3-point Hermite integration of differential equations (Q912575)

Hermite methods approximate a function by using its values and derivatives of one or more orders at a specified set of points. The authors propose that \(y'=F(x,y)\) can be solved by approximating a linear combination of integrals of F(x,y) on two adjacent intervals by a 3-point Hermite approximation at the three endpoints. Coefficients of the method are obtained using Taylor series expansions of \(F(x+\xi_ i,y)=F(\xi)\). This will yield the order of accuracy claimed when F varies with x alone. In an earlier paper [Comput. Methods Appl. Mech. Eng. 39, 19-224 (1983; Zbl 0501.65036)], the authors propose 2-point Hermite methods which are stable because they are essentially single step discrete methods. In contrast, the 3-point methods require two steps, and as such are stable only if the second step is not larger than the first. Even for equal steps, the methods are weakly stable and hence inappropriate for problems with exponentially decaying solutions. The authors give explicit formulations of two possibilities for linear problems. The numerical examples use constant coefficient differential equations, suitably illustrating the implementation in this case. Numerical evidence of the order could have been given by solving the example problems over a fixed interval with a sequence of decreasing steplengths. It is not clear how the methods would be implemented to solve nonlinear problems. Before this approach can be accepted as a useful tool, its implementation and range of application must be clarified. A more complete analysis as well as numerical evidence of accuracy and stability for linear and nonlinear problems is needed. While some comparisons occur in the earlier paper, comparisons with methods implemented for control of local truncation error (such as a formula pair of explicit Runge-Kutta methods) would identify the utility of the proposed methods.