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.


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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