Nonmatching Grid Technique for Highly Accurate Control Volume Pade-type Differences (CVPD)

摘要

The motivation for the present work was basically the necessity to simulate time-dependent incompressible flows in stirred tanks with turbine-like agitators. The computational meshes for these problems are inevitably nonmatching due to the complicated geometry and the sliding possibilities. In order to obtain the high quality solutions on relatively coarse meshes we consider the highly accurate Pade-type differences are as a basic cell-centered discretization for the systems of the conservation laws. The Pade approximations are characterized by high accuracy and very favorable spectral properties, hence to balance the overall accuracy the grid coupling technique should be very accurate. To our opinion this requirement practically rules out the use of overlapping methods based on interpolation, which is very cumbersome and inaccurate in general coordinates. We consider an approach which can be viewed as method with one layer of cells overlap. This shared layer is crossed by the interface surface, where the interface fluxes are treated as unknown variables and, accordingly, the continuity of the vector of states across the interface is imposed as the surface integral equalities. Evidently in this approach the unknown fluxes can be interpreted as the Lagrange Multipliers (LM's) for the cross-interface continuity constraint. Similar methods are widely used now in the framework of finite elements, in most cases with application to elliptic problems. Generally "scalar" LM's or normal components of the fluxes are considered as unknowns, but in this work we treat all components of the fluxes as unknowns. This choice is quite natural when we formulate the surface integral equalities using the vector surface differential ds as compared to the scalar surface differential ds in the conventional approach. The resulting grid coupling technique ensures higher order local truncation error, stability, discrete conservation and geometric con-servation.