Hybrid finite element/volume method for shallow water equations

Abstract

SUMMARYA hybrid numerical scheme based on finite element and finite volume methods is developed to solve shallow water equations. In the recent past, we introduced a series of hybrid methods to solve incompressible low and high Reynolds number flows for single and two-fluid flow problems. The present work extends the application of hybrid method to shallow water equations. In our hybrid shallow water flow solver, we write the governing equations in non-conservation form and solve the non-linear wave equation using finite element method with linear interpolation functions in space. On the other hand, the momentum equation is solved with highly accurate cell-center finite volume method. Our hybrid numerical scheme is truly a segregated method with primitive variables stored and solved at both node and element centers. To enhance the stability of the hybrid method around discontinuities, we introduce a new shock capturing which will act only around sharp interfaces without sacrificing the accuracy elsewhere. Matrix-free GMRES iterative solvers are used to solve both the wave and momentum equations in finite element and finite volume schemes. Several test problems are presented to demonstrate the robustness and applicability of the numerical method.

CaMEL and ADCIRC Storm Surge Models—A Comparative Study

“…The finite element formulation in Equation (16) is discretized using linear functions for triangles or bilinear functions for quadrilaterals. For more details on the solution technique, refer to Aliabadi et al [16].…”

“…An implicit cell-centered finite volume method is used to solve the momentum equations, Equation (13), to obtain an intermediate velocity field. The node-based Galerkin finite element method is used to solve the continuity equation, Equation (16), to obtain the water elevation. The water elevation is used to update the velocity field.…”

“…Similar to any iterative methods, the performance of the GMRES solver is highly dependent on the preconditioning technique. For the FV method, Equation (13), an approximate Lower Upper-Symmetric Gauss-Seidel (LU-SGS) preconditioner is used, and for the FE method, Equation (16), a diagonal preconditioner is used. The GMRES solver has excellent convergence characteristics, and requires only matrix-vector multiplication, i.e., there is no need to store the huge sparse Jacobian matrix and thus the memory requirement is significantly reduced.…”

“…Step 2: Solve Equation (16) to get the incremental water elevation, h at cell nodes using continuous Galerkin finite element methods. The linear system of equations is solved by the GMRES solver.…”

Profiling and Evaluation of Implicit and Explicit Storm Surge Models

“…In ADCIRC, the water elevation and velocity are obtained by solving the depth-integrated continuity equation in Generalized Wave-Continuity Equation (GWCE), and the momentum equations in its 2DDI or 3D forms, respectively. On the other hand, CaMEL is developed recently by Akbar and Alibadi [2] and Aliabadi et al [10]. CaMEL uses hybrid finite element and finite volume techniques to solve the conservative equations implicitly with the focus of being numerically stable.…”