ABSTRACT
Mathematically, an approximate solution of a differential equation may be developed using a Taylor series expansion with a specified number of terms. If only one Taylor series expansion term is used, the method is first order accurate, whereas if n Taylor series expansion terms are used, the method is n-th order accurate. The order of the method is especially important when the increment is integrated in subincrements (substeps), as the accuracy rises in line with the number of subincrements to the power equal to the order of the method. So if the method is first order accurate, integration with twice as many subincrements leads to a solution which is only twice as accurate, whereas for a second order method the solution will be four times more accurate (see e.g. Labert 1973. Shampine 1994, Deuflhard & Bornemann 2002).
