Beschreibung:
In this paper, we introduce a new approach for constructing robust well-balanced (WB) finite-volume methods for nonconservative one-dimensional hyperbolic systems of nonlinear partial differential equations. The WB property, namely, the ability of the scheme to exactly preserve physically relevant steady-state solutions is enforced using a flux globalization approach according to which a studied system is rewritten in an equivalent quasi-conservative form with global fluxes. To this end, one needs to incorporate nonconservative product terms into the global fluxes. The resulting system can then be solved using a Riemann-problem-solver-free central-upwind (CU) scheme. However, a straightforward integration of the nonconservative terms would result in a scheme capable of exactly preserving very simple smooth steady states only and failing to preserve discontinuous steady states naturally arising in the nonconservative models.In order to ameliorate the flux globalization based CU scheme, we evaluate the integrals of the nonconservative product terms using the technique introduced in [M. J. Castro D{\'{\i}}az, A. Kurganov and T. Morales de Luna, ESAIM Math. Model. Numer. Anal., 53 (2019), pp. 959--985], where a path-conservative central-upwind scheme (PCCU) was introduced. This results in a new WB flux globalization based PCCU scheme, which is much more accurate and robust than both the original PCCU scheme and the straightforward flux globalization based CU scheme. This is illustrated on an example of the nonconservative system describing fluid flows in nozzles with variable cross-sections. We only consider a simple isentropic case, which, however, admits nontrivial discontinuous steady states and thus serves as an excellent test problem to demonstrate superiority of the proposed WB flux globalization based PCCU scheme