an:05736070
Zbl 1201.35022
B??rger, Raimund; Karlsen, Kenneth H.; Towers, John D.
An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections
EN
SIAM J. Numer. Anal. 47, No. 3, 1684-1712 (2009).
00263351
2009
j
35A35 35L65 65M06 35L45
flux connection; adapted entropy; entropy solution of type \((A, B)\); Heaviside function; scalar monotone difference scheme
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.
Qin Mengzhao (Beijing)