Implicit two-stage Lagrangian–Euler difference scheme for gasdynamics calculations using concerted approximations to mass and momentum balance equations

We propose a variant of the two-stage Eulerian–Lagrangian difference scheme for gas dynamics calculations in the two-dimensional approximation. Energy balance equation in the original gas-dynamic system of equations is written in the “entropy” form — with respect to the gas medium internal energy. The initial equations of continuity and momentum are divergent-conservative, therefore they are approximated as conservative balance equations. Using the constructed difference approximations to gasdynamics equations we derive a kinetic energy balance equation that does not contain nonphysical source terms which in some other difference constructions can appear due to the method of differencing. Such sources usually are proportional to the time step, and may accumulate during the calculation. The kinetic energy balance can be combined with that of the internal energy thus producing the equation which is a gasdynamic form of the total energy conservation law. In accordance with the physical processes occurring inside the gas flow an internal energy and kinetic energy can pass through the mutual transformations without the influence of “source terms” caused by approximation effects.