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An implicit time-integration scheme provides solution to the compressible Navier-Stokes equations describing supersonic internal flows. Line relaxation is written in terms of primitive variables in the scheme. Central differencing with artificial dissipation provides the linearized implicit terms which are a simple approximation of the flux splitting. Effective propagation of shocks and handling interactions with the internal viscous flow features, present in an arc-heated plasma jet reactor used in chemical vapor deposition, is possible.