An alternating order algorithm based on stages of Ceschino’s method

About the authors:

Evgeny A. Novikov, Head Researcher, Institute of Computational Modeling, Siberian Branch of Russian Academy of Science, Dr. Phys. and Math. Sci., Professor
Alexander A. Zakharov, Dr. Sci (Tech.), Professor, Secure Smart City Information Technologies Department, University of Tyumen; a.a.zakharov@utmn.ru

Abstract:

This paper investigates methods for the numerical solution of stiff problems with large dimension. Using the estimation of the largest eigenvalue of the Jacobi matrix, we create an inequality in order to control the stability of a Cescino numerical scheme with second-order accuracy. In order to integrate a variable step, we propose a formula which allows one to predict the next step in time. On the basis of this formula, we develop a method with first-order accuracy with extended stability range. This method allows stabilized behaviour of step integration at the stage of solution exactly where stability plays a crucial role. This makes it possible to remove restrictions on the possibility of using explicit methods for solving stiff problems. We formulate an algorithm for the numerical solution of stiff problems of variable order, which uses the irregular step in time with an additional control of stability of the numerical integration scheme. This paper demonstrates solutions of stiff problems associated with numerical simulations of ethane pyrolysis, which confirm an increase in efficiency due to the use of variable order.

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