Gear method (backward differentiation formulas) MASTER STUDENT ELECTRONIC SCIENCE AND ENGINEERING 2021/10/20 History & application First introduced by Charles F. Curtiss and Joseph o. Hirschfelder in 1952. Used for the solution of stiff differential equations Stiff systems can be exemplified by problems: of chemical kinetics nonstationary processes in complex electric circuits systems emerging while solving equations of heat conduction and diffusion movement of celestial bodies and satellites plasticity physics Ordinary differential equations (ODE) Types: soft, stiff, illconditioned and rapidly oscillating Variables systems may be of different orders or change within the interval of integration by orders of magnitude The modelling of a complex physical process speeds of local processes may vary significantly Stiff equations are equations where certain implicit methods perform better, than using classical explicit ones like Euler or Adams methods One of the major problems associated with the use almost of all the explicit methods lies in choosing the size of the integration step h, which provides the stability of the computational scheme Gear method implicit methods take the following form Where q determines the order of the method, the constants βi correspond to the chosen order of the method The construction of the multistep methods is based on the polynomial of the degree q. It requires the calculations of the q units of the initial values Y1, Y2, . . . , Yn The polynomial can be represented Some constants αi and βi in equation can take zero values. When β1 = = β2 = : : : = βq = 0, it is possible to construct backward differentiation formulas. Practical Comparison between the time variations of the presented numerical methods and experimental averaged bed temperatures. Comparison of discretization, accuracy, process of time, and numbers of steps for the following numerical methods: RKfixed, R45, and Gear BDW Advantages & Disadvantages + can obtain higher calculation precision and efficiency + can change step size automatically + validated especially for stiff differential equations + takes fewer calculation times in each step for solving implicit equations than most other numerical calculation methods - it requires at each time step the solution of a nonlinear system of equations References Katarzyna Zwarycz-Makles , and Dorota Majorkowska-Mech, 2018. Gear and Runge–Kutta Numerical Discretization Methods in Differential Equations of Adsorption in Adsorption Heat Pump. Y. X. Wang and J. M. Wen 2006. Gear Method for Solving Differential Equations of Gear Systems. M. Semenov, 2011. Analyzing the absolute stability region of implicit methods of solving ODEs. E. Suli, 2014. Numerical Solution of Ordinary Differential Equations. THANK YOU FOR ATTENTION!