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PresentationsMulti-implicit methods of solution of stiff ODE systemsVolgograd State University, University Avenue 100, Volgograd, 400062, Russia Stiff problems are found in many areas of science and technology, including chemical reaction kinetics, hydrodynamics of multiphase flow applications in biology, etc. In the numerical solution of such problems raises a number of problems related to stability, accuracy and computational complexity of the methods used. In this paper we consider a new family of difference schemes, which is an extension of a family of multi-implicit schemes with second derivative [1]. The presence of the Jacobi matrices in the finite difference scheme allows to increase the order of approximation of the scheme without increasing the computational complexity of its implementation. In applications of numerical methods for stiff systems is a central problem of the stability of methods. The requirement of absolute stability leads to implicit difference methods. But for them, there are limitations, for example, order the Dahlquist barrier: an implicit linear multistep methods of order higher than the second cannot be A-stable. However, for very stiff problems, the condition of A-stability may be insufficient. They require L-stable methods.
In the classical multi-step methods use information at more than one points. In contrast, in this paper, we consider methods that use several subsequent points, each of which is written separate differential equation. This approach leads to the construction of a family of so-called multi-implicit methods with second derivative (m-Implicit Second Derivative). mISD-scheme is a system of difference equations. The order of approximation of the individual equations can be varied while maintaining the order of the scheme as a whole, that allows to expand the family mISD-schemes. In this paper we detail the expanded three-parameter family for 2ISD schemes. It is shown that among the many A-stable 2ISD schemes there are two families: the family of L-stable schemes and collection schemes of high accuracy for linear problems. Conducted testing of these difference schemes for linear and nonlinear problems with varying degrees of stiffness. Dependences of the integral error of the numerical solution the size of the integration step as well demonstrate the stability and precision of the proposed 2ISD diagrams.
Literature 1. Vasilev E. I., Vasilyeva T. A., An extended family of double-implicit methods for stiff systems of differential equations. – Bulletin Volgograd State University, Series 1 Mathematics. Physics, N3 (28), 2015. P.34-43.
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