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Hybrid Methods for Solution of Ordinary Differential Equations of Stiff and/or Oscillating Circuits

Keywords:

A.M. Pilipenko, V.N. Biryukov


Ordinary differential equations (ODE) of RF circuits are divided on stiff and oscillating. For efficient solution of stiff ODEs L-stable numerical methods are required while A(π/2)-stable methods are used for solving oscillation ODEs. Multistage fully implicit Runge-Kutta methods and A(π/2)-stable high order methods are probably the most effective for specified problems respectively. Unfortunately, oscillating problems are often stiff also. In this paper properties of hybrid methods combining A-, P- and L-stability of different methods are discussed. The hybrid 2-nd order method is generated by exponential fitted method if its constant weight ratio becomes a function of a step size. The hybrid fourth order methods are based on known implicit one-step two-stage Runge-Kutta methods: symmetric fourth order method and fully implicit second order method. The non linear scheme of this method is a function of a step size also. Hybrid methods’ efficiency was proved by means of the comparative analysis of canonical and hybrid methods’ global errors while solving test problems – stiff and oscillating ODE. Hybrid methods combine properties of the well-stable low order methods and more accurate methods with low stability. Hybrid methods make it possible to acquire simultaneously P- and L-stability, which is impossible for one-step implicit methods with the Padé error function.
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