Improving the accuracy of a variable step-size method for solving stiff Lyapunov differential equations

A method based on the matrix generalization of the second-order backward differentiation formula is proposed for computing the solutions of stiff Lyapunov differential equations. The proposed method efficiently computes the solutions by exploiting the matrix structure of the Lyapunov differential equations and by allowing the step-size of integration to increase automatically. Such a variable step-size feature shortens the computational time when the transient part of the solution becomes insignificant. This proposed method was encoded as a MATLAB^® script file and was tested on several Lyapunov differential equations with known closed-form solutions. This method produced accurate solutions. It was observed that the step-size of integration could increase automatically. Also when it was in or near the steady state, the errors of the computed solutions remained very small even for very large step-sizes of integration. This proposed method was also compared with the matrix version of the backward Euler method proposed. It was found that the new method produced solutions with accuracy an order of magnitude higher than the matrix backward Euler method.