Exponential fitting BDF algorithms: Explicit and implicit 0-stable methods

We present BDF-type formulas capable of the exact integration (with only round-off errors) of differential equations, the solutions of which belong to the space generated by the linear combinations of 〈1,eAx,xeAx〉〈1,eAx,xeAx〉. Plots of their 0-stability regions in terms of λλ are provided. We will see that the explicit method is 0-stable with many parameters λhλh. Plots of their regions’ absolute stability that include all the negative real axis are provided. Numerical examples show the efficiency of the proposed codes, specially when we are integrating stiff problems with the explicit method.