Studying the evolution of the convective burning process before and during the thermonuclear runaway in a white dwarf is crucial in order to measure the enrichment of the hydrogen envelope by convective overshoot. Recent numerical simulations, that start when the temperature at the base of the envelope is close to 10^8 K, show that in a few hundreds seconds the temperature grows up to 2 10^8 K. At this time the runaway takes place. Our simulations, performed by running a high order of accuracy code, with low numerical viscosity, show that care must be taken in the choice of the initial and boundary conditions. We have observed, in fact, the onset of fast convective instabilities that are driven by boundary effects and affect the dynamics of the pre-runaway phase. We plan, as a next step, to take the initial equilibrium with a peak of temperature close to 10^7 K, that corresponds to earlier and less unstable phase of the white dwarf evolution.