform Asymptotic Stability of Strang's Explicit Compact Schemes for Linear Advection

We consider a family of explicit compact schemes for advection in one dimension. The order is arbitrarily high. These stencils may be called Strang’s stencils after the seminal work of Strang [J. Math. Phys., 41 (1962), pp. 147–154]. We prove that odd order schemes are stable in all Lq under CFL one. The strategy of the proof is similar to the one of Thomée [J. Differential Equations, 1 (1965), pp. 273–292] with a careful verification that all sharp estimates on the amplification factor are independent of the CFL number. This is possible based on a general representation formula for the amplification factor. Numerical results in one dimension confirm the analysis.