TY - JOUR
T1 - A Well-Balanced Runge-Kutta Discontinuous Galerkin Method for Multilayer Shallow Water Equations with Non-Flat Bottom Topography
AU - Izem , Nouh
AU - Seaid , Mohammed
JO - Advances in Applied Mathematics and Mechanics
VL - 3
SP - 725
EP - 758
PY - 2022
DA - 2022/02
SN - 14
DO - http://doi.org/10.4208/aamm.OA-2020-0364
UR - https://global-sci.org/intro/article_detail/aamm/20282.html
KW - Discontinuous Galerkin method, well-balanced discretization, Runge-Kutta scheme, multilayer shallow water equations, free-surface flows, mass exchange, wind-driven flows, strait of Gibraltar.
AB - <p style="text-align: justify;">A well-balanced Runge-Kutta discontinuous Galerkin method is presented
for the numerical solution of multilayer shallow water equations with mass exchange
and 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 Galerkin
method is a locally conservative finite element method whose approximate solutions
are discontinuous across the inter-element boundaries. The well-balanced property is
achieved using a special discretization of source terms that depends on the nature of
hydrostatic 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 a
high-order version of upwind finite volume solvers and it offers attractive features for
the numerical solution of conservation laws for which standard finite element methods
fail. To deal with the source terms we also implement a high-order splitting operator
for the time integration. The accuracy of the proposed Runge-Kutta discontinuous
Galerkin method is examined for several examples of multilayer free-surface flows
over both flat and non-flat beds. The performance of the method is also demonstrated
by comparing the results obtained using the proposed method to those obtained using
the incompressible hydrostatic Navier-Stokes equations and a well-established kinetic
method. The proposed method is also applied to solve a recirculation flow problem in
the Strait of Gibraltar.</p>