  In the paper, we propose a new effective mathematical formulation and
resulting universal numerical algorithm capable of tackling various HF models
in the framework of a unified approach. The presented numerical scheme is not
limited to any particular elasticity model or crack propagation regime. Its
basic assumptions are: i) proper choice of independent and dependent variables
(with the direct utilization of a new one - the reduced particle velocity), ii)
tracing the fracture front by use of the speed equation which can be integrated
in a closed form and sets an explicit relation between the crack propagation
speed and the coefficients in the asymptotic expansion of the crack opening,
iii) proper regularization techniques, iv) improved temporal approximation, v)
modular algorithm architecture. The application of the new dependent variable,
the reduced particle velocity, instead of the usual fluid flow rate,
facilitates the computation of the crack propagation speed from the local
relation based on the speed equation. As a result, the position of the crack
front is accurately determined from an explicit formula derived from the speed
equation. The underlying ideas employed in the algorithm are combined together
producing a robust and efficient numerical scheme. Its performance is
demonstrated using classical examples of 1D models for hydraulic fracturing:
PKN and KGD under various fracture propagation regimes. Solution accuracy is
verified against dedicated analytical benchmarks and other solutions available
in the literature. Most of the ideas developed here, can be directly extended
to more general 2D and 3D cases.
