The FVC Scheme on Unstructured Meshes for the Two-Dimensional Shallow Water Equations

Abstract

The fluid flow transport and hydrodynamic problems often take the form of hyperbolic systems of conservation laws. In this work we will present a new scheme of finite volume methods for solving these evolution equations. It is a family of finite volume Eulerian-Lagrangian methods for the solution of non-linear problems in two space dimensions on unstructured triangular meshes. The proposed approach belongs to the class of predictor-corrector procedures where the numerical fluxes are reconstructed using the method of characteristics, while an Eulerian method is used to discretize the conservation equation in a finite volume framework. The scheme is accurate, conservative and it combines advantages of the modified method of characteristics to accurately solve the non-linear conservation laws with a finite volume method to discretize the equations. The proposed Finite Volume Characteristics (FVC) scheme is also non-oscillatory and avoids the need to solve a Riemann problem. Several test examples will be presented for the shallow water equations. The results will be compared to those obtained with the Roe.

A well balanced fvc scheme for 2d shallow water flows on unstructured triangular meshes

“…The method consists of two steps and can be interpreted as a predictor-corrector approach. The first step deals with the classical finite volume method, whereas in the second step, the reconstruction of the numerical flux by constructing the intermediate state using the characteristics method [28].…”

“…In this subsection, we present a generalization of this scheme that was introduced in the preliminary works [16,28], the first was done in the Cartesian mesh and the second was done without taking into account the source term related to bathymetry and the balance produced by this term. For the corrector stage, we will use 2D finite volume formalism described in Section 3.1.…”

“…Notice that h l , h k and h j are the average values of h, respectively, on the triangle T l , T k and T j . Summing up, using the fact that (N xij = −N xji ) so, the discretization (28) gives…”

A highly efficient finite volume method with a diffusion control parameter for hyperbolic problems

“…The proposed scheme belongs to a class of numerical schemes that incorporates the method of characteristics in reconstructing numerical ux. This approach is called the Finite Volume Characteristics (FVC) scheme and has been proposed by Benkhaldoun and Seaïd in [10] and used in the context of shallow water ow [11][12][13][14].…”

Manapy: MPI-Based framework for solving partial differential equations using finite-volume on unstructured-grid

“…Different models have been developed using Manapy with Master and PhD. students (2D/3D Poisson equation, 2D Shallow Water [21,21], 2D Shallow Water Magnetohydrodynamics [22], 2D/3D Streamer discharge [23,24], 2D Transient, incompressible, Navier-Stokes solver using the PISO algorithm [25]).…”