On nonlinear difference and differential equations

• DOI: 10.1007/bf01934923
• Publication Date: 2005-07-30T22:56:58Z
• Authors: Gustaf Söderlind
• Keywords: Ode; Constant (computer programming); Constant coefficients

A generalization of the logarithmic norm to nonlinear operators, the Dahlquist constant is introduced as a useful tool for the estimation and analysis of error propagation in general nonlinear first-order ODE's. It is a counterpart to the Lipschitz constant which has similar applications to difference equations. While Lipschitz constants can also be used for ODE's, estimates based on the Dahlquist constant always give sharper results. The analogy between difference and differential equations is investigated, and some existence and uniqueness results for nonlinear (algebraic) equations are given. We finally apply the formalism to the implicit Euler method, deriving a rigorous global error bound for stiff nonlinear problems.