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

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 33, Issue: 1, page 129-156
- ISSN: 0764-583X

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topChainais-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 -

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