In this paper, some new approaches are proposed for moving boundary/interface problems, particularly for the Stefan problems and the problem of unstable crystal growth. We will focus on the issues of accuracy and speed-up. A modiied Crank-Nicolson method which is second order accurate and stable is developed. The ADI (alternating implicit directional method) is also developed to speed up the si...

This paper presents new geometrical flow equations for the theoretical modeling of biomolecular surfaces in the context of multiscale implicit solvent models. To account for the local variations near the biomolecular surfaces due to interactions between solvent molecules, and between solvent and solute molecules, we propose potential driven geometric flows, which balance the intrinsic geometric...

A new version of the alternating directions implicit (ADI) iteration for the solution of large-scale Lyapunov equations is introduced. It generalizes the hitherto existing iteration, by incorporating tangential directions in the way they are already available for rational Krylov subspaces. Additionally, first strategies to adaptively select shifts and tangential directions in each iteration are...

A multi-cluster computational environment with mixed-mode (MPI + OpenMP) parallelism for estimation of unknown regional electrical conductivities of the human head, based on realistic geometry from segmented MRI up to 256 voxels resolution, is described. A finite difference multi-component alternating direction implicit (ADI) algorithm, parallelized using OpenMP, is used to solve the forward pr...

We introduce a class of alternating direction implicit (ADI) methods, based on approximate factorizations of backward differentiation formulas (BDFs) of order p 2, for the numerical solution of two-dimensional, time-dependent, nonlinear, convection-diffusion partial differential equation (PDE) systems in Cartesian domains. The proposed algorithms, which do not require the solution of nonlinear ...

In this paper, a finite-difference time-domain method that is free of the constraint of the Courant stability condition is presented for solving electromagnetic problems. In it, the alternating direction implicit (ADI) technique is applied in formulating the finite-difference time-domain (FDTD) algorithm. Although the resulting formulations are computationally more complicate than the conventio...

We introduce a class of alternating direction implicit (ADI) methods, based on approximate factorizations of backward differentiation formulas (BDF) of order p ≥ 2, for the numerical solution of two-dimensional, timedependent, nonlinear, convection-diffusion PDE systems in Cartesian domains. The proposed algorithms, which do not require solution of nonlinear systems, additionally produce soluti...

In this paper, we extend the Sun and Zhang’s [24] work on high order finite difference method, which is based on the Richardson extrapolation technique and an operator interpolation scheme for the one and two dimensional steady convection diffusion equations to the three dimensional case. Firstly, we employ a fourth order compact difference scheme to get the fourth order accurate solution on th...