Solving stiff Lyapunov differential equations

We propose a method based on the matrix generalization of the backward differentiation formula for solving stiff Lyapunov differential equations. This method turns a Lyapunov differential equation into an algebraic Lyapunov equation so that the structure of the original equation can be exploited. The Hessenberg-Schur method can be used to obtain the numerical solution of the algebraic equation. This approach is applied to several stiff Lyapunov differential equations with known closed-form solutions. The computed solution is compared with the closed-form solution and the relative error of the computed solution is small in each of the cases