The fully nonlinear and weakly dispersive Green-Naghdi model for shallow water waves of large amplitude is studied. The original model is first recast under a new formulation more suitable for numerical resolution. An hybrid finite volume and finite difference splitting approach is then proposed. The hyperbolic part of the equations is handled with a high-order finite volume scheme allowing for...

1. GENERAL FORM OF FINITE VOLUME METHODS We consider vertex-centered finite volume methods for solving diffusion type elliptic equation (1) −∇ · (K∇u) = f in Ω, with suitable Dirichlet or Neumann boundary conditions. Here Ω ⊂ R is a polyhedral domain (d ≥ 2), the diffusion coefficient K(x) is a d× d symmetric matrix function that is uniformly positive definite on Ω with components in L∞(Ω), and...

Computational modelling has received significant attention in the research community over the past few decades due to its pronounced contribution in better understanding of the nature, as well as in the development of advanced technologies. The modelling of more and more complex transport physical systems helps the community to address important issues, like identification of environmental prob...

The fractional step method is a technique that results in a computationally-efficient implementation of Navier–Stokes solvers. In the finite element-based models, it is often applied in conjunction with implicit time integration schemes. On the other hand, in the framework of finite difference and finite volume methods, the fractional step method had been successfully applied to obtain predicto...

1. Lax equivalence theorem 1 2. Abstract error analysis 2 3. Application: Finite difference method 3 4. Application: Finite element method 4 5. Application: Conforming Discretization of Variational Problems 5 6. Application: Perturbed Discretization 6 7. Application: Nonconforming finite element methods 7 8. Application: Finite volume method 7 9. Application: Superconvergence of linear finite e...

In this work, we present a novel method to approximate stiff problems using a finite volume (FV) discretization. The stiffness is caused by the existence of a small parameter in the equation which introduces a boundary layer. The semi-analytic method consists in adding in the finite volume space the boundary layer corrector which encompasses the singularities of the problem. We verify the stabi...

The weighted essentially non-oscillatory (WENO) methods are a popular high-order spatial discretization for hyperbolic partial differential equations. Typical treatments of WENO methods assume a uniform mesh. In this paper we give explicit formulas for the finite-volume, fifth-orderWENO (WENO5) method on non-uniform meshes in a way that is amenable to efficient implementation. We then compare t...