\(L q\)-decreasing monotonic schemes with complex coefficients and applications to complicated PDE systems (Q1339336)

\(L_ q\)-decreasing monotonic schemes with complex coefficients for solving a system of partial differential equations are considered. At present there are good methods for solving separate equations or special groups of equations. Nevertheless, the solution of a system of many equations (describing processes of transfer, heat conduction, diffusion, melting, acoustics and gas dynamic flows of chemical reacting components) remains a difficult task. The complex Rosenbrock scheme (monotonic and \(L_ 2\)-decreasing) has certain advantages. \(L_ q\)-decreasing means that stiff components of the difference-differential equations will decrease for large time step \(\tau\) as well as for small \(\tau\); montonicity means that the numerical solution for the stiff components will be a monotonically decreasing function, as the exact solution. Rosenbrock schemes with complex coefficients (RCC) are applied for numerical solutions of parabolic equations (heat conduction), transfer equations (neutron or light propagation), acoustic systems and gas dynamics problems. Numerical calculations of gas dynamics equations show that the results are much more acceptable than those for acoustic and transfer problems. In this paper the difficulties of comparison with the exact solution are considered. The RCC method can be used successfully for wide classes of problems.