Workshop on numerical methods for system engineering problems

The paper presents the comparative study on numerical methods of Euler method, Improved Euler method and fourth-order Runge-Kutta method for solving the engineering problems and applications. The three proposed methods are quite efficient and practically well suited for solving the unknown engineering problems. This paper aims to enhance the teaching and learning quality of teachers and students for various levels. At each point of the interval, the value of y is calculated and compared with its exact value at that point. The next interesting point is the observation of error from those methods. Error in the value of y is the difference between calculated and exact value. A mathematical equation which relates various functions with its derivatives is known as a differential equation. It is a popular field of mathematics because of its application to real-world problems. To calculate the exact values, the approximate values and the errors, the numerical tool such as MATLAB is appropriate for observing the results. This paper mainly concentrates on identifying the method which provides more accurate results. Then the analytical results and calculates their corresponding error were compared in details. The minimum error directly reflected to realize the best method from different numerical methods. According to the analyses from those three approaches, we observed that only the error is nominal for the fourth-order Runge-Kutta method.

Download Full-text

Convergence Stability of Through-Flow Analyses via the Implementation of the Secant and Bisection Iteration Schemes

Volume 4: Cycle Innovations; Industrial and Cogeneration; Manufacturing Materials and Metallurgy; Marine ◽

10.1115/gt2009-59760 ◽

2009 ◽

Author(s):

Vassilios Pachidis ◽

Ioannis Templalexis ◽

Pericles Pilidis

Keyword(s):

Numerical Methods ◽

Numerical Solution ◽

Numerical Scheme ◽

Linear Equations ◽

Original Form ◽

Through Flow ◽

Engineering Problems ◽

Non Linear ◽

Newton Raphson ◽

Non Linear Equations

One of the most frequently encountered problems in engineering is dealing with non-linear equations. For example, the solution of the full Radial Equilibrium Equation (REE) in Streamline Curvature (SLC) through-flow methods is a typical case of a scientific analysis associated with a complex mathematical problem that can not be handled analytically. Various schemes are used routinely in scientific studies for the numerical solution of mathematical problems. In simple cases, these methods can be applied in their original form with success. The Newton-Raphson for example is one such scheme, commonly employed in simple engineering problems that require an iterative solution. Frequently however, the analysis of more complex phenomena may fall beyond the range of applicability of ‘textbook’ numerical methods, and may demand the design of more dedicated algorithms for the mathematical solution of a specific problem. These algorithms can be empirical in nature, developed from scratch, or the combination of previously established techniques. In terms of robustness and efficiency, all these different schemes would have their own merits and shortcomings. The success or failure of the numerical scheme applied depends also on the limitations imposed by the physical characteristics of the computational platform used, as well as by the nature of the problem itself. The effects of these constraints need to be assessed and taken into account, so that they can be anticipated and controlled. This manuscript discusses the development, validation and deployment of a convergence algorithm for the fast, accurate and robust numerical solution of the non-linear equations of motion for two-dimensional flow fields. The algorithm is based on a hybrid scheme, combining the Secant and Bisection iteration methods. Although it was specifically developed to address the computational challenges presented by SLC-type of analyses, it can also be used in other engineering problems. The algorithm was developed to provide a mid-of-the-range option between the very efficient but notoriously unstable Newton-Raphson scheme and other more robust, but less efficient schemes, usually employing some sort of Dynamic Convergence Control (DCC). It was also developed to eliminate the large user intervention, usually required by standard numerical methods. This new numerical scheme was integrated into a compressor SLC software and was tested rigorously, particularly at compressor operating regimes traditionally exhibiting convergence difficulties (i.e. part-speed performance). The analysis showed that the algorithm could successfully reach a converged solution, equally robustly but much more efficiently compared to a hybrid Newton-Raphson scheme employing DCC. The performance of these two schemes, in terms of speed of execution, is presented here. Typical error histories and comparisons of simulated results against experimental are also presented in this manuscript for a particular case-study.

Download Full-text