A numerical method for optimizing the gas flow field in a centrifuge

The performance of a centrifuge which is used in the uranium enrichment industry depends strongly on the gas flow field inside the device. The flow, governed by linearized hydrodynamic equations, is activated by proper boundary conditions which are controllable functions. A numerical method is developed to determine these functions on the boundaries in order to optimize the separative performance. By expanding the control functions in a finite base, the problem is reduced to minimizing a function with respect to a finite number of variables and solved by the simplex method.     

The Local Artificial Boundary Conditions for Numerical Simulations of the Flow Around a Submerged Body

We consider in this work the numerical approximations of the two-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface. Several vertical segments are introduced as the upstream and the downstream artificial boundaries, where a sequence of high-order local artificial boundary conditions are proposed. Then the original problem is solved in a finite computational domain, which is equivalent to a variational problem. The numerical approximations for the original problem are obtained by solving the variational problem with the finite element method. The numerical examples show that the artificial boundary conditions given in this work are very effective.     

Numerical solution of a nonlocal identification problem for nonlinear ion transport

A numerical method for the solution of a parameter identification problem in a nonlinear non-self-adjoint two-point boundary value problem with an additional nonlocal condition defining the parameter is presented. The equation arises in the modelling of an experiment known as chronoamperometry for the study of kinetics and mass-transfer in electrochemical events. The algorithm is based on the reformulation of the identification problem as a nonlinear fixed-point problem involving the concentration flux of the reduced species. The linearized boundary value problem is shown to have a unique solution with the unknown parameter uniquely determined by the flux. The linearized BVP is solved using finite differences and the fixed-point is found using the α-bisection method. The results of computational experiments are presented and their physical significance is discussed.     

Discrete radiation boundary conditions for a semi-infinite layer of fluid

The paper deals with the discrete formulation of radiation boundary conditions for a layer of fluid. The problem is examined with the help of the finite difference method. The proposed radiation boundary enables us to replace an infinite layer by a finite domain. The conditions ensure near equivalence between the infinite layer and the proposed finite model. The method is consistent itself and operates on a finite number of points. The results of numerical solutions are in good agreement with the results of analytical solutions of the problem.     

High-order accurate computations for unsteady aerodynamics

A high-order accurate finite difference scheme is used to perform numerical studies on the benefit of high-order methods. The main advantage of the present technique is the possibility to prove stability for the linearized Euler equations on a multi-block domain, including the boundary conditions. The result is a robust high-order scheme for realistic applications. Convergence studies are presented, verifying design order of accuracy and the superior efficiency of high-order methods for applications dominated by wave propagation. Furthermore, numerical computations of a more complex problem, a vortex-airfoil interaction, show that high-order methods are necessary to capture the significant flow features for transient problems and realistic grid resolutions. This methodology is easy to parallelize due to the multi-block capability. Indeed, we show that the speedup of our numerical method scales almost linearly with the number of processors.     

Finite deformation of a rotating orthotropic cylinder with linear elasticity

The effects of finite deformation upon a rotating, orthotropic cylinder with linear elasticity is investigated. The governing equation and boundary conditions form a non-linear two-point boundary-value problem. A numerical integration technique is used in conjunction with the related initial-value problem and Newton's method to obtain approximate solutions. The problem and method of solution provide an example for treatment of non-linear boundary conditions. The stresses induced by finite deformation and the influence of the degree of orthotropy are studied.     

Operator-splitting methods for the 2D convective Cahn–Hilliard equation

In this work, we present operator-splitting methods for the two-dimensional nonlinear fourth-order convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. The full problem is split into hyperbolic, nonlinear diffusion and linear fourth-order problems. We prove that the semi-discrete approximate solution obtained from the operator-splitting method converges to the weak solution. Numerical methods are then constructed to solve each sub equations sequentially. The hyperbolic conservation law is solved by efficient finite volume methods and dimensional splitting method, while the one-dimensional hyperbolic conservation laws are solved using front tracking algorithm. The front tracking method is based on the exact solution and hence has no stability restriction on the size of the time step. The nonlinear diffusion problem is solved by a linearized implicit finite volume method, which is unconditionally stable. The linear fourth-order equation is solved using a pseudo-spectral method, which is based on an exact solution. Finally, some numerical experiments are carried out to test the performance of the proposed numerical methods.     

A new numerical approach to a non-steady filtration problem

A free-boundary transient problem of seepage flow is studied from a numerical standpoint. From a suitable formulation of the problem in terms of variational inequality we introduce a new numerical approach of the implicit type and based on the finite element method. In this approach the problem is solved on a fixed region and the position of the free boundary is automatically found as part of the solution of the problem; so it is not necessary to solve a succession of problems with different positions of the free boundary. We prove stability and convergence for the approximate solution and we give several numerical results. Work supported by C. N. R. of Italy through the Laboratorio di Analisi Numerica of Pavia.     

Multilevel projection algorithm for solving obstacle problems

Obstacle problems are nonlinear free boundary problems and the computation of approximate solutions can be difficult and expensive. Little work has been done on effective numerical methods of such problems. This paper addresses some aspects of this issue. Discretizing the problem in a continuous piecewise linear finite element space gives a quadratic programming problem with inequality constraints. A new method, called the multilevel projection (MP) method, is established in this paper. The MP algorithm extends the multigrid method for linear equations to nonlinear obstacle problems. The convergence theorems of this method are also proved. A numerical example presented shows our error estimate is sharp and the MP algorithm is robust.     

Front-Tracking Finite Difference Methods for the Valuation of American Options

This paper is concerned with the numerical solution of the American option valuation problem formulated as a parabolic free boundary/initial value model. We introduce and analyze a front-tracking finite difference method and compare it with other commonly used techniques. The numerical experiments performed indicate that the front-tracking method considered is an efficient alternative for approximating simultaneously the option value and free boundary functions associated with the valuation problem.     

A time semi-exponentially fitted scheme for chemotaxis-growth models

In this work, we develop a new linearized implicit finite volume method for chemotaxis-growth models. First, we derive the scheme for a simplified chemotaxis model arising in embryology. The model consists of two coupled nonlinear PDEs: parabolic convection-diffusion equation with a logistic source term for the cell-density, and an elliptic reaction-diffusion equation for the chemical signal. The numerical approximation makes use of a standard finite volume scheme in space with a special treatment for the convection-diffusion fluxes which are approximated by the classical Il’in fluxes. For the time discretization, we introduce our linearized semi-exponentially fitted scheme. The paper gives a comparison between the proposed scheme and different versions of linearized backward Euler schemes. The existence and uniqueness of a numerical solution to the scheme and its convergence to a weak solution of the studied system are proved. In the last section, we present some numerical tests to show the performance of our method. Our numerical approach is then applied to a chemotaxis-growth model describing bacterial pattern formation.     

Analysis of finite element methods for the Brinkman problem

The parameter dependent Brinkman problem, covering a field of problems from the Darcy equations to the Stokes problem, is studied. A mathematical framework is introduced for analyzing the problem. Using this uniform a priori and a posteriori estimates for two families of finite element methods are proved. Nitsche’s method for imposing boundary conditions is discussed.     

On finite element approximation of general boundary value problems in nonlinear elasticity

This paper deals with the approximate solution of the general boundary value problem in nonlinear elasticity by the finite element displacement method. Under usual conditions which also guarantee the existence of locally unique solutions the quasi-optimal convergence inL ² andL ^∞ is shown for displacement fields and stresses. Furthermore a projective Newton method is considered which reduces the solution of the nonlinear continuous problem to the successive solution of a sequence of linearized problems of increasing dimension. It is proved that this procedure is well defined and also converges with quasi-optimal rates.     

Boundary conditions and estimates for the linearized Navier-Stokes equations on staggered grids

In this paper, we consider the linearized Navier-Stokes equations in two dimensions under specified boundary conditions. We study both the continuous case and a discretization using a second-order finite difference method on a staggered grid and derive estimates for both the analytic solution and the approximation on staggered grids. We present numerical experiments to verify our results.     

Multiscale methods for the valuation of American options with stochastic volatility

This paper deals with the efficient valuation of American options. We adopt Heston's approach for a model of stochastic volatility, leading to a generalized Black–Scholes equation called Heston's equation. Together with appropriate boundary conditions, this can be formulated as a parabolic boundary value problem with a free boundary, the optimal exercise price of the option. For its efficient numerical solution, we employ, among other multiscale methods, a monotone multigrid method based on linear finite elements in space and display corresponding numerical experiments.     

An inversion method for parabolic equations based on quasireversibility

This paper is concerned with a new method to solve a linearized inverse problem for one-dimensional parabolic equations. The inverse problem seeks to recover the subsurface absorption coefficient function based on the measurements obtained at the boundary. The method considers a temporal interval during which time dependent measurements are provided. It linearizes the working equation around the system response for a background medium. It is then possible to relate the inverse problem of interest to an ill-posed boundary value problem for a differential-integral equation, whose solution is obtained by the method of quasireversibility. This approach leads to an iterative method. A number of numerical results are presented which indicate that a close estimate of the unknown function can be obtained based on the boundary measurements only.     

基于有限元的一类stefan问题数值模型研究

该文给出基于有限元方法的一类一维stefan问题的数值求解过程及算法.模型的建立基于已知的相变界面和固定边界处测得的温度和热流.模型的精度通过与Neumann获得的解析解的比较而得到验证.文中所讨论的模型可以用于反Stefan问题中自由边界的实时跟踪或者控制.最后,比较了已有的有限元模型,给出了仿真结果.     

Convergence and instability in PCG methods for bordered systems

Bordered almost block diagonal systems arise from discretizing a linearized first-order system of n ordinary differential equations in a two-point boundary value problem with nonseparated boundary conditions. The discretization may use spline collocation, finite differences, or multiple shooting. After internal condensation, if necessary, the bordered almost block diagonal system reduces to a standard finite difference structure, which can be solved using a preconditioned conjugate gradient method based on a simple matrix splitting technique. This preconditioned conjugate gradient method is “guaranteed” to converge in at most 2n + 1 iterations. We exhibit a significant collection of two-point boundary value problems for which this preconditioned conjugate gradient method is unstable, and hence, convergence is not achieved.     

Continuum solvation model: Computation of electrostatic forces from numerical solutions to the Poisson-Boltzmann equation

A rigorous formulation of the solvation forces (first derivatives) associated with the electrostatic free energy calculated from numerical solutions of the linearized Poisson-Boltzmann equation on a discrete grid is described. The solvation forces are obtained from the formal solution of the linearized Poisson-Boltzmann equation written in terms of the Green function. An intermediate region for the solute-solvent dielectric boundary is introduced to yield a continuous solvation free energy and accurate solvation forces. A series of numerical tests show that the calculated forces agree extremely well with finite-difference derivatives of the solvation free energy. To gain a maximum efficiency, the nonpolar contribution to the free energy is expressed in terms of the discretized grid used for the electrostatic problem. The current treatment of solvation forces can be used to introduce the influence of a continuum solvation model in molecular mechanics calculations of large biological systems.     

A sixth order method for the integration of two point boundary value problems

A finite difference method for obtaining sixth order accurate approximation to the solution of the two-point linear boundary value problem is given. The convergence of the method is proved. Numerical results for a typical problem are tabulated and in each case the observed error is compared with its theoretical estimate. The numerical results are compared with those obtained from an earlier method of the author and the method of Noumerov.