Progress and Challenge of Limiting Strategy for Higher-Order CFD Methods

This paper presents the recent progress of multi-dimensional limiting process (MLP) and discuss some issues for further improvements. MLP, which has been originally developed in finite volume method (FVM), provides an accurate, robust and efficient oscillation-control mechanism in multiple dimensions for linear reconstruction. This limiting philosophy can be hierarchically extended into higher-order Pn approximation/reconstruction. The resulting algorithm, called the hierarchical MLP, facilitates the capturing of detailed flow structures while maintaining the formal order-ofaccuracy in smooth region and providing accurate non-oscillatory solutions across discontinuous region. This algorithm has been originally developed within the modal DG framework and extended into a nodal framework, most notably the FR/CPR framework. Troubled-cells are detected by applying the MLP condition and smooth extrema detector. The final accuracy is then determined by the projection procedure and the hierarchical MLP limiting step. Through extensive numerical analyses and computations ranging from scalar conservation laws to fluid systems, it is observed that the proposed limiting approach yields the outstanding performances in capturing compressible inviscid and viscous flow features. Some challenging issues are also mentioned to improve and extend the current approach for higher-order simulations of high-Reynolds number compressible flows.