  The paper presents novel results, obtained on the basis of the modified
theory of hydraulic fractures (HF). The theory underlines significance of the
speed equation. When applied to numerical simulation of HF, the theory reveals
three distinct issues: (i) modeling the central part of a HF; (ii) modeling the
near-front zone; and (iii) tracing changes in the shape of a fracture contour.
Modeling the central part leads to a stiff system of ODE in time, what strongly
complicates its integration. For explicit schemes, it requires small time steps
to meet the CFL condition. For implicit schemes, it requires proper
preconditioners. The gains and flaws of the two strategies are discussed. It is
noted that a rough spatial mesh may be used in the central part. Modeling the
near-front zone reveals the vital role of the speed equation for HF modeling by
any method. Its asymptotic analysis has resulted in the fundamental concept of
the universal asymptotic umbrella. For the near-front zone, it is also
established that a notable part of the zone adjacent to the front propagates
virtually as a simple wave. This implies that the CFL condition of stability
for this zone is much less restrictive than for the central part of the
fracture. Of essence is also that the wave-like propagation of the near front
zone makes preferable upwind schemes of time stepping. On whole, the analysis
implies that explicit time stepping may be complementing, competitive and even
superior over implicit integration. Tracing changes in the front shape appears
merely in 3D problems when the contour changes its form. The problem, being
essentially geometrical, it may be solved separately by various methods,
including fast marching, level set and the simplest string/marker methods. In
1D cases, it does not arise at all.
  Quantitative estimations and numerical examples illustrate the theoretical
conclusions.
