261 ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2009, Vol. 49, No. 2, pp. 261–278. © Pleiades Publishing, Ltd., 2009. Original Russian Text © A.B. Al’shin, E.A. Al’shina, A.G. Limonov, 2009, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2009, Vol. 49, No. 2, pp. 270–287. 1. INTRODUCTION Stiff systems nearly inevitably arise in the mathematical simulation of problems with many processes on different time scales (see [1]). Examples are problems describing chemical and neutron reactions, nonsta- tionary processes in electric circuits, and many others. They are traditionally difficult to solve numerically, and special numerical methods have to be developed for them. Stiff stability. Beginning in the 1950s, special implicit methods were developed for stiff problems and a number of additional properties were stated that must be satisfied by the desired schemes. Consider the Dahlquist problem (1.1) When λ  –1, the exact solution to Eq. (1.1) decays rapidly and monotonically. For any linear scheme, the transition to the next time level while solving problem (1.1) has the form = R(τλ)y , where R(ξ) is called the growth function or the stability function. Definition 1. A scheme is said to be A-stable if |R(ξ)| ≤ 1 for Re ξ ≤ 0; i.e., the numerical solution is stable in the same range of λ as the exact solution to problem (1.1). If a scheme is not at least A-stable, it is not applicable to stiff problems. It is desirable that the stability function also decay rapidly as Re λ –∞. For this reason, the concept of Lp-stability is introduced. Definition 2. A scheme is said to be Lp-stable if it is A-stable and R(ξ) = O(ξ –p ) as |ξ| ∞. The stiffer the problem (where the measure of stiffness is |λτ|), the more advantageous the Lp-stable schemes with large p. For high-order schemes, a useful generalization is Lpα-stability (| R(ξ)| ≤ 1 for 90° + α ≤ argξ ≤ 270° – α and R(ξ) = O(ξ –p ) as |ξ| ∞). As a rule, the stiff character of the numerical solution to the Cauchy problem for a system of ordinary differential equations is manifested only in certain domains depending on the particular statement of the problem. A key role is played by the stability of the scheme on stiff intervals and by the approximation accu- racy on soft intervals. dy dt ----- λ y , 0 t T , y 0 () ≤ ≤ y 0 , y exact t () y 0 e λt . = = = y ˆ Two-Stage Complex Rosenbrock Schemes for Stiff Systems A. B. Al’shin a , E. A. Al’shina a , and A. G. Limonov b a Institute of Mathematical Modeling, Russian Academy of Sciences, pl. Miusskaya 4a, Moscow, 125047 Russia e-mail: elena@alshina@gmail.com b Moscow State Institute of Electronic Engineering (Technical University), Zelenograd, Moscow, 124498 Russia Received February 26, 2008; in final form, June 6, 2008 Abstract—New two-stage Rosenbrock schemes with complex coefficients are proposed for stiff sys- tems of differential equations. The schemes are fourth-order accurate and satisfy enhanced stability requirements. A one-parameter family of L1-stable schemes with coefficients explicitly calculated by formulas involving only fractions and radicals is constructed. A single L2-stable scheme is found in this family. The coefficients of the fourth-order accurate L4-stable scheme previously obtained by P.D Shirkov are refined. Several fourth-order schemes are constructed that are high-order accurate for linear problems and possess the limiting order of L-decay. The schemes proposed are proved to con- verge. A symbolic computation algorithm is developed that constructs order conditions for multistage Rosenbrock schemes with complex coefficients. This algorithm is used to design the schemes proposed and to obtain fifth-order accurate conditions. DOI: 10.1134/S0965542509020067 Keywords: two-stage complex Rosenbrock schemes, stiff systems of ordinary differential equations, Lp-stable schemes, A-stability.