A Well-Balanced Runge-Kutta Discontinuous Galerkin Method for Multilayer Shallow Water Equations with Non-Flat Bottom Topography (Advances in Applied Mathematics and Mechanics, 2022)
A well-balanced Runge-Kutta discontinuous Galerkin method is presentedfor the numerical solution of multilayer shallow water equations with mass exchangeand non-flat bottom topography. The governing equations are reformulated as a nonlinear system of conservation laws with differential source forces and reaction terms.Coupling between the flow layers is accounted for in the system using a set of exchange relations. The considered well-balanced Runge-Kutta discontinuous Galerkinmethod is a locally conservative finite element method whose approximate solutionsare discontinuous across the inter-element boundaries. The well-balanced property isachieved using a special discretization of source terms that depends on the nature ofhydrostatic solutions along with the Gauss-Lobatto-Legendre nodes for the quadrature used in the approximation of source terms. The method can also be viewed as ahigh-order version of upwind finite volume solvers and it offers attractive features forthe numerical solution of conservation laws for which standard finite element methodsfail. To deal with the source terms we also implement a high-order splitting operatorfor the time integration. The accuracy of the proposed Runge-Kutta discontinuousGalerkin method is examined for several examples of multilayer free-surface flowsover both flat and non-flat beds. The performance of the method is also demonstratedby comparing the results obtained using the proposed method to those obtained usingthe incompressible hydrostatic Navier-Stokes equations and a well-established kineticmethod. The proposed method is also applied to solve a recirculation flow problem inthe Strait of Gibraltar.