2016 / March Volume 11 No.1
Convergence Rate of the Glimm Scheme
| Published Date |
2016 / March
|
|---|---|
| Title | Convergence Rate of the Glimm Scheme |
| Author | |
| Keyword | |
| Download | |
| Pagination | 235-300 |
| 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. |
| AMS Subject Classification |
35L65.
|
| Received |
2015-10-20
|
| Accepted |
2015-10-20
|