A Boltzmann-based finite volume algorithm for surface water flows on cells of arbitrary shapes

An explicit two-dimensional conservative finite volume model for shallow water equations is formulated and tested. The algorithm for the mass and momentum fluxes at the control surface of the finite volume is obtained from the solution of the Bhatnagar-Gross-Krook (BGK) Boltzmann equation. Unlike classical methods, BGK schemes do not require an ad-hoc splitting of advection and diffusion. The BGK scheme is second order in both time and space. The formulation of the BGK algorithm is performed for a cell of arbitrary irregular shape, but the test cases are conducted using a structured grid of quadrilateral cells. Two approximate Riemann solvers, the HLLC scheme and the two-stage Hancock-HLLC scheme, where HLL stands for Harten, Lax and van Leer and C stands for contact discontinuity, are also considered. The second-order accuracy of HLL and Hancock-HLLC schemes is obtained by MUSCL approach, where MUSCL is the acronym for Monotone Upstream-centered Schemes for Conservation Laws. The data reconstruction for all three schemes is carried out by the Van Leer limiter. The test cases involve strong shocks and expansion waves. The accuracy of the schemes are measured using an absolute error norm and a waviness error norm. The HLLC scheme is highly oscillatory for Courant number larger than 0.5, while the BGK and the Hancock-HLLC schemes are applicable for Courant numbers as high as 1.0. For a fixed value of the central processing unit (CPU) time, the absolute error of the Hancock-HLLC is slightly smaller than that of the BGK while the waviness error of the BGK is quite close to that of Hancock-HLLC. This is because (i) the Hancock-HLLC is a two-step method while the BGK is a single-step method (i.e., the Hancock-HLLC requires storage of intermediate variables, but the BGK does not), and (ii) the Hancock-HLLC schemes requires larger number of grid points than the BGK scheme for the same level of accuracy. For example, to achieve an absolute error of 0.01, the BGK requires about 600 grid points while the Hancock-HLLC requires about 800 grid points. Both the BGK and Hancock-HLLC schemes have similar convergence properties. Unlike exact or approximate Riemann solvers, BGK fluxes accounts for both waves and diffusion. The ability of the BGK scheme to model diffusion is illustrated using a viscous flow problem. Excellent agreement between the analytical and computed viscous flow solution is found. Although the BGK and Hancock-HLLC schemes perform similarly for hyperbolic problems, BGK schemes have the added advantage of being able to solve hyperbolic-parabolic problems without the need for an ad-hoc operator splitting. This is important given that the artificial splitting of advection and diffusion is known to cause artificial widening in shear layers and introduces artificial transient in regions with sharp gradients. Such problems arise when the splitting operation fails to faithfully represent the correct coupling between the physics of advection and the physics of waves