Simplified $L^ \infty$ estimates for difference schemes for partial differential equations

Abstract:

${L^\infty }$ estimates for one-step difference approximations to the Cauchy problem for $\partial u/\partial t = \partial u/\partial x$ are proven by means of simple ${L^2}$-techniques. It is shown that, provided the difference approximation is stable in ${L^2}$ (and not necessarily ${L^\infty }$) and accurate of order $r$, the error in approximating smooth solutions is $O({h^r})$. This has been proven by Hedstrom and Thomée using Fourier multipliers and Besov spaces. The present paper shows how convergence rates in ${L^\infty }$ can be recovered using simple techniques (such as the Fourier inversion formula). The methods of Hedstrom and Thomée give sharper results when the difference scheme diverges. The present paper exploits the fact that estimates between ${L^\infty }$ and ${L^1}$ are frequently easier to obtain than between ${L^\infty }$ and ${L^\infty }$.