Entropy Dissipation in Finite Difference Schemes

Abstract

We are concerned with hyperbolic systems of conservation laws

$$ {\partial _t}u + {\partial _x}\int {\left( u \right)} = 0, $$

((1))

where $D: = \left\{ {\left( {x,t} \right) \in \mathbb{R}\; \times \mathbb{R}\_ + } \right\} \ni \left( {x,t} \right)\xrightarrow{u}u\left( {x,t} \right) \in \Omega \subset {\mathbb{R}^m}\Omega$ , Ω being the open state space, and $\Omega \ni u\xrightarrow{f}f\left( u \right) \in {\mathbb{R}^m}$ is a nonlinear flux function of class [C ²]^m(Ω; ℝ^m). Without restriction we assume f(0) = 0. We consider the Cauchy problem for (1) with initial function u ₀ ∈ [L ^∞∩ BV _loc]^m (ℝ; Ω). A weak solution is a function u∈ [L ^∞∩ BV _loc]^m (D; Ω) for wich

$$\int\limits_D {\left\{ {u{\partial _t}\phi + f\left( u \right){\partial _x}\phi } \right\}} dxdt + \int\limits_\mathbb{R} {\left\{ {{u_0}\left( x \right)\phi \left( {x,0} \right)} \right\}} dx = 0$$

holds for all Ф ∈ C ¹₀ ]^m (D; ℝ^m). Since weak solutions are not uniquely determined we call a solution admissible, if an entropy inequality

$$ {\partial _t}\eta \left( u \right) + {\partial _x}q\left( u \right) \leqslant 0$$

((2))

in the sense of distributions for any entropy pair $\Omega \ni u\xrightarrow{\eta }\eta \left( u \right) \in \mathbb{R},\Omega \ni u\xrightarrow{q}q\left( u \right) \in \mathbb{R}$ , where η is a strictly convex entropy and q an entropy flux satisfying the compatibility condition

$${\nabla _u}\eta \left( u \right){\nabla _u}f\left( u \right) = {\nabla _u}q\left( u \right)$$

((3))

.