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Godunov-type methods for conservation laws with a flux function discontinuous in space. (English) Zbl 1081.65082
Let \(f,g:I\rightarrow\mathbb{R};\) \(I\subset\mathbb{R}\) be continuous and define the flux function \(F(x,u)=H(x)f(u)+(1-H(x)))g(x);\quad H\) being the Heaviside function. The authors consider the problem: \[ \begin{aligned} \frac{\partial u}{\partial t}+\frac{\partial}{\partial x}F(x,u) =0, &\quad x\in\mathbb{R}, \quad t>0, \\ u(x,0)=u_{0}(x), &\quad x \in \mathbb{R}. \end{aligned} \] Since this problem does not in general possess a continuous solution they define a weak one in \(L_{loc}^{\infty}(\mathbb{R}\times\mathbb{R}_{+})\). Special attention is given to the case where the flux functions \(f,g\) intersect. The paper presents a Godunov type method for the numerical solution and proves the convergence of this method. A consequence of this convergence theorem is an existence theorem for the conservation laws under consideration.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
76S05 Flows in porous media; filtration; seepage
35R05 PDEs with low regular coefficients and/or low regular data
76M20 Finite difference methods applied to problems in fluid mechanics
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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