In this paper we analyze the convergence of a numerical scheme for a class of degenerate parabolic problems. Such problems are often used to model reactions in porous media, and involve a nonlinear, possibly vanishing diffusion. The scheme considered here involves the Kirchhoff transformation coupled with the regularization of the nonlinearity, and is based on the Euler implicit time stepping a...

This article is concerned with new strategies with which explicit time-stepping procedures of PDE-based restoration models converge with a similar efficiency to implicit algorithms. Conventional explicit algorithms often require hundreds of iterations to converge. In order to overcome the difficulty and to further improve image quality, the article introduces new spatially variable constraint t...

In this paper we propose and analyze a Discontinuous Galerkin method for a linear parabolic problem with dynamic boundary conditions. We present the formulation and prove stability and optimal a priori error estimates for the fully discrete scheme. More precisely, using polynomials of degree p ≥ 1 on meshes with granularity h along with a backward Euler time-stepping scheme with time-step ∆t, w...

This paper introduces BFEMP, a new approach for monolithically coupling the Material Point Method (MPM) with Finite Element (FEM) through barrier energy-based particle–mesh frictional contact using variational time-stepping formulation. The fully implicit time integration of coupled system is recast into barrier-augmented unconstrained nonlinear optimization problem. A modified line-search Newt...

Time integration of advection dominated advection–diffusion problems on refined meshes can be a challenging task, since local refinement lead to severe time step restriction, whereas standard implicit stepping is usually hardly suitable for treating terms. We show that exponential integrators an efficient, yet conceptually simple, option in this case. Our comparison includes three and one conve...

In this paper we derive an a posteriori error estimate for the numerical approximation of the solution of a system modeling the flow of two incompressible and immiscible fluids in a porous medium. We take into account the capillary pressure, which leads to a coupled system of two equations: parabolic and elliptic. The parabolic equation may become degenerate, i.e., the nonlinear diffusion coeff...

A numerical model is presented for computation of unsteady two-fluid interfaces in nonlinear porous media flow. The nonlinear Forchheimer equation is included in the Navier-Stokes equations for porous media flow. The model is based on capturing the interface on a fixed mesh domain. The zero level set of a pseudo-concentration function, which defines the interface between the two fluids, is gove...

The Computation and Modeling Engineering Laboratory (CaMEL), an implicit solver-based storm surge model, has been extended for use on high performance computing platforms. An MPI (Message Passing Interface) based parallel version of CaMEL has been developed from the previously existing serial version. CaMEL uses hybrid finite element and finite volume techniques to solve shallow water conservat...

Leaping methods provide for efficient and approximate time stepping of chemical reaction systems modeled by continuous time discrete state stochastic dynamics. We investigate the application of leaping methods for ‘‘small number and stiff’’ systems, i.e. systems whose dynamics involve different time scales and have some molecular species present in very small numbers, specifically in the range ...