Comparing the accuracy of backward differentiation formulas for solving Lyapunov differential equations

This paper describes the results in the comparison of the accuracy of the first through fifth order backward differentiation formulas for solving Lyapunov differential equations. Each of the five formulas was encoded as a MATLAB^reg script file and was used to solve two stiff Lyapunov differential equations with known closed-form solutions. In solving the Lyapunov differential equations by the backward differentiation formulas, the differential equations were manipulated into algebraic Lyapunov equations so that numerically stable methods could be applied to obtain the solutions. The findings were that first, the BDF approach is a viable one for computing the solutions of stiff Lyapunov differential equations. Second, a higher order backward differentiation formula may not produce more accurate numerical solutions but a higher order formula is computationally more costly than a lower order one. Third, the BDF approach allowed step-size of integration to increase continually without loss of accuracy when the transient components were waning away.