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  5. Using the Broyden method in computer modeling of transients

Using the Broyden method in computer modeling of transients

I. Husarova, O. Solovj’ev
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Description: The article substantiates the relevance of modeling non-stationary non-isothermal gas flow regimes in a pipeline section. The analysis of existing works on modeling gas flow regimes, where authors of these articles use various methods for solving a system of differential equations in partial derivatives of hyperbolic type, obtained from the general equations of gas dynamics, is carried out. A mathematical model of a non-stationary non-isothermal gas flow regime along a pipeline section has been developed. The finite difference method with implicit finite difference grid was applied. It becomes necessary to solve a nonlinear system of finite difference equations after applying this method. Newton and Broyden methods were used to solve this system of nonlinear equations. Newton's method has a quadratic convergence rate, but it requires calculating the Jacobi matrix for each iteration. The Broyden method has a superlinear rate of convergence, but it was promising to reduce the total time to obtain a numerical solution to the problem of modeling the transition regime by using the approximation of the Jacobi matrix. A numerical experiment has been carried out for the task of connecting a large consumer to a gas flow system. A comparison is made for the effectiveness in implementation of Newton and Broyden methods in the context of the given task. It is shown that the Newton method is the most effective for solving the problem. Although both methods can later be used in the application of other finite-difference grid or in the simulation of linear sections of the gas transmission system. The information from this study can be used to potentially improve methods for modeling non-stationary non-isothermal gas mode in a gas transportation pipe network in order to increase the efficiency of decisions made in emergency situations.


Keywords: pipeline section, non-stationary non-isothermal gas flow regime, mathematical model, system of differential equations, finite difference method in partial derivatives, system of nonlinear equations, Newton method, Broyden method

References

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Reference:
 Husarova, I.H. and Soloviov, O.M. (2019), “Zastosuvannia metodu Broidena pry kompiuternomu modeliuvanni perekhidnykh rezhymiv techii hazu” [Using the Broyden method in computer modeling of transients], Information Processing Systems, Vol. 2(157), pp. 33-39. https://doi.org/10.30748/soi.2019.157.04.