Simulation of hypersonic viscous flows around a cone-delta-wing combination by an implicit method with multigrid acceleration

Abstract

There is a general tendency to use a time marching approach for steady state solutions in fluid dynamics because of its robustness in following true physical processes. Implicit methods can generally yield faster convergence to steady state than explicit ones by taking larger time steps, which are not subjected to the severe CFL stability criterion(see, e.g.,[]]). Even quadratic convergence will be achieved if the implicit time marching can reduce to Newton iteration for the corresponding steady state problem when ~t~. One of these "perfect" implicit operators has recently been constructed by Mulder and van Leer[2] for one-dimensional problems in gas dynamics. Unfortunately b this perfect implicit operator is usually very difficult and sometimes even impossible to construct. Furthermore the way forward into multidimensional problems is blocked by the fact that no efficient direct inversion of the unfactored multidimensional implicit operator exists and approximate fact~rization is often introduced. Therefore we should make the errors from the approximate factorization balance those from the dlscretlzation of the continuous model. This imposes a limitation on time steps and prevents the convergence of the implicit method from achieving the fast rate achieved in one-dlmension. Too large time steps can only create severe errors instead of yielding quadratic convergence.