Modelling one-dimensional unsteady flows in ducts: Symmetric finite difference schemes versus galerkin discontinuous finite element methods

Abstract

The Euler equations for one-dimensional unsteady flows in ducts have been solved resorting to classical symmetric shock-capturing methods with second-order accuracy and to the recent discontinuous Galerkin finite-element method, with second- and third-order accuracy. In particular, the finite difference techniques adopted are the two-step Lax—Wendroff method and the MacCormack predictor—corrector method, with the addition of the flux corrected transport (FCT) or of the Davis nonupwind TVD scheme to suppress the spurious oscillations in the vicinity of discontinuous solutions. The finite-element method adopted is based on the weak formulation of the Euler equations, which are solved by introducing a discontinuous finite-element space discretization. A dissipative mechanism has been considered to supplement the FEM with a “discontinuity capturing” operator, adding a “viscous like” term to damp minor numerical overshoots arising in proximity of steep gradients of the solution. The numerical tests chosen to carry out a comparison between these schemes are the shock-tube problem and the shock—turbulence interaction problem. Both the test cases considered show the superiority of third-order FEM calculations, whereas the comparison between the computer run times points out the greater computational effort required.