A third-order subcell finite volume gas-kinetic scheme for the Euler and Navier-Stokes equations on triangular meshes

Highlights

• A subcell finite volume gas-kinetic scheme is developed on triangular meshes.

• A constrained least-square reconstruction is adopted to improve the efficiency.

• The inviscid and viscous fluxes are coupled and computed uniformly.

• It is third-order accurate in space and time with a single stage and without using quadrature points.

• The efficiency is higher than the third-order k-exact gas-kinetic scheme.

Abstract

A third-order gas-kinetic scheme (GKS) based on the subcell finite volume (SCFV) method is developed for the Euler and Navier-Stokes equations on triangular meshes, in which a computational cell is subdivided into four subcells. The scheme combines the compact high-order reconstruction of the SCFV method with the high-order flux evolution of the gas-kinetic solver. Different from the original SCFV method using weighted least-square reconstruction along with a conservation correction, a constrained least-square reconstruction is adopted in SCFV-GKS so that the correction is not necessary and continuous polynomials inside each main cell can be reconstructed. For flows with large gradients, a compact hierarchical WENO limiting is adopted. In the gas-kinetic flux solver, the inviscid and viscous flux are coupled and computed simultaneously. Moreover, the spatial and temporal evolution are coupled nonlinearly which enables the developed scheme to achieve third-order accuracy in both space and time within a single stage. As a result, no quadrature point is required for the flux evaluation and the multi-stage Runge-Kutta method is unnecessary as well. A third-order k-exact finite volume GKS is also constructed with the help of k-exact reconstruction. Compared to the k-exact GKS, SCFV-GKS is compact, and with less computational cost for the reconstruction which is based on the main cell. Besides, a continuous reconstruction can be adopted in SCFV-GKS among subcells in smooth flow regions, which results in less numerical dissipation and less computational cost for flux evolution. Thus the efficiency of SCFV-GKS is much higher. Numerical tests demonstrate the strong robustness, high accuracy and efficiency of SCFV-GKS in a wide range of flow problems from nearly incompressible to supersonic flows with strong shock waves.

Introduction

In the past decades, high-order CFD methods on unstructured meshes have attracted many attentions due to the advantage of high accuracy and efficiency, especially for complex geometries. A lot of high-order methods have been proposed for solving the Euler and Navier-Stokes (NS) equations, such as the Discontinuous Galerkin (DG) method [1], [2], [3], [4], [5], correction procedure via reconstruction (CPR) method [6], [7], [8] and ${\mathrm{P}}_{\mathrm{N}}{\mathrm{P}}_{\mathrm{M}}$ [9], [10] method, etc. Based on the finite volume (FV) method, many high-order schemes have been developed and investigated extensively, typically including the k-exact method [11], [12], [13], ENO method [14], [15] and WENO [16], [17], [18], [19] method. The FV schemes are relatively simpler, more robust near strong discontinuities and may allow larger CFL numbers, when compared with DG methods [20], [21], [22]. However, since only cell-averaged values are available, a large stencil is usually required for high-order reconstruction, which leads to the difficulty of high-order boundary conditions, cache missing, etc [23]. It is more difficult for a non-compact stencil to choose and search for farther cells.

To avoid a large stencil in high-order FV methods, the conservation of spatial derivatives on the face-neighboring cells, or the minimization of the derivatives difference across cell interfaces can be additionally taken into account, resulting in the compact least-square reconstruction and the variational reconstruction [24], [25], [26]. A large linear equation system should be solved to obtain the derivatives, where an iterative method can be adopted and combined into the iteration of implicit schemes to effectively reduce the computational cost. By taking the point values at cell vertexes as additional variables, the multi-moment finite volume scheme has been developed with high-order compact reconstruction, in which the cell averaged values are solved via the finite volume formulation, while the point values are computed based on the differential form of the governing equations [27], [28], [29].

Another way to develop a compact scheme is to subdivide a cell into a set of subcells, based on which high-order reconstruction can be performed compactly. It was originally adopted by Wang et al. in the spectral volume (SV) method [30], [31], [32], [33]. Nevertheless, since the number of subcells is strictly equal to the number of unknowns on each main cell, how to subdivide main cells becomes another difficulty, especially for quadrilateral and hexahedral meshes.

The subcell finite volume (SCFV) method [34] proposed by Pan et al. is similar to SV method considering the use of subcells and the FV discretization. The main difference between them is that, SCFV method extends the stencil to face-neighboring main cells and adopts a weighted least-square reconstruction. In this way, the partition of subcells is easier and more flexible. Theoretically, by a suitable subdivision of subcells, SCFV method can achieve arbitrarily high-order accuracy with a compact stencil. So far, this method has been applied to solve the Euler equations on two-dimensional quadrilateral meshes [20]. Riemann solvers are adopted to compute the numerical flux, combined with the multi-stage Runge-Kutta (R-K) method for updating solutions. Nonetheless, the weighted least-square reconstruction is unable to conserve the averaged solutions on subcells directly. To fix this problem, a simple correction is applied to the solution polynomial on each subcell, which introduces jumps across interfaces.This correction can improve the stability and the resolution of discontinuities. However, it also reduces the efficiency of flux evaluation, compared to SV method.

Besides the reconstruction, the flux solver is also very important for a FV method. The Riemann solver based on the Euler equations is usually adopted, along with a central scheme for the viscous term when solving the NS equations. Based on the mesoscopic Bhatnagar-Gross-Krook (BGK) model, the gas-kinetic scheme (GKS) provides an alternative way to approach the NS solutions [35]. The NS equations can be recovered through the first-order Chapman-Enskog expansion of the gas distribution function, based on which, as well as a Taylor expansion in space and time, the time evolving solution can be obtained through the integral solution of BGK equation. GKS adopts this local solution to compute the flux at a cell interface, achieving the second order accuracy in both space and time through a single stage. Besides, the time evolving solution provides a dynamic adjustment of numerical dissipation inherently. Furthermore, the inviscid and viscous effects are coupled and computed simultaneously, which is also different from traditional Riemann solvers. They guarantee GKS a good balance between the accuracy and robustness, especially in solving hypersonic viscous flows [36].

With the help of a second-order Taylor expansion as well as the local integral solution of BGK equation, a third-order accurate time evolving solution can be explicitly obtained, and thus a third-order multi-dimensional GKS (HGKS or HBGK) can be developed [37], [38], [39]. The solution describes the flow evolution from a piecewise discontinuous initial state around a cell interface to the equilibrium state, in which the high-order spatial and temporal derivatives of the gas distribution function are coupled nonlinearly. Based on this high-order evolution model, a series of high-order GKS have been developed, such as the DG-GKS [40], SV-GKS [33], CPR-GKS [41], CLS-GKS [42], etc. With a third-order Taylor expansion, a fourth-order GKS can be constructed as well [43]. In particular, by using the time-dependent gas distribution function to update flow variables at cell interfaces which are borrowed in the reconstruction, the compact GKS offers a novel way to develop high-order compact FV methods [21], [44]. Recently, a two-stage fourth-order time-stepping method [45] has been proposed for the Lax-Wendroff type flux solvers. It provides a promising way to develop high-order GKS which is very efficient by adopting the second-order gas-kinetic flux solver [46], [47], [48]. Besides, for the time integration of the hyperbolic conservation law, a multi-stage multi-derivative method (MSMD) has been proposed. Then a family of MSMD-GKS has been developed [47], as well as the compact scheme with spectral-like resolution [48]. In viscous flows, it is still necessary to introduce high-order terms to the gas distribution function for better time accuracy, which are derived from the high-order gas evolution model.

The objective of the present study is to develop an efficient and robust high-order scheme on triangular meshes, denoted as SCFV-GKS, by combining the simple and compact reconstruction of the SCFV method with the high-order gas-kinetic flux solver. Different from the original SCFV method, a constrained least-square reconstruction [49] is adopted so that the averaged solutions on subcells can be conserved automatically. As a result, no correction is required and the solution polynomial can be continuous inside each main cell so that the flux for interfaces between subcells can be computed more efficiently. Furthermore, thanks to the high-order space-time evolution mechanism, the flux at a cell interface can be directly integrated along the tangential direction to avoid the Gauss integration, as well as temporally integrated within a time step so that the multistage R-K scheme is unnecessary. When simulating supersonic flows, a suitable limiting procedure is necessary, which is usually time consuming. Thus, it is more efficient to adopt the single stage time-stepping method.

This paper is organized as follows. In Section 2, the construction of the current method is introduced, including the high-order reconstruction based on the SCFV method, as well as the k-exact method, the hierarchical limiting procedure and the high-order gas-kinetic flux solver. Numerical tests are presented in Section 3 with several benchmark tests to demonstrate the performance of the current method in a wide range of flow problems. The last section draws the conclusions.