Variable structure algorithm applying 3-stage methods of Runge-Kutta and Rosenbrock types

Background. The Cauchy problem for large-scale stiff systems arises in simulation of physical and chemical processes, in approximation of partial differential equations by a system of ordinary differential equations and in plenty of other important applications. Taking into consideration a large number of factors in model development leads to expansion of a class of problems, determined by stiff systems of high dimension. The complexity of practical problems leads to additional requirements to computational algorithms. Materials and methods. Dealing with high dimensionality of a stiff system of ordinary differential equations, the main computational expenses concern the Jacobi matrix decomposition. In some algorithms one may use freezing of the Jacobi matrix, i.e. applying the same matrix over several integration steps. The problem of freezing is solved rather easy in those methods, the stages of which are computed using some iterative processes. For non-iterative numerical formulas the freezing is quite a difficult problem. In this study, the costs reduction was achieved by combining explicit and L-stable methods using the criterion of stability in calculations. Results. The author has created an algorithm of variable structure integration, based on the explicit scheme of the Runge-Kutta type and the L-stable method of the Rosenbrock type. Both schemes have the third order of accuracy. An efficient numerical formula was chosen according to the criterion of stability at each step of integration. Estimation of maximum eigen value, which is necessary to switch between the methods, for explicit numerical schemes was determined by power iterations using already computed stages, and using the Jacobi matrix norm for the Rosenbrock type method. The researcher also formulated inequalities for accuracy and stability control. The article adduces the results of calculations. Conclusions. The integration algorithm is aimed at solving stiff problems of high dimension. Numerical results confirm the efficiency of the constructed algorithm.