Supercomputers and the mathematical modeling of high complexity problems

This paper is a review of many works carried out by members of our scientific school in past years. The general principles of constructing numerical algorithms for high-performance computers are described. Several techniques are highlighted and these are based on the method of splitting with respect to physical processes and are widely used in computing nonlinear multidimensional processes in fluid dynamics, in studies of turbulence and hydrodynamic instabilities and in medicine and other natural sciences. The advances and developments related to the new generation of high-performance supercomputing in Russia are presented.