Simulating Rotating Shallow Water Flows By BGK Scheme

The shallow water equation is an usual model to describe the flows in river, ocean, and atmosphere etc. In this paper, a finite volume gas-kinetic BGK scheme is derived to simulate rotating shallow water flows.Abstract vii 1 Introduction 1 1.1 Shallow water equation . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Boltzmann and BGK model . . . . . . . . . . . . . . . . . . . 1 2 Gas-Kinetic BGK Scheme 5 2.1 KFVS scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 1st-order KFVS . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 2nd-order KFVS . . . . . . . . . . . . . . . . . . . . . 6 2.2 KFVS Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 BGK scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.1 1st-order BGK . . . . . . . . . . . . . . . . . . . . . . 10 2.3.2 2nd-order BGK . . . . . . . . . . . . . . . . . . . . . . 11 2.4 BGK Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 A well-balanced condition 17 3.1 1D case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 2D case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4 A second-order BGK Scheme for shallow water ‡ows 27 4.1 non-rotating shallow water ‡ows . . . . . . . . . . . . . . . . . 27 4.2 Rotating shallow water ‡ows . . . . . . . . . . . . . . . . . . . 31 4.2.1 On ‡at topography bottom . . . . . . . . . . . . . . . 31 4.2.2 On wary bottom . . . . . . . . . . . . . . . . . . . . . 35 5 Conclusion 3