A family of Anadromic numerical methods for matrix Riccati differential equations

Abstract

Matrix Riccati Differential Equations (MRDEs) X′ = A21 −XA11 +A22X −XA12X, X(0) = X0, where Aij ≡ Aij(t), appear frequently throughout applied mathematics, science, and engineering. Naturally, the existing conventional Runge-Kutta methods and linear multi-step methods can be adapted to solve MRDEs numerically. Indeed, they have been adapted. There are a few unconventional numerical methods, too, but they are suited more for time-invariant MRDEs than time-varying ones. For stiff MRDEs, existing implicit methods which are preferred to explicit ones require solving nonlinear systems of equations (of possibly much higher dimensions than the original problem itself of, for example, implicit Runge-Kutta methods), and thus they can pose implementation difficulties and also be expensive. In the past, the property of an MRDE which has been most preserved is the symmetry property for a symmetric MRDE; many other crucial properties have been discarded. Besides the symmetry property, our proposed methods also preserve two other important properties — Bilinear Rational Dependence on the initial value, and a Generalized Inverse Relation between an MRDE and its complementary MRDE. By preserving the generalized inverse relation, our methods are accurately able to integrate an MRDE whose solution has singularities. By preserving the property of bilinear dependence on the initial value, our methods also conserve the rank of change to the initial value and a solution’s monotonicity property. Our methods are anadromic,1meaning if an MRDE is integrated by one of our methods from t=τ to τ+θ and then integrated backward from t=τ+θ to τ using the same method, the value at t=τ is recovered in the absence of rounding errors. This implies that our methods are necessarily of even order of convergence. For time-invariant MRDEs, methods of any even order of convergence are established, while for time-varying MRDEs, methods of order as high as 10 are established; but only methods of order up to 6 are stated in detail. Our methods are semi-implicit, in the sense that there are no nonlinear systems of matrix equations to solve, only linear ones, unlike any pre-existing implicit method. Given the availability of high quality codes for linear matrix equations, our methods can easily be implemented and embedded into any application software package that needs a robust MRDE solver. Numerical examples are presented to support our claims. 1We coined “anadromic” from the Greek roots “ανα-” (back up) and “δρoμoσ” (act of running) of “anadromous” (which describes fish that must return from the ocean to spawn in the same streams whence they hatched) after we had tried to describe our numerical methods with words such as “reflexive” [27, 28], “symmetric” (which is still used today in the literature, e.g., [22]), and “reversible” which turned out to be overused. Received by the editor Received August 25, 2009 and, in revised form, October 30, 2010. 2010 Mathematics Subject Classification. Primary 65L05.