Study of stiff differential equations of mathematical description of izomerization of pentane-hexane cut process

This paper provides a brief overview of the existing definitions of a stiff system of differential equations. Further, on the example of the accepted scheme of chemical transformations of the catalytic isomerization process of the pentane-hexane fraction, the stiffness of the system of differential equations was studied. In the course of the work, a method for studying the direct kinetic problem for stiffness is presented. In the Matlab software, the results of solving a system of differential equations by five methods (solvers) were compared. The given method can be tried for solving other problems of chemical kinetics.

A Class of Hybrid Multistep Block Methods with A–Stability for the Numerical Solution of Stiff Ordinary Differential Equations

Recently, block backward differentiation formulas (BBDFs) are used successfully for solving stiff differential equations. In this article, a class of hybrid block backward differentiation formulas (HBBDFs) methods that possessed A –stability are constructed by reformulating the BBDFs for the numerical solution of stiff ordinary differential equations (ODEs). The stability and convergence of the new method are investigated. The methods are found to be zero-stable and consistent, hence the method is convergent. Comparisons between the proposed method with exact solutions and existing methods of similar type show that the new extension of the BBDFs improved the stability with acceptable degree of accuracy.

A Seven-Step Block Multistep Method for the Solution of First Order Stiff Differential Equations

In this paper, a seven-step block method for the solution of first order initial value problem in ordinary differential equations based on collocation of the differential equation and interpolation of the approximate solution using power series have been formed. The method is found to be consistent and zero-stable which guarantees convergence. Finally, numerical examples are presented to illustrate the accuracy and effectiveness of the method. Keywords: Power series, Collocation, Interpolation, Block method, Stiff.

Study on Banded Implicit Runge–Kutta Methods for Solving Stiff Differential Equations

The implicit Runge–Kutta method with A-stability is suitable for solving stiff differential equations. However, the fully implicit Runge–Kutta method is very expensive in solving large system problems. Although some implicit Runge–Kutta methods can reduce the cost of computation, their accuracy and stability are also adversely affected. Therefore, an effective banded implicit Runge–Kutta method with high accuracy and high stability is proposed, which reduces the computation cost by changing the Jacobian matrix from a full coefficient matrix to a banded matrix. Numerical solutions and results of stiff equations obtained by the methods involved are compared, and the results show that the banded implicit Runge–Kutta method is advantageous to solve large stiff problems and conducive to the development of simulation.

On a Variational Method for Stiff Differential Equations Arising from Chemistry Kinetics

For the approximation of stiff systems of ODEs arising from chemistry kinetics, implicit integrators emerge as good candidates. This paper proposes a variational approach for this type of systems. In addition to introducing the technique, we present its most basic properties and test its numerical performance through some experiments. The main advantage with respect to other implicit methods is that our approach has a global convergence. The other approaches need to ensure convergence of the iterative scheme used to approximate the associated nonlinear equations that appear for the implicitness. Notice that these iterative methods, for these nonlinear equations, have bounded basins of attraction.

A new comprehensive tribo-dynamic analysis for lubricated translational joints in low-speed two-stroke marine engines

In low-speed two-stroke marine engines, the effect of connecting rod inertia is important for the tribo-dynamics of crosshead slipper-guide and piston skirt-liner. However, this has not been considered in previous research. Therefore, a new tribo-dynamic model that considers the connecting rod inertia is presented for the two lubricated translational joints, and the lubricant viscosity–temperature properties are accurately incorporated into the model. The modified extended backward differentiation formulate method is used to solve the nonlinear stiff differential equations effectively. The simulation results show that the effect of connecting rod inertia on the dynamic characteristics of two translational joints is mainly observed during the latter half of the upward stroke. For the crosshead slipper, the amplitude of the transverse velocity is increased by 30% at a crank angle of approximately 300° compared to the early model that ignored the connecting rod inertia. With the increase of the connecting rod mass, the secondary motions of both the crosshead slipper and the piston skirt are increased. Furthermore, a decrease of the connecting rod mass can reduce the friction losses of the engine.