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

Authors:
Bruno Després
Affiliations:
bruno.despres@cea.fr
Venue:
SIAM Journal on Numerical Analysis
Year:
2009

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Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications

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Abstract

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 $L^q$ 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.