Convergence study of an implicit multidomain approximation for the compressible Euler equations

Abstract

The influence of grid overlapping on the convergence to a steady state is studied for a time-dependent multidomain difference approximation of a hyperbolic initial boundary value problem. Implicit dissipative difference schemes for interior points and explicit matching conditions at grid interfaces are considered. For a scalar model equation, it is proved that when the total number of interior mesh points is large, the convergence speed is an increasing function of the overlapping length. The convergence rate of the corresponding single domain treatment is recovered for a sufficiently large overlapping length. For a particular scheme, a quantitative analysis shows the existence of an optimal overlapping length, equal to the CFL number, for which the multidomain scheme converges as well as and sometimes even better than the single domain one in terms of the CPU time. Numerical experiments on a quasi-one-dimensional supersonic flow in a duct show also that a proper choice of the overlapping length ensures the same convergence rate as the one in the single domain calculation. Further applications to transonic flow calculations over single and two-element airfoils reveal the good convergence property of the overlapping treatment even for problems containing shocks.