Domain decomposition methods are studied in a scalable parallel solver for the Cahn-Hilliard equation in 3D. The discretization is based on a stabilized implicit cell-centered finite difference scheme together with an adaptive time-stepping strategy. A Newton-KrylovSchwarz algorithm is applied to solve the nonlinear system of equations arising at each time step. In the Schwarz preconditioner, w...

The Poisson-Nernst-Planck equations with generalized Frumkin-Butler-Volmer boundary conditions (PNP-FBV) describe ion transport Faradaic reactions, and have applications in a number of fields. In this article, we develop an adaptive time-stepping scheme for the solution PNP-FBV system based on two methods: fully-implicit (VSBDF2) method, semi-implicit (VSSBDF2) method. We present simulations un...

An implicit preconditioned multigrid algorithm is developed for the efficient solution of two-dimensional, lowfrequency unsteady turbulent Navier-Stokes calculations on highly stretched meshes. The efficiency of the approach derives from three key attributes: 1) an implicit time discretization that allows the time step to be determined solely by the resolution requirements of the unsteady pheno...

Solidi cation of a pure material from its undercooled melt is considered, taking into account e ects of surface tension, kinetic undercooling and anisotropy. The resulting crystals exhibit complicated dendritic shapes. To avoid explicit tracking of the solid-liquid interfaces a phase eld model is employed. A di use interface of arti cial width is introduced and the solidi cation is modeled by a...

This study concerns numerical methods for efficiently solving the Richards equation where different weak formulations and computational techniques are analyzed. The spatial discretizations based on standard or mixed finite element methods. Different implicit semi-implicit temporal discretization of second-order accuracy studied. To obtain a linear system schemes, we propose using extrapolation ...

A method for relaxing the CFL-condition, which limits the time step size in explicit methods in computational fluid dynamics, is presented. The method is based on re-formulating explicit methods in matrix form, and considering them as a special-Jacobi iteration scheme that converge efficiently if the CFLnumber is less than unity. By adopting this formulation, one can design various solution met...

This paper focuses on increasing the accuracy of low-order four-node quadrilateral finite elements for the transient analysis of wave propagation. Modified integration rules, originally proposed for time-harmonic problems, provide the basis for the proposed technique. The modified integration rules shift the integration points to locations away from the conventional Gauss or Gauss-Lobatto integ...

In this article we present robust, efficient and accurate fully implicit time-stepping schemes and nonlinear solvers for systems of reaction-diffusion equations. The applications of reaction-diffusion systems is abundant in the literature, from modelling pattern formation in developmental biology to cancer research, wound healing, tissue and bone regeneration and cell motility. Therefore, it is...