Mathematical modeling is a tool of the prime importance for the run-off formation study. It gains in importance under condition of a sparse and out-of date monitoring network typical of majority of the Siberia regions. In analyzing the spatial-temporal laws of the run-off formation in large river basins, the main problems are associated with the absence of a basic hydrological model capable of treating real hydrological and hydro-chemical observation data.

A method of solution for variable-depth flows in arbitrarily-shaped domains are currently underestimated. The water level rise and fall can result in a change of the shape of an area due the islands submerging, sandbanks emerging, inundation of floodplains, etc. This requires the formulation of the edge problem with unknown boundaries, tracing through all wetted perimeters, and foreseeing the formation of new internal boundaries in the form of islands due to the shallow water places drying up. Conventional methods of solving such problems (e.g., the fictitious domain method) encounter severe algorithmic difficulties associated with problem degeneracy under condition of water layer disappearance, with development of multilogic systems, and with excessive intellectualization of a program.

Comprehensive methods of computational mathematics allow us to avoid these difficulties through the use of monotone numerical schemes. The mono-tonicity property itself provides the non-negativity of calculated values such as a water layer thickness h, the Celsius temperature, admixture concentration. The application of monotone scheme, for example, h, ensures the fulfilment of the relation h ≥ 0 within the whole definition domain including shallow waters and dry places where h = 0. Hence, such an approach does not require building the algorithm of non-calculated areas formation, but allows us to assume h = 0 for dry places and to integrate the equations through the whole domain. The external boundary in this case can be located along the watershed line or assigned according to topography peculiarities.

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