Existence and Uniqueness of Solutions of the Algebraic Equations in the Bdf Methods Department of Mathematics Existence and Uniqueness of Solutions of the Algebraic Equations in the Bdf Methods

The paper presents a new result for the existence and uniqueness of solutions of the system of nonlinear algebraic equations obtained from the application of the BDF methods to stii initial value problems in ordinary diierential equations. Expressed in terms of the local truncation error, starting errors in the BDF and contractivity properties of the diierential equation, the result is used to explain how with a certain form of local error control existence and uniqueness is guaranteed. Although the form of local error control is a theoretical procedure it relates well to the error control procedures used in practical codes and provides insight into how at a typical step the BDF yields solution values. A potential diiculty in trying to compute low accuracy numerical solutions is also identiied.