Multistep Methods Integrating Ordinary Differential Equations on Manifolds

This paper presents a family of generalized multistep methods that evolves the numerical solution of ordinary differential equations on configuration spaces formulated as homogeneous manifolds. Any classical multistep method may be employed as an invariant method, and the order of the invariant method is as high as in the classical setting. We present numerical results that reflect some of the properties of the multistep methods. AMS Subject Classification: 65L06, 34A50 Key Words: geometric integration, multistep methods, numerical integration of ordinary differential equations on manifolds, numerical analysis, Lie groups, homogeneous spaces 1 Introduction Classical multistep methods that solve the initial value problem y 0 = f(t; y); y(0) = y 0 2 R d ; (1) are discussed in a number of texts, see e.g. [7, 8, 11]. We write a general k-step method in the form k X i=0 ff i y n+i = h k X i=0 fi i f(t n+i ; yn+i ); n = 0; 1; : : : ; (2) where ff j and fi j , j = 0; 1; : : : ..