E.F.F. Botta

The electric potential of a macromolecule in a solvent: A fundamental approach

Abstract A general numerical method is presented to compute the electric potential for a macromolecule of arbitrary shape in a solvent with nonzero ionic strength. The model is based on a continuum description of the dielectric and screening properties of the system, which consists of a bounded internal region with discrete charges and an infinite external region. The potential obeys the Poisson equation in the internal region and the linearized Poisson-Boltzmann equation in the external region, coupled through appropriate boundary conditions. It is shown how this three-dimensional problem can be presented as a pair of coupled integral equations for the potential and the normal component of the electric field at the dielectric interface. These equations can be solved by a straightforward application of boundary element techniques. The solution involves the decomposition of a matrix that depends only on the geometry of the surface and not on the positions of the charges. With this approach the number of unknowns is reduced by an order of magnitude with respect to the usual finite difference methods. Special attention is given to the numerical inaccuracies resulting from charges which are located close to the interface; an adapted formulation is given for that case. The method is tested both for a spherical geometry, for which an exact solution is available, and for a realistic problem, for which a finite difference solution and experimental verification is available. The latter concerns the shift in acid strength (pK-values) of histidines in the copper-containing protein azurin on oxidation of the copper, for various values of the ionic strength. A general method is given to triangulate a macromolecular surface. The possibility is discussed to use the method presented here for a correct treatment of long-range electrostatic interactions in simulations of solvated macromolecules, which form an essential part of correct potentials of mean force.

Matrix Renumbering ILU: An Effective Algebraic Multilevel ILU Preconditioner for Sparse Matrices

In this paper a multilevel-like ILU preconditioner is introduced. The ILU factorization generates its own ordering during the elimination process. Both ordering and dropping depend on the size of the entries. The method can handle structured and unstructured problems. Results are presented for some important classes of matrices and for several well-known test examples. The results illustrate the efficiency of the method and show in several cases near grid independent convergence.

Nested grids ILU-decomposition (NGILU)

A preconditioning technique is described which shows, in many cases, grid-independent convergence. This technique only requires an ordering of the unknowns based on the different levels of multigrid, and an incomplete LU-decomposition based on a drop tolerance. The method is demonstrated on a variety of well-known elliptic test problems including strongly varying coefficients, advective terms and grid refinement.

On Local Relaxation Methods and Their Application to Convection-Diffusion Equations

Abstract This paper discusses local relaxation (LR) methods which can be regarded as generalizations of the successive overrelaxation (SOR) method. The difference is that within an LR method the relaxation factor is allowed to vary from equation to equation. A number of existing methods are found to be in fact special LR methods. Moreover, based on SOR theory, a new LR method is developed. The performance of LR methods is illustrated by applying them to central difference approximations of convection-diffusion equations. It is found that equations with small diffusion coefficients can be handled without difficulty. For equations with strongly varying coefficients, and for nonlinear equations, a properly selected LR method can be significantly more efficient than the optimum SOR method. As a special example, a 16 × 16 driven cavity problem for a Reynolds number of 10 6 can be solved in just a few seconds on a modern computer.

An efficient code to compute non-parallel steady flows and their linear stability

Abstract A simple, fast and efficient algorithm to compute steady non-parallel flows and their linear stability in parameter space is described. The pseudo-arclength continuation method is used to trace branches of steady states as one of the parameters is varied. To determine the linear stability of each state computed, a generalized eigenvalue problem of large order is solved. Only a prescribed number of eigenvalues, those closest to the imaginary axis, are calculated by a combination of a complex mapping and the Simultaneous Iteration Technique. The underlying linear systems are solved with preconditioned Bi-CGSTAB. It is shown that it is possible to deal efficiently with (discretized) problems with O (10 5 ) degrees of freedom. As an application, the bifurcation structure of steady two-dimensional Rayleigh-Benard flows in large rectangular containers (up to aspect ratio 20) is computed. We show how the results connect up with those obtained with weakly nonlinear theory and extend these into the nonlinear regime. Main aim is to investigate whether pattern selection occurs through the occurrence of saddle node bifurcations creating intervals of unique steady states. It turns out that these intervals do not exist; multiple stable states continue to exist at large aspect ratio over a large range of Rayleigh numbers. In addition, the bifurcation structure provides no answer why the ‘preferred’ wavelength increases with increasing Rayleigh number, as observed in experiments.

A modified SOR method for the poisson equation in unsteady free-surface flow calculations

Convergence difficulties that sometimes occur if the successive overrelaxation (SOR) method is applied to the Poisson equation on a region with irregular free boundaries are analyzed. It is shown that these difficulties are related to the treatment of the free boundaries and caused by the appearance of complex eigenvalues in the system of discrete equations, when standard centered differences are used. After a modification of this system of equations such that the complex eigenvalues become small, a modified SOR method is presented where two relaxation factors are used alternately. The method leads to fast convergence without requiring specific information about the complex eigenvalues.

How fast the Laplace equation was solved in 1995

On the occasion of the third centenary of the appointment of Johann Bernoulli at the University of Groningen, a number of linear systems solvers for some Laplace-like equations have been compared during a one-day workshop. CPU times of several advanced solvers measured on the same computer (an HP-755 workstation) are presented, which makes it possible to draw clear conclusions about the performance of these solvers.

Grid-independent convergence based on preconditioning techniques

Today numerical calculations are no longer restricted to a class of simple problems, but cope with complicated simulations and complex geometries. In many situations the accuracy of the numerical solution is determined by the limited amount of computer power and memory. Therefore much attention has been given to the development of numerical methods for solving the large sparse system of equations Ax = b obtained by discretising some partial differential equation. Since direct methods require much computer storage and CPU-time, a large variety of iterative methods has been derived. In this paper we will focus on iterative methods like MICCG and algebraic multigrid. Gustafsson [1] has shown that for several problems the CPU-time using MICCG is O(N 5/4) in 2 dimensions and O(N 7/6) for 3D-problems, where N is the total number of unknowns. Multigrid methods perform even better and for a large class of problems they have an optimal order of convergence: the amount of work and storage is proportial to the number of unknowns N. However, due to the required proper smoothers and the restriction and prolongation operators at each level, the implementation of multigrid for practical problems is much more complicated than that of MICCG. Here we look for a combination of these properties: an incomplete LU-decomposition such that the preconditioned system can be solved with the optimal computational complexity O(N) by a conjugate gradient-like method. The basic idea behind this preconditioning technique is the same as in multigrid methods. In Section 2 a preconditioning technique is described which uses a partition of the unknowns based on the sequence of grids in multigrid.

The Numerical Solution of the Navier-Stokes Equations for Laminar, Incompressible Flow past a Parabolic Cylinder

SummaryThe numerical method of solution of van de Vooren and Dijkstra [1] for the semi-infinite flat plate has been extended to the case of the parabolic cylinder. Results are presented for the skin friction, the friction drag, the pressure and the pressure drag. The drag coefficients have been checked by means of an application of the momentum theorem.

Fluid flow induced by a rotating disk of finite radius

The boundary-layer equations outside a rotating disk of radius a have been solved. It is shown that it is unnecessary to take special precautions for the sudden change in boundary conditions at the edge of the disk except if one is interested in the flow at distances which are smaller than about 10−3a from the edge. The behaviour of the flow at large distances from the disk is investigated analytically with results which are confirmed by the numerical computations.