Estimating the Domain of Absolute Stability of a Numerical Scheme Based on the Method of Solution Continuation with Respect to a Parameter for Solving Stiff Initial Value Problems

The modeling of physical and technological processes often involves solving stiff initial value problems. In most cases, their exact solution is difficult to find, while numerical schemes sometimes fail to produce a sufficiently accurate solution in acceptable computation time. Moreover, for some classes of problems, numerical solution schemes are unsuitable because of their insufficient stability. This paper deals with numerical methods based on solution continuation with respect to arguments of various types that make it possible to enhance the stability of explicit numerical schemes. Most frequently, the used best argument is hardly applicable to problems in which the integral curves grow at a superpower or nearly exponential rate. Previously, the authors proposed a modification of the best argument that alleviates these disadvantages. In the present paper, we estimate the domain of absolute stability of the explicit Euler scheme as applied to problems transformed to a modified best argument of special form and refine the proof of a similar estimate for initial value problems transformed to the best argument. A test initial value problem is used to verify the resulting theoretical estimates and to analyze the application of the modified best argument of solution continuation.