Novikov E.A.  

Algorithms for stiff problems integration using non-uniform numerical schemes

In modeling the kinetics of chemical reactions, the calculation of electronic circuits, simulation of emergencies in large electric grids and other critical applications, there is a problem of numerical solution of stiff systems. Major trends in the construction of numerical methods are associated with the expansion of their capacity to meet the challenges of high dimensionality. The mathematical formulation of practical problems constantly refined, leading to an increase in dimension as well as the right to the complication of the system of differential equations. In many cases, the calculations required to carry out the so-called engineering precision - about 1% or less. This is due to the fact that the measurement of the constants in the right-hand side of the system of differential equations, is often carried out rather rudely. Sometimes this calculation accuracy is satisfactory from the point of view of the target. It is known that the order of approximation of the numerical scheme should be combined with the required accuracy of calculations.

Modern methods for solving stiff problems, as a rule, use the calculation and treatment of the Jacobian matrix of the system of differential equations. In the case of large-scale systems of efficient numerical methods almost completely determined by the decomposition of the matrix. To improve the efficiency of calculations in a number of algorithms used freezing Jacobi matrix, that is, the use of a matrix by a few steps of integration. An analog of freezing is to use in the calculation of integration algorithms, based on implicit and L-stable methods with an automatic choice of the numerical scheme. It built a clear two-step scheme of Runge-Kutta methods and L-stable (2,1)-method of the second order of accuracy. Based on the stages of the explicit numerical method is based first order formula with up to 8 intervals of stability. The algorithm of integration of variable order and step in which the selection of the most efficient numerical scheme is implemented at each step using the inequalities for the stability control. Numerical results confirm the efficiency of the constructed algorithm.