Central-upwind schemes for the system of shallow water equations with horizontal temperature gradients

We introduce a central-upwind scheme for oneand two-dimensional systems of shallow-water equations with horizontal temperature gradients (the Ripa system). The scheme iswell-balanced, positivity preserving and does not develop spurious pressure oscillations in the neighborhood of temperature jumps, that is, near the contact waves. Such oscillations would typically appear when a conventional Godunov-type finite volume method is applied to the Ripa system, and the nature of the oscillation is similar to the ones appearing at material interfaces in compressible multifluid computations. The idea behind the proposed approach is to utilize the interface tracking method, originally developed in Chertock et al. (M2ANMath Model Numer Anal 42:991–1019, 2008) for compressiblemultifluids. The resulting scheme is highly accurate, preserves two types of “lake at rest” steady states, and is oscillation free across the temperature jumps, as it is illustrated in a number of numerical experiments. Mathematics Subject Classification (2000) 76M12 · 65M08 · 86A05 · 35L65 · 35L67 The work of A. Chertock was supported in part by the NSF Grants DMS-1115682 and DMS-1216974 and by the ONR Grant N00014-12-1-0832. The research of A. Kurganov was supported in part by the NSF Grants DMS-1115718 and DMS-1216957 and by the ONR Grant N00014-12-1-0833. A. Chertock (B) Department of Mathematics, North Carolina State University State University, Raleigh, NC 27695, USA e-mail: [email protected] A. Kurganov · Y. Liu Mathematics Department, Tulane University, New Orleans, LA 70118, USA e-mail: [email protected] Y. Liu e-mail: [email protected] 123 Author's personal copy