WENO schemes for balance laws with spatially varying flux

In this paper we construct numerical schemes of high order of accuracy for hyperbolic balance law systems with spatially variable flux function and a source term of the geometrical type. We start with the original finite difference characteristicwise weighted essentially nonoscillatory (WENO) schemes and then we create new schemes by modifying the flux formulations (locally Lax-Friedrichs and Roe with entropy fix) in order to account for the spatially variable flux, and by decomposing the source term in order to obtain balance between numerical approximations of the flux gradient and of the source term. We apply so extended WENO schemes to the one-dimensional open channel flow equations and to the one-dimensional elastic wave equations. In particular, we prove that in these applications the new schemes are exactly consistent with steady-state solutions from an appropriately chosen subset. Experimentally obtained orders of accuracy of the extended and original WENO schemes are almost identical on a convergence test. Other presented test problems illustrate the improvement of the proposed schemes relative to the original WENO schemes combined with the pointwise source term evaluation. As expected, the increase in the formal order of accuracy of applied WENO reconstructions in all the tests causes visible increase in the high resolution properties of the schemes.