Lagged diffusivity fixed point iteration for solving steady-state reaction diffusion problems

The paper concerns with the computational algorithms for a steady-state reaction diffusion problem. A lagged diffusivity iterative algorithm is proposed for solving resulting system of quasilinear equations from a finite difference discretization. The convergence of the algorithm is discussed and the numerical results show the efficiency of this algorithm

On the Lagged Diffusivity Method for the solution of nonlinear finite difference systems

In this paper, we extend the analysis of the Lagged Diffusivity Method for nonlinear, non-steady reaction-convection-diffusion equations. In particular, we describe how the method can be used to solve the systems arising from different discretization schemes, recalling some results on the convergence of the method itself. Moreover, we also analyze the behavior of the method in case of problems presenting boundary layers or blow-up solutions

A nonlinearity lagging for the solution of nonlinear steady state reaction diffusion problems

This paper concerns with the analysis of the iterative procedure for the solution of a nonlinear reaction diffusion equation at the steady state in a two dimensional bounded domain supplemented by suitable boundary conditions. This procedure, called Lagged Diffusivity Functional Iteration (LDFI)-procedure, computes the solution by "lagging'' the diffusion term. A model problem is considered and a finite difference discretization for that model problem is described.Furthermore, properties of the finite difference operator are proved. Then, sufficient conditions for the convergence of the LDFI-procedure are given. At each stage of the LDFI-procedure a weakly nonlinearalgebraic system has to be solved and the simplified Newton-Arithmetic Mean method is used. This method is particularly well suited for implementation on parallel computers.Numerical studies show the efficiency, for different test functions, of the LDFI-procedure combined with the simplified Newton-Arithmetic Mean method. Better results are obtained when in the reaction diffusion equation also a convection term is present

the arithmetic mean solver in lagged diffusivity method for nonlinear diffusion equations

Th is paper deals with the solution of nonlinear system arising fro m finite difference discretization of nonlinear diffusion convection equations by the lagged diffusivity functional iteration method co mbined with d ifferent inner iterative solvers. The analysis of the whole procedure with the splitt ing methods of the Arith met ic Mean (AM) and of the Alternating Group Exp licit (A GE) has been developed. A comparison in terms of number of iterations has been done with the BiCG-STA B algorith m. So me nu merical experiments have been carried out and they seem to show the effectiveness of the lagged diffusivity procedure with the Arithmet ic Mean method as inner solver

ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing

We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the H−1-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation

Spray combustion experiments and numerical predictions

The next generation of commercial aircraft will include turbofan engines with performance significantly better than those in the current fleet. Control of particulate and gaseous emissions will also be an integral part of the engine design criteria. These performance and emission requirements present a technical challenge for the combustor: control of the fuel and air mixing and control of the local stoichiometry will have to be maintained much more rigorously than with combustors in current production. A better understanding of the flow physics of liquid fuel spray combustion is necessary. This paper describes recent experiments on spray combustion where detailed measurements of the spray characteristics were made, including local drop-size distributions and velocities. Also, an advanced combustor CFD code has been under development and predictions from this code are compared with experimental results. Studies such as these will provide information to the advanced combustor designer on fuel spray quality and mixing effectiveness. Validation of new fast, robust, and efficient CFD codes will also enable the combustor designer to use them as additional design tools for optimization of combustor concepts for the next generation of aircraft engines

Linear Diffusion Acceleration for Neutron Transport Problems

Nuclear engineers are interested in solutions of the Neutron Transport Equation (NTE), with the goal of improving the safety and efficiency of reactors and critical nuclear systems. Complex simulations are used to obtain detailed solutions of the NTE, and can require immense computational resources to execute. A variety of methods have been developed to ease the computational burden of simulating full-scale, whole-core reactor problems. Among these is transport acceleration, which improves the convergence rate of iterative transport calculations. In addition to the use of acceleration methods, certain approximations are often made when solving the NTE. The 2D/1D approximation is used to generate a 3D solution of the NTE by iteratively solving coupled 2D radial and 1D axial equations. This method is one of the foundational techniques used in the neutronics code MPACT. Also, the Transport-Corrected P0 (TCP0) approximation for neutron scattering is often used in reactor analysis codes to simplify higher-order scattering physics. Unfortunately, both of these approximations allow for non-positive flux solutions of the NTE. More importantly, some spatial discretizations of the NTE also permit negative solutions. Under certain conditions, this can cause instability for nonlinear acceleration methods such as Coarse Mesh Finite Difference (CMFD). In this thesis, we propose a novel acceleration scheme called Linear Diffusion Acceleration (LDA) that does not possess the nonlinearities present in CMFD. This thesis work presents LDA as an alternative acceleration scheme to CMFD. As the name suggests, the LDA method is linear with respect to the scalar flux. Therefore, LDA is not susceptible to the same nonlinear modes of numerical failure as CMFD. In addition, LDA is shown to possess similar convergence properties as CMFD for practical problems that have no negative scalar fluxes. Transport acceleration with LDA allows for the use of some of the aforementioned approximations, in which the positivity of the scalar flux is not guaranteed. Fourier analysis of CMFD and LDA is performed to compare the theoretical convergence rates of the two methods for simple, spatially-heterogeneous problems. In addition, simple and practical case studies are presented in which CMFD fails due to nonlinearity. For these cases, LDA is shown to retain stability. Certain other advantages of LDA, which are a consequence of its mathematical structure, are also discussed.PHDNuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169898/1/zdodson_1.pd