$U_i^{new}=U_i^{old}+dtF_i(\overline{U^{old}},\over line{U^{new}})$ (A5)
where $F_i$ - function in the right part of the equations, and $\overline{U_i}$ - variables.
, $U_i^{new}$ as I have understood, these are predicted value of all variables from, for example, Euler's method. Then it is necessary to calculate variables and to compare them with predicted, repeating algorithm before achievement of necessary accuracy. But how to calculate if in the right part there are the same $U_i^{new}$, for me not clearly? In any way I don't understand the dependence $F_i(\overline{U^{old}},\overline{U^{new}})$ of two variables means? How to use this formula in calculations? I will be glad to any concrete algorithms or the references to it.