Application of AD-based Quasi-Newton Methods to Stiff ODEs

Systems of stiff ordinary differential equations (ODEs) can be integrated properly only by implicit methods. For that purpose, one usually has to solve a system of nonlinear equations at each time step. This system of equations may be solved by variants of Newton’s method. The main computing effort lies in forming and factoring the Jacobian or a suitable approximation to it. We examine a new approach of constructing an appropriate quasi-Newton approximation for solving stiff ODEs. The method makes explicit use of tangent and adjoint information that can be obtained using the forward and the reverse modes of algorithmic differentiation (AD). We elaborate the conditions for invariance with respect to linear transformations of the state space and thus similarity transformations of the Jacobian. We present one new updating variant that yields such an invariant method. Numerical results for Runge-Kutta methods and linear multi-step methods are discussed.