Analytical integration and a numerical method for 1-D compressible flow with diffusive transport

A new analytic integral and a numerical method are presented for 1-D compressible flow; the method is suitable for problems in which diffusive scales are of the same order as the length scale of the flow. The conventional, Eulerian formulation of the compressible flow equations is converted to a Lagrangian formulation by coordinate transform. Thereafter, the balance of linear momentum is integrated once analytically. This procedure yields a set of two nonlinear partial differential equations for the particle position and total energy fields; these equations are easily solved using finite differences. As a consequence of the analytic integration, the system is characterized by easily controlled numerical dispersion and dissipation for a very broad range of Reynolds or Mach numbers. Solutions are presented for the well-known piston problem; comparison with available analytical results is excellent.