Error estimator and adaptive moving grids for finite volumes schemes

A new adaptation technique has been developed for two- and three-dimensional grids, based on relocation of the mesh nodes. The flow is governed by the Euler equations expressed in an Euler-Lagrange form and the discretization uses a first-order finite volume scheme based on Roe's approximate Riemann solver. The grid is unstructured, and its motion and evolution obey the physical and geometrical conservation laws. The error is evaluated by a reconstructed, higher order solution, which is used to derive a forcing term to drive the motion of the grid nodes. This is formulated so as to uniformly distribute the error, while keeping the same grid connectivity. The development is made for two- and three-dimensional grids, although results are presented only for two-dimensional flows. HE adaptation of the grid to the flow characteristics is widely recognized as a method to improve the solution accuracy and the overall computing efficiency of numerical schemes. The moti- vation is that since the grid is normally generated before knowing the flow behavior, it is possible that the nodes distribution does not correspond to the solution requirements. The adaptivity process is made up of two key elements: an error estimator and an adaptation procedure. The simplest error estimator, used for finite volume schemes,1'2 is based on the physical behavior of the flowfield, more specifically on the gradient of the density or the Mach number. The adaptation consists in refining and coarsening the mesh in regions of higher or lower gradients respectively. The projection method developed by Zienkiewicz and Zhu3'4 is based on the observation that for finite elements discretization, the solution is continuous, whereas the derivatives are discontinuous at the interface of two elements. Those derivatives are projected in a space of continuous functions and the norm of the difference be- tween this projection and the derivatives is used as a measure of the computational error. For some problems, the fluxes, instead of derivatives, are physically continuous in all of the domain and, con- sequently, more appropriate to be projected in a space of continuous functions.5 Babuska and Rheinboldt6 propose an error estimator based on the solution of a local variational problem, where the unknowns are the approximation errors. The advantage consists in the reduction of the size of the linear system to be solved. In compressible flows, with shock waves or contact discontinu- ities, the variables can no longer be considered continuous. The computed errors based on the solution gradient increase unbound- edly with the adaptation of the discontinuity. This acts as permanent source for grid refinement and could produce wrong results.7 Paillere et al. 8 use a wave model to evaluate the error for com- pressible flows. The model combines the effects of the divergence and rotational of the velocity. To reduce the influence of disconti- nuities, the error is multiplied by a power of a characteristic length of the grid cell. The adaptation procedure also considers a minimal local grid size.