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Convergence Rate of the Glimm Scheme
by Stefano Modena   Stefano Bianchini

Vol. 11 No. 1 (2016) P.235~P.300

ABSTRACT

In this paper we prove that there exists a random sequence $\theta_i$ for the Glimm scheme such that the approximate solution $u^\epsilon(t)$ converges to the exact semigroup solution $S_t \bar u$ of the strictly hyperbolic system of conservation laws \begin{equation*} u_t + f(u)_x = 0, \qquad u(t=0) = \bar u \end{equation*} as follows: for all $T \geq 0$ it holds \begin{equation*} \lim_{\varepsilon \to 0} \frac{\|u^\varepsilon (T) - S_T \bar u\|_1}{\sqrt{\varepsilon } |\log \varepsilon |} = 0. \end{equation*} This result is the extension of the analysis of [8] to the general case, when no assumptions on the flux $f$ are made besides strict hyperbolicity. As a corollary, we obtain a deterministic version of the Glimm scheme for the general system case, extending the analysis of [14].

The analysis requires an extension of the quadratic interaction estimates obtained in [3] in order to analyze interaction occurring during an interval of time.

KEYWORDS
Conservation laws, interaction functionals, Glimm scheme.

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 35L65.

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