Some algorithms designed for high-precision nonlinear monotonic finite-difference schemes are reported. These algorithms are applied for solving the boundary value problem of the set of hyperbolic equations in complex multiply connected integration domains and network structures (graphs). It allows to simulate some important tasks for medical biomechanics and ionospheric plasma.

The results demonstrated in the first part will be accompanied with examples related to several physiological processes in health, disease and some external disturbances: trachea-bronchial tree ventilation; blood flow in systemic and pulmonary circulation (including their interaction); mass transport in respiratory and cardiovascular systems; clots formation in the blood flow; cataract excision surgery via laser and ultrasonic devices simulation in order to determine consequences and possibly affected zones due to surgical aggression; simulations of the consequences and possibly affected zones due to craniocerebral traumas etc. Mathematical abstraction of these processes results in complex nonlinear models. That requires original unconventional approaches to overcome typical complexities arising in computational solving of the tasks mentioned above.

Numerical algorithm developed for the 3D magnetogasdynamics equations (MGD-model includes magnetic field diffusion) will be demonstrated in the second part. Software implementation of this algorithm allows to simulate physical processes going with upper atmospheric explosion in the magnetosphere of the Earth (for the altitudes м0 in the range 100 - 1000 km and initial energy release е0=10e19 – 10e22 erg) within the multicomponent (plasma of the explosion products and atmospheric air), one-velocity, one-temperature approximation. Numerical simulations of the plasma violent disturbances evolution in the circumterrestrial space were performed for the time periods limited by several hundreds of seconds. Substantial impact of exponential atmosphere and geomagnetic field heterogeneity as well as values of м0 and е0 to the plasma flow structure was discovered in the series of numerical experiments.