Zeka Mazhar

A PROCEDURE FOR THE TREATMENT OF THE VELOCITY-PRESSURE COUPLING PROBLEM IN INCOMPRESSIBLE FLUID FLOW

A new block implicit procedure (BID) is introduced which utilizes a simple incomplete decomposition of the matrix resulting from the discretization of the momentum and mass conservation equations for incompressible fluid flow problems. In contrast to the conventional methods, the new method is not of the segregated type, and does not require an explicit equation for pressure. The complete, coupled block system is solved in its primitive form. In this way, mass and momentum conservation are satisfied simultaneously at all grid points, while pressure is calculated implicitly. Only a couple of overall iterations are required for the treatment of the nonlinearities of the problem. Tests show that the new procedure converges fast for any E value (in an E-factor formulation), and therefore virtually the E-factor formulation is not necessary.

On asymptotical behavior of solution of Riccati equation arising in linear filtering with shifted noises

In this paper we consider a linear signal system together with the two linear observation systems. The observation systems differ from each other by the noise processes. The noise of one of them is a constant shift in time of the signal noise. In the other one the shift is neglected. Respectively, we consider two best estimates of the signal corresponding to two different observation systems. The following problem is investigated: whether the effect of the shift on the best estimate becomes negligible as time increases. This leads to a comparison of the asymptotical behaviors of the solutions of respective Riccati equations. It is proved that under a certain relation between the parameters, the effect of the shift is not negligible.

Boundary Value Problems Arising in Kalman Filtering

The classic Kalman filtering equations for independent and correlated white noises are ordinary differential equations (deterministic or stochastic) with the respective initial conditions. Changing the noise processes by taking them to be more realistic wide band noises or delayed white noises creates challenging partial differential equations with initial and boundary conditions. In this paper, we are aimed to give a survey of this connection between Kalman filtering and boundary value problems, bringing them into the attention of mathematicians as well as engineers dealing with Kalman filtering and boundary value problems.

On the preconditioning conjugate gradient method for the solution of 9-point elliptic difference equations

The Preconditioned Conjugate Gradient (PCG) method is implemented to solve the system with 9 diagonal entries generated from a 9-point finite difference approximation of the self-adjoint elliptic partial differential equation using an incomplete matrix decomposition. A simpler matrix decomposition for such a linear system is also proposed with a main advantage that it preserves the symmetry of the original matrix, and is easy to implement. Results of the numerical experiments and comparison with other iterative methods are presented.

A novel fully implicit block coupled solution strategy for the ultimate treatment of the velocity–pressure coupling problem in incompressible fluid flow

ABSTRACT Two new extremely robust, fully implicit coupled solution procedures (FICS-1 and FICS-2) are presented for the ultimate solution of the notorious velocity–pressure coupling problem arising in incompressible fluid flow problems. Based on a previous idea of the author, the algebraic coupled system of equations resulting from the discretization of the momentum and mass conservation equations is taken in its primitive form. A special incomplete decomposition technique is applied to the block matrix of the algebraic system, requiring only two defect vectors in the defect matrix. With the new mechanism applied, the mass and momentum conservations are satisfied simultaneously at all points of the solution region and at each step of the solution process. In this way, the effect of any change in a dependent variable is sensed immediately at all of the points in the solution region. Contrary to the almost outdated segregated-type approaches, the new procedures do not require any explicit equation for pressure, so that the laborious tasks of formulation and solution of any Poisson-type equations are avoided. The procedures are not pressure-based. They are very simple to formulate and implement. The strong coupling preserved and the full implicitness of the algorithm involved helps in treating the nonlinearities most efficiently through a couple of overall block solutions. Tests on the two procedures presented in this work show that up to at least 20 times faster convergence rates can be achieved, compared with any of the segregated-type procedures, which accounts for a 95% reduction in computing time. The procedures may converge even when no relaxation is applied, but they may converge faster if some optimal relaxation is applied. With these properties, the procedures presented seem to provide a breakthrough in the area of computational fluid mechanics.

Finite Difference Formulations

This chapter is devoted to introduce, in full, the discretization process for the governing equations, utilizing some enhanced versions of the approximations derived in Chap. 2.

The Solution Procedure: Block Incomplete Decomposition

This is the main chapter in which the fully implicit clock-coupled solution strategy is introduced. First a discussion on the advantages of the block system are presented, followed by a general incomplete decomposition process. Then, the technique is applied to the block-coupled system and the relevant equations are derived. Based on the new technique, possible alternatives are presented leading to a family of procedures. The storage requirements and complexity of the algorithms are also presented.

Fully implicit, coupled procedures in computational fluid dynamics : an engineer's resource book

Preface -- Chapter 1 Introduction -- 1.1 Scope of the Book -- 1.2 Outline of the Book -- Chapter 2 Preliminaries -- 2.1 Quadratic Interpolation -- 2.2 Approximations using Lagrangian Polynomial -- 2.3 Approximations using Taylor Series -- 2.4 General Elimination Technique for Linear Systems -- 2.5 Solution Techniques for Special Linear Systems -- Chapter 3 Governing Differential Equations -- 3.1 Governing Equations -- 3.2 Characteristics of the Governing Equations -- 3.3 The Velocity-Pressure Coupling Problem -- Chapter 4 Finite Difference Formulations -- 4.1 Manipulation of the Momentum Equations -- 4.2 Grid Arrangement for the Solution -- 4.3 Profile Assumptions for the Discretizations -- 4.4 Discretization of the Governing Equations -- 4.5 A Discussion on the Profile Assumptions -- Chapter 5 Preparations For Solution -- 5.1 The Solution Region -- 5.2 Boundary Conditions -- 5.3 Incorporating Relaxation -- Chapter 6 Assembling The Discretized Equations Into A Block Matrix System -- 6.1 The Numbering Scheme -- 6.2 Construction of the Block Matrix System -- 6.3. Disadvantages of the Block Matrix -- Chapter 7 The Solution Procedure: Block Incomplete Decomposition -- 7.1 Properties and Advantages of the Block Matrix -- 7.2 General Incomplete Decomposition -- 7.3 An Incomplete Decomposition of the Block System (BIP) -- 7.4 The Block Solution Procedure -- 7.5 Complete Solution of the Flow Field -- 7.6 A Family of Procedures: BIPEN, FICS-1, FICS-2 -- 7.7 Storage Requirements and Complexity -- 7.8 The Simplest Case (Simple Implicit Coupled Solution- SICS) -- Chapter 8 Applications And Testing -- 8.1 Benchmark Fluid Flow Problems -- 8.2 Testing Criteria -- 8.3 Performance Analysis and Comparisons -- 8.4 A Discussion of the Mechanism of the Procedures -- 8.5 Comparison with the Segregated-Type Procedures -- 8.6 Convergence Characteristics and Performances of SICS and SIMPLER: A Relative Comparison -- Chapter 9: Special Cases -- 9.1 Time-Dependent Problems -- 9.2 Stoke’s Flow Equations -- 9.3 Turbulent Flows and Heat Transfer -- 9.4 Adaptation to Existing Codes -- 9.5 Three-Dimensional Problems -- Chapter 10 Concluding Remarks -- Appendix A: A Critical Survey of Literature—an Adventure Into Perfection -- Appendix B: Segregated Solution Procedures: Simple And Simpler -- Appendix C: Fortran Subroutines — Blocksolfics2 and Blocksolsics -- References -- Nomenclature -- Index.

Governing Differential Equations

In this chapter, the set of basic differential equations governing field problems is introduced. This is followed by a review of the special features of these equations and the difficulties involved in their solutions. Special emphasis is given to the crucial velocity-pressure coupling problem by verifying the need for the new viewpoint applied in the proposed procedures.

Applications and Testing

In this chapter, the procedures presented in the previous chapters are applied on certain, well established benchmark problems. Convergence, stability and cost comparisons are provided in detail for problems of various sizes. Comparison of one of the selected procedures with a classical segregated one (SIMPLER) is also presented. A discussion on the mechanism of the procedures both numerically and physically is provided to answer the question of why the procedures converge so fast and they are so stable.