Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate

Chainais-Hillairet, Claire. "Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate." ESAIM: Mathematical Modelling and Numerical Analysis 33.1 (2010): 129-156. <http://eudml.org/doc/197585>.

@article{Chainais2010,
abstract = { In this paper, we study some finite volume schemes for the nonlinear hyperbolic equation $\{u_t\}(x,t)+\mbox\{div\}F(x,t,u(x,t))=0$ with the initial condition $u_\{0\}\in\{L^\infty\}(\mathbb\{R\}^N)$. Passing to the limit in these schemes, we prove the existence of an entropy solution $u\in\{L^infty\}(\mathbb\{R\}^N\times\mathbb\{R\}_+)$. Proving also uniqueness, we obtain the convergence of the finite volume approximation to the entropy solution in $L^p_\{loc\}(\mathbb\{R\}^N\times\mathbb\{R\}_+)$, 1 ≤ p ≤ +∞. Furthermore, if $\{u_0\}\in \{L^\infty\}\cap\mbox\{BV\}_\{loc\}(\mathbb\{R\}^N)$, we show that $u\in\mbox\{BV\}_\{loc\}(\mathbb\{R\}^N\times\mathbb\{R\}_+)$, which leads to an “$h^\{\frac\{1\}\{4\}\}$” error estimate between the approximate and the entropy solutions (where h defines the size of the mesh). },
author = {Chainais-Hillairet, Claire},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {finite volume method; error estimate; nonlinear conservation laws; convergence; entropy solution},
language = {eng},
month = {3},
number = {1},
pages = {129-156},
publisher = {EDP Sciences},
title = {Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate},
url = {http://eudml.org/doc/197585},
volume = {33},
year = {2010},
}

TY - JOUR
AU - Chainais-Hillairet, Claire
TI - Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 33
IS - 1
SP - 129
EP - 156
AB - In this paper, we study some finite volume schemes for the nonlinear hyperbolic equation ${u_t}(x,t)+\mbox{div}F(x,t,u(x,t))=0$ with the initial condition $u_{0}\in{L^\infty}(\mathbb{R}^N)$. Passing to the limit in these schemes, we prove the existence of an entropy solution $u\in{L^infty}(\mathbb{R}^N\times\mathbb{R}_+)$. Proving also uniqueness, we obtain the convergence of the finite volume approximation to the entropy solution in $L^p_{loc}(\mathbb{R}^N\times\mathbb{R}_+)$, 1 ≤ p ≤ +∞. Furthermore, if ${u_0}\in {L^\infty}\cap\mbox{BV}_{loc}(\mathbb{R}^N)$, we show that $u\in\mbox{BV}_{loc}(\mathbb{R}^N\times\mathbb{R}_+)$, which leads to an “$h^{\frac{1}{4}}$” error estimate between the approximate and the entropy solutions (where h defines the size of the mesh).
LA - eng
KW - finite volume method; error estimate; nonlinear conservation laws; convergence; entropy solution
UR - http://eudml.org/doc/197585
ER -