Preserving monotonicity in the numerical solution of Riccati differential equations (Q1923279)

Consider the matrix Riccati differential equation (RDE) \(X'(t)= A(t)X(t)+ X (t)A^T(t) -X(t)B(t)X(t) +C(t)\), \(X(0)=X_0\), where all matrices are in \(\mathbb{R}^{n \times n}\), bounded, piecewise continuous and \(B(t)\), \(C(t)\), \(X_0\geq 0\). As is well known the solutions of RDE are monotone with respect to properly ordered data \(A(t)\), \(B(t)\), \(C(t)\), \(X_0\). It is shown that the backward Euler integration method preserves this type of monotonicity. It is also proved that any one-step method or strictly stable multistep method that preserves monotonicity in the numerical solution of RDE is of order at most one. Runge-Kutta integration schemes are investigated in particular in connection with their property to preserve monotonicity.