Novikov E.A.  

An Additive Method of Second Order for Solving Stiff Problems

For numerical solution of the Cauchy problem for stiff systems of ordinary differential equations are commonly used L-stable methods. In case of high-dimensional problems for methods with an unlimited stability region, total computational costs are almost completely determined by time of calculation and decomposition of a Jacobian matrix of an original system. Many integration algorithms use freezing of the Jacobian matrix, i.e. using same matrix on several steps. This can significantly reduce computational cost. It is the most natural to use it in iterative methods for solving ordinary differential equations, in which the matrix doesn't affect an order of accuracy of the numerical scheme, but only determines speed of convergence of an iterative process. This approach is widely used in implementation of semi-explicit and implicit Runge-Kutta methods, multistep methods of Adams and Gear types. For non-iterative methods freezing or some other approximation of the Jacobian matrix is much more intricate problem. In these methods, the Jacobian matrix affects an order of accuracy of the numerical scheme, and therefore any of its perturbations can lead to loss of accuracy order. It should be noted that non-iterative methods are simple in terms of implementation on a computer and, therefore, attractive to many users. Here is constructed a four-stage method of second order of accuracy, allowing different types of approximation of a Jacobian matrix. An error estimation and an inequality for control of precision are obtained. Numerical results confirm an efficiency and effectiveness of the algorithm of integration.