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An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections. (English) Zbl 1201.35022
The authors consider the numerical approximation of solutions to the initial value problem
\begin{aligned} u_t + {\mathcal F} (x, u)_x =0 \quad&\text{for }(x, t) \in\mathbb R\times (0,T),\\ u(x,0)= u_0(x) \quad&\text{for }x \in \mathbb R, \end{aligned}
${\mathcal F} (x, u):= H(x) f (u) +\big(1-H(x)\big)g(u)= \begin{cases} f(u) &\text{for }x \geq 0,\\ g(u)& \text{for }x < 0.\end{cases}$ The main contribution of this paper is a scalar monotone difference scheme, for which the authors prove convergence to an entropy solution of type $$(A, B)$$. The scheme is simple in the sense that no $$2\times 2$$ Riemann solver is required. It takes the form of an explicit conservative marching formula on a rectangular grid, where the numerical flux for all cells is the Engquist-Osher (EO) flux, with the exception of the cell interface that is associated with the flux discontinuity, and for which a specific interface flux is used. The interface flux, which is based on a novel modification of the EO flux, is designed to preserve certain steady-state solutions. Some numerical examples are presented.

##### MSC:
 35A35 Theoretical approximation in context of PDEs 35L65 Hyperbolic conservation laws 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35L45 Initial value problems for first-order hyperbolic systems
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