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Order of convergence of minmod-type schemes

A 2003 Preprint by B. Popov and O. Trifonov

  • 2003:21
  • A class of non-oscillatory numerical methods for solving nonlinear scalar conservation laws in one space dimension is considered. Non-oscillatory schemes are based on minmod limiters and the standard second order representatives are the staggered Nessyahu-Tadmor scheme and the usual TVD2 scheme. It is well known that the $L _ p$-error of monotone finite difference methods for the linear advection equation is exactly $\frac{1}{2}$ for initial data in $W^1(L _ p),$ $1\leq{p}\leq\infty$. For a second or higher order non-oscillatory schemes very little is known because they are nonlinear even for the simple advection equation. In this paper, in the case of a linear advection equation with monotone initial data, it is shown that the order of the $L _ 2$-error for the standard second order minmod-type schemes is at least $\frac{5}{8}$ in contrast to the exact $\frac{1}{2}$ order for any formal first order scheme.

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