Nabil Nassif

On the numerical solution of schroedinger's radial equation☆

Abstract Higher-order difference schemes are considered for the numerical solution of Schroedingers radial equation. They are a family of difference equations which are extensions of the well-known Numerov difference equation and give highly convergent approximate solutions, the least being O(h6) compared to O(h4) in the Numerov equation. An algorithm to find eigenvalues and eigenfunctions using one-directional “shooting” is discussed. The stability and convergence of these schemes are also discussed. An example and numerical results are given, and the order of convergence which is estimated from the results is found to be close to the theoretical value.

Adaptive time-step with anisotropic meshing for incompressible flows

This paper presents a method of combining anisotropic mesh adaptation and adaptive time-stepping for Computational Fluid Dynamics (CFD). First, we recall important features of the anisotropic meshing approach using a posteriori estimates relying on the length distribution tensor approach and the associated edge based error analysis. Then we extend the proposed technique to contain adaptive time advancing based on a newly developed time error estimator. The objective of this paper is to show that the combination of time and space anisotropic adaptations with highly stretched elements can be used to compute high Reynolds number flows within reasonable computational and storage costs. In particular, it will be shown that boundary layers, flow detachments and all vortices are well captured automatically by the mesh. The time-step is controlled by the interpolation error and preserves the accuracy of the mesh adapted solution. A Variational MultiScale (VMS) method is employed for the discretization of the Navier-Stokes equations. Numerical solutions of some benchmark problems demonstrate the applicability of the proposed space-time error estimator. An important feature of the proposed method is its conceptual and computational simplicity as it only requires from the user a number of nodes according to which the mesh and the time-steps are automatically adapted.

Sufficient Conditions for Converging Drift-Diffusion Discrete Systems. Application to The Finite Element Method

In this paper we generalize the abstract results of Mock and Marcowich [13, 12] for convergence of discrete Van Roosbroeck systems [12, 13, 17], to the case when the solutions are typically in W1,4-e and not in H2. These conditions are verified on finite element discretizations. Error estimates are derived when the solution is unique. Due to the singularity at the flat angles, these estimates in the H1 norm are only O(h1/2). The techniques that are presented are broad and may be applied to other type of discretizations.

Determination of the Mechanical Properties of a Solid Elastic Medium from a Seismic Wave Propagation Using Two Statistical Estimators

In this paper, we study an inverse problem consisting in the determination of the mechanical properties of a layered solid elastic medium in contact with a fluid medium by measuring the variation of the pressure in the fluid while propagating a seismic/acoustic wave. The estimation of mechanical parameters of the solid is obtained from the simulation of a seismic wave propagation model governed by a system of partial differential equations. Two stochastic methods, Markov chain Monte Carlo with an accelerated version and simultaneous perturbation stochastic approximation, are implemented and compared with respect to cost and accuracy.

Sliced-Time computations with re-scaling for blowing-up solutions to initial value differential equations

In this paper, we present a new approach to simulate time-dependent initial value differential equations which solutions have a common property of blowing-up in a finite time. For that purpose, we introduce the concept of “sliced-time computations”, whereby, a sequence of time intervals (slices) {[Tn−1, Tn]| n≥1} is defined on the basis of a change of variables (re-scaling), allowing the generation of computational models that share symbolically or numerically “similarity” criteria. One of these properties is to impose that the re-scaled solution computed on each slice do not exceed a well-defined cut-off value (or threshold) S. In this work we provide fundamental elements of the method, illustrated on a scalar ordinary differential equation y′ = f(y) where f(y) verifies

On the existence and uniqueness of solutions for a class of a quasi-variational inequalities to solve reverse-biased semi-conductor devices

\int_0^\infty {f(y)dy} < \infty

A Simulation Model for the Physiological Tick Life Cycle

. Numerical results on various ordinary and partial differential equations are available in [7], some of which will be presented in this paper.

Introduction to computational linear algebra

SummaryThis paper discusses existence and uniqueness for solutions of quasi-variational inequalities where the obstacle function w=M(z) satisfies an elliptic equation of the form ∇-exp [kz] ∇ exp[−kw]=0. Verifying Joly-Mosco hypotheses for existence appear to depend on an a-priori estimate on ‖∇w‖∞. Thus it was possible to obtain existence and uniqueness for certain cases of the applied biased potentials.

An Adaptive Parallel-in-Time Method with Application to a Membrane Problem

In this paper and following an approach used by two of the authors in (Nassif, N.R., Sheaib, D. (2009) On spectral methods for scalar aged-structured population models.) [5], we present a mathematical model for the tick life cycle based on the McKendrick Partial Differential Equation (PDE). Putting this model using a semi-variational formulation, we derive a Petrov–Galerkin approximation to the solution of the McKendrick PDE, using finite element semi-discretizations that lead to a system of ordinary differential equations in time which computations are carried out using an Euler semi-implicit scheme. The resulting simulations allow us to investigate and understand the dynamics of tick populations. Numerical results are presented illustrating in a realistic way the basic features of the computational model solutions.

Determination and sensitivity analysis of the seismic velocity of a shallow layer from refraction traveltimes measures

Basic Linear Algebra Subprograms: BLAS An Introductory Example Matrix Notations IEEE Floating Point Systems and Computer Arithmetic Vector-Vector Operations: Level-1 BLAS Matrix-Vector Operations: Level-2 BLAS Matrix-Matrix Operations: Level-3 BLAS Sparse Matrices: Storage and Associated Operations Basic Concepts for Matrix Computations Vector Norms Complements on Square Matrices Rectangular Matrices: Ranks and Singular Values Matrix Norms Gauss Elimination and LU Decompositions of Matrices Special Matrices for LU Decomposition Gauss Transforms Naive LU Decomposition for a Square Matrix with Principal Minor Property (pmp) Gauss Reduction with Partial Pivoting: PLU Decompositions MATLAB Commands Related to the LU Decomposition Condition Number of a Square Matrix Orthogonal Factorizations and Linear Least Squares Problems Formulation of Least Squares Problems: Regression Analysis Existence of Solutions Using Quadratic Forms Existence of Solutions through Matrix Pseudo-Inverse The QR Factorization Theorem Gram-Schmidt Orthogonalization: Classical, Modified, and Block Solving the Least Squares Problem with the QR Decomposition Householder QR with Column Pivoting MATLAB Implementations Algorithms for the Eigenvalue Problem Basic Principles QR Method for a Non-Symmetric Matrix Algorithms for Symmetric Matrices Methods for Large Size Matrices Singular Value Decomposition Iterative Methods for Systems of Linear Equations Stationary Methods Krylov Methods Method of Steepest Descent for spd Matrices Conjugate Gradient Method (CG) for spd Matrices The Generalized Minimal Residual Method The Bi-Conjugate Gradient Method Preconditioning Issues Sparse Systems to Solve Poisson Differential Equations Poisson Differential Equations The Path to Poisson Solvers Finite Differences for Poisson-Dirichlet Problems Variational Formulations One-Dimensional Finite-Element Discretizations Bibliography Index Exercises and Computer Exercises appear at the end of each chapter.