A note on \(L^{\infty }\) bounds and convergence rates of summation-by-parts schemes
The author proves that summation-by-parts finite difference schemes approximating linear initial boundary value problems are stable in \(L^\infty\). He considers \[ \begin{aligned} u_t &= P(\partial_x) u + F(x,t),\quad 0\leq x \leq L, \quad t\geq 0,\\u(x,0) &= f(x),\\ L_0(\partial_x,u(0,t),g_0(t)) &= 0,\\ L_1(\partial_x,u(L,t),g_1(t)) &= 0,\end{aligned} \] where \(F,f, g_0, g_1\) are smooth and compatible to sufficiently high order. Together with the standard \(L^2\) bound, the \(L^\infty\) bound is sufficient for optimal convergence rates by applying the results of \textit{M. Svärd} and \textit{J. Nordström} [J. Comput. Phys. 218, No. 1, 333-352 (2006; Zbl 1123.65088)].
scientific article; zbMATH DE number 6363090
scientific article