A time-stepping procedure for Ẋ=A1X+XA2+D, X(0)=C

We develop an expression for the exact solution of the matrix differential problem
\dot{X} = A_{1} X + XA_{2} + D, X(0) = C
based on variation of parameters and use this to devise the time-stepping relation
X(t+h)=e^{A_{1}h}\{X(t)+\int\liminf{0}\limsup{h}e^{-A_{1}s}De^{-A_{2}s}ds\}e^{A_{2}h}
. We modify a procedure of Van Loan to effect efficient computation of all the terms necessary to advance the solution in time according to this relation. We consider some alternatives when sparsity is a concern. A numerical example of our procedure is included.