A conjugate gradient algorithm for solving the Galerkin-characteristic approximation of interfacial flows

摘要

We present a conjugate gradient algorithm for solving the Galerkin-characteristic approximation of interfacial flows. The governing equations are the incompressible Navier-Stokes for two fluids separated with an interface in the computational domain. We consider a level set method to track the interface in these equations. The method combines advantages of the semi-Lagrangian method to accurately solve the convection-dominated flow problems with a finite element method for space discretization of the governing equations. It can be interpreted as a fractional-step technique where the transport part and the Stokes part are treated separately. A limiting procedure is implemented for the reconstruction of numerical solutions at the departure points. The implementation of the proposed Galerkin-characteristic method differs from its Eulerian counterpart in the fact that it is applied during each time step, along the characteristic curves rather than in the time direction. Therefore, due to the Lagrangian treatment of convection, the standard Courant-Friedrichs-Levy condition is relaxed and the time truncation errors are reduced in the Stokes part. To solve the generalized Stokes problem we implement a conjugate gradient algorithm. This method avoids projection techniques and does not require any special correction for the pressure. The focus is on constructing efficient algorithms with a large stability region to solve interfacial flow problems. We verify the method for a passive transport of a slotted cylinder and for the benchmark problem of rising bubbles. We also present numerical results for a problem of barotropic flow in the Strait of Gibraltar. The conjugate gradient algorithm has been found to be feasible and satisfactory.