High-order -methods

Implicit methods are the natural choice for solving stiff systems of ODEs. Rosenbrock methods are a class of linear implicit methods for solving such stiff systems of ODEs. In the Rosenbrock methods the exact Jacobian must be evaluated at every step. These evaluations can make the computations costly. By contrast, W -methods use occasional calculations of the Jacobian matrix. This makes the W -methods popular among the class of linear implicit methods for numerical solution of stiff ODEs. However, the design of high-order W -methods is not easy, because as the order of the W -methods increases, the number of order conditions of the W -methods increases very fast. In this paper, we describe an approach to constructing high-order W -methods.