CHAPTER ONE

INTRODUCTION

BACKGROUND OF THE STUDY

Modern turbomachinery design highly relies on computational fluid dynamics(CFD) for analyzing its performance by solving the steady and unsteady Euler equations or Reynolds-averaged Navier-Stokes(RANS) equations. In numerical analysis, the robustness and the convergence rate of involved solution methods are vitally important for design engineers, as they largely dictate the accomplishment of a design task. Typically, there are two kinds of numerical methods for pseudo-time integration: explicit methods and implicit methods. The representative explicit methods are the multi-stage Runge-Kutta methods (Jameson et al., 1981). Their advantages lie in the low memory consumption, low CPU cost per iteration, and easy implementation. Because of this, they have been very popular in the CFD community. The biggest disadvantage of the explicit methods is their conditional stability. To achieve a stable solution, the time step is limited by the Courant-Fredrichs-Levy (CFL) stability condition. This time step limit is manifested by flow field analysis at off-design conditions and harmonic balance analysis with a large maximum grid-reduced frequency (Hall et al., 2013).

            The implicit methods other hand, often have higher memory consumption and higher CPU cost per iteration, and are much more difficult to implement, when compared with the explicit methods. The biggest advantage of the implicit methods is better stability. Thus a much bigger time step can be allowed in an analysis leading to reduced overall time cost. The extended stability of an implicit method depends on the approximation of the system Jacobian matrix. A better approximation generally leads to higher stability but incurs more implementation complexity and computing resources. The Lower-Upper Symmetric Gauss-Seidel(LU-SGS) method is a popular implicit method, as it represents a good compromise between the advantages and the disadvantages of implicit methods. In a steady analysis, its maximum allowable Courant number is theoretically infinite, though anything more than 1,000 does not make any meaningful difference to the convergence rate of a solution. The LU-SGS method can also be used as a preconditioner for the Runge-Kutta schemes to achieve a better convergence rate and stability (Rossow, 2006). When it comes to the harmonic balance equation system, the implementation of the LU-SGS method is quite difficult because solutions at different time instants are coupled. To reduce the programming complexity, the Jacobian matrix of the time spectral source term is excluded from the LU-SGS method and it is implicitly integrated using a block Jacobi(BJ) method (Wang and Huang, 2017). Although numerical experiments demonstrate that this decoupled implicit treatment of the time spectral source term can increase solution stability, a thorough stability analysis of this numerical method has not yet been reported in the open literature. There are two main methods for assessing the stability and the convergence properties of a given numerical scheme. One is well known as the von Neumann method developed by Charney et al. (1950), which has been widely used to analyze the stability of a linear equation system with constant coefficients and periodic boundary conditions. A von Neumann analysis shows that the first-order forward temporal integration scheme is unconditionally unstable for the linearized Burger’s equation in the harmonic balance form when a first-order upwind scheme is used for the spatial discretization (Hall et al., 2013). However, the stability of an equation system can be impacted by inlet, outlet, slip wall boundary conditions, and so on. The effect of these boundary conditions on the stability of an equation system can only be analyzed by using the matrix or eigen-analysis method. The Laplace equation and a two-equation model hyperbolic system were examined using the matrix method with various boundary conditions and stretched grids (Roberts and Swanson, 2005). Eriksson and Rizzi (1985) investigated the solution behaviour of a center scheme for two-dimensional Euler equations. The beneficial effects of local time-step scaling and artificial dissipations were studied using an eigen-analysis. This paper presents the stability analysis results of solution methods for steady and harmonic balance form Euler equations.

The solution methods involve a central scheme with blended second- and fourth-order artificial dissipations for the spatial discretization and a Runge-Kutta scheme for pseudo-time integration. The LU-SGS method and LU-SGS/BJ method are used as the implicit residual smoothing for the steady and harmonic balance solutions, respectively. Both the von Neumann and the matrix methods are used to analyze the stability of the involved solution methods. Based on the stability analysis results, the impact of the boundary conditions and the block Jacobi method on solution stability and convergence rate can be analyzed, providing insights on how to obtain a solution fast and robustly. Though the stability analysis is carried out based on the twodimensional Euler equations, the conclusions can be readily extended to the three-dimensional RANS equations as demonstrated by using a Laval nozzle and the NASA rotor 37.

STATEMENT OF THE PROBLEM

The problem statement for the stability analysis of matrix decomposition in solving linear systems (Ax=b) centers on evaluating how round-off errors, arising from finite-precision floating-point arithmetic, propagate through the decomposition process (LU,QR, Cholesky) and affect the accuracy of the computed solution x̂. Specifically, the goal is to determine if the decomposition algorithm is backward stable, meaning the calculated solution x̂ is the exact solution to a nearby problem  (A+E)x̂=b, where the perturbation E is small relative to A.

AIM AND OBJECTIVES OF THE STUDY

The study seeks to carry out a numerical stability analysis of matrix decomposition in solving linear systems. The objectives of the study are:

1. To use Gauss-Seidel Method in numerical stability analysis of matrix decomposition in solving linear syetems
2. To apply the Successive Overrelaxation Method in matrix decomposition in solving linear system
3. To determine the factors affecting numerical stability analysis of matrix decomposition in solving linear systems   
4. To compare various numerical stability analysis methods for solving linear systems
5. To identify the factors affecting numerical stability analysis methods for solving linear systems

SIGNIFICANCE OF THE STUDY

This study will be of immense benefit to other researchers who intend to know more on this topic and can also be used by non-researchers to build more on their work.

This study contributes to knowledge and could serve as a bench mark or guide for other work or study. It avails individuals with information about numerical stability analysis of matrix decomposition in solving linear systems

SCOPE OF THE STUDY

The study covers on numerical stability analysis of matrix decomposition in solving linear systems

DEFINITION OF TERMS

Matrix: A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns, serving as a fundamental tool in mathematics for representing linear equations, transformations, and data. Widely used in engineering, physics, and computer graphics, matrices enable complex calculations through defined operations like addition and multiplication

Linear systems: Linear systems are collections of two or more linear equations sharing common variables, representing straight lines, planes, or hyperplanes. They are characterized by proportionality and superposition, where the output is directly proportional to the input. Solutions can be unique, infinite, or non-existent (consistent/inconsistent)