A Numerical Study of Integrated Linear Reconstruction for Steady Euler Equations Based on Finite Volume Scheme (Advances in Applied Mathematics and Mechanics, 2024)
Towards the solution reconstruction, one of the main steps in Godunov typefinite volume scheme, a class of integrated linear reconstruction (ILR) methods hasbeen 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 playan 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 steadystate problems. The numerical framework to solve the steady Euler equations consistsof a Newton iteration for the linearization, and a geometric multigrid solver for thederived linear system. It is found that even for a shock free problem, the convergenceof residual towards the machine precision can not be obtained by directly using theILR. 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 post-processing 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 extratwo operations for ILR are very efficient. Several numerical examples are presented toshow the effectiveness of the proposed scheme for the steady state problems.