A Numerical Study of Integrated Linear Reconstruction for Steady Euler Equations Based on Finite Volume Scheme

Towards the solution reconstruction, one of the main steps in Godunov type finite volume scheme, a class of integrated linear reconstruction (ILR) methods has been developed recently, from which the advantages such as parameters free and maximum principle preserving can be observed. It is noted that only time-dependent problems are considered in the previous study on ILR, while the steady state problems play an important role in applications such as optimal design of vehicle shape. In this paper, focusing on the steady Euler equations, we will extend the study of ILR to the steady state problems. The numerical framework to solve the steady Euler equations consists of a Newton iteration for the linearization, and a geometric multigrid solver for the derived linear system. It is found that even for a shock free problem, the convergence of residual towards the machine precision can not be obtained by directly using the ILR. With the lack of the differentiability of reconstructed solution as a partial explanation, a simple Laplacian smoothing procedure is introduced in the method as a postprocessing technique, which dramatically improves the convergence to steady state. To prevent the numerical oscillations around the discontinuity, an efficient WENO reconstruction based on secondary reconstruction is employed. It is shown that the extra two operations for ILR are very efficient. Several numerical examples are presented to show the effectiveness of the proposed scheme for the steady state problems.