An improved approximate Newton method for implicit Runge–Kutta formulas

Implicit Runge–Kutta (IRK) methods (such as the ss-stage Radau IIA method with s=3,5s=3,5, or 7) for solving stiff ordinary differential equation systems have excellent stability properties and high solution accuracy orders, but their high computing costs in solving their nonlinear stage equations have seriously limited their applications to large scale problems. To reduce such a cost, several approximate Newton algorithms were developed, including a commonly used one called the simplified Newton method. In this paper, a new approximate Jacobian matrix and two new test rules for controlling the updating of approximate Jacobian matrices are proposed, yielding an improved approximate Newton method. Theoretical and numerical analysis show that the improved approximate Newton method can significantly improve the convergence and performance of the simplified Newton method.

► A novel construction of a new approximate Jacobian matrix, JnewJnew, is proposed.
► Each related linear system can be solved efficiently by the same linear system splitting scheme as the one used in the simplified Newton method.
► It is proved that the improved approximate Newton method can have a smaller linear convergence factor than the simplified Newton method.
► Two simple test rules, called Test-Rule 1 and Test-Rule 2, are also proposed.
► The improved approximate Newton method and Test-Rule 1 and Test-Rule 2 were adapted to the Fortran program package RADAU.