Elsevier

Journal of Computational Physics

Volume 396, 1 November 2019, Pages 161-192
Journal of Computational Physics

Compact high order finite volume method on unstructured grids IV: Explicit multi-step reconstruction schemes on compact stencil

https://doi.org/10.1016/j.jcp.2019.06.054Get rights and content

Abstract

In the present paper, a multi-step reconstruction procedure is proposed for high order finite volume schemes on unstructured grids using compact stencil. The procedure is a recursive algorithm that can eventually provide sufficient relations for high order reconstruction in a multi-step procedure. Two key elements of this procedure are the partial inversion technique and the continuation technique. The partial inversion can be used not only to obtain lower order reconstruction based on existing reconstruction relations, but also to regularize the existing reconstruction relations to provide new relations for higher order reconstructions. The continuation technique is to extend the regularized relations on the face-neighboring cells to current cell as additional reconstruction relations. This multi-step procedure is operationally compact since in each step only the relations defined on a compact stencil are used. In the present paper, the third and fourth order finite volume schemes based on two-step quadratic and three-step cubic reconstructions are studied.

Introduction

High order methods have shown great capability in the simulation of flows with multi-scale structures [1]. To handle complicated geometries, various high order numerical methods on the unstructured grids have been developed such as the finite volume (FV) methods [2], [3], [4], [5], [6], [7], [8], [9], [56], [57], discontinuous Galerkin (DG) methods [10], [11], [12], [13], [14], spectral volume (SV)/spectral difference (SD) methods [15], [16], [17], [18], [19], [20], [21], PNPM procedure [22], [23], [24] and the hybrid FV/DG methods [25], [26].

Historically, the high order FV methods on the unstructured grids were among the numerical schemes that received earliest attention since they are simpler to construct and implement. The key point of FV schemes is to reconstruct high order representation of the solutions in each cell or control volume. The k-exact FV method was developed by Barth and Frederickson [2]. ENO and WENO schemes were then developed [6], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36]. The high order FV schemes usually require a large number of cells in the stencil of the reconstruction procedure. The large stencil problem is a serious drawback of the traditional FV schemes. It may result in algorithmic complexity [37] in searching and identification of the stencil, large memory requirement to store the stencil and the reconstruction coefficients [25], difficulty in boundary treatment, and impairment of the scalability of the code in parallel computation [51]. Therefore, the large stencil may pose a major obstacle for the application of the high order FV schemes in very large scale engineering simulations.

To overcome this problem, Wang and Ren developed the compact least squares reconstruction (CLSR) scheme [38], [39] on compact stencil. This method is constructed by requiring the variable and its derivatives on the control volume of interest to conserve their averages on the face-neighboring cells. To ensure the non-singularity of the reconstruction procedure, the variational reconstruction (VR) procedure for the high order FV schemes on the unstructured grids was recently proposed [40]. This method can be also applied on a compact stencil and more importantly, can be proved to be non-singular on general shaped unstructured grids. Both CLSR and VR are implicit, and a large system of linear equations should be solved. To design an efficient solution procedure, the CLSR and VR are computed using a certain iterative solution procedure. When solving unsteady flows, they should be coupled with implicit dual time stepping procedure so that only one iteration is performed in each pseudo time step. Using this approach, the FV schemes based on CLSR and VR can be as efficient as those based on the k-exact reconstruction when the implicit time stepping schemes are used. However, when using the explicit time stepping schemes, these methods become less efficient because of the implicit nature of these schemes. Therefore, it is desirable to develop an explicit reconstruction algorithm on a compact stencil which can be readily applied in both explicit and implicit time marching schemes.

We notice some approaches with above-mentioned property have been proposed. For example, Yang et al. [41] used the Gauss-Green theorem successively to obtain high order distribution of the solution in a control volume. However, it is not able to prove if this approach has the property of k-exactness. Haider et al. [42] developed the Coupled Least Squares reconstruction by approximating derivatives from higher order to lower order. However, on the unstructured grids, this approach is depended on some claims that are not easy to prove and is very complicated. Chiravalle et al. [52] developed a two-step reconstruction, which is quite convenient to obtain 3rd order accuracy schemes. The main drawbacks of this method is that the accuracy is restricted to 3rd order.

In the present paper, the FV scheme based on a multi-step reconstruction (MSR) procedure is proposed on 1D mesh and 2D triangular mesh. The MSR is a recursive algorithm which can eventually provide sufficient relations for high order reconstruction in a multi-step procedure. Two key elements of this procedure are the partial inversion technique (PIT) and the continuation technique (CT). The PIT can be used not only to obtain lower order reconstruction based on existing reconstruction relations, but also to regularize the existing reconstruction relations to provide new relations for higher order reconstructions. The CT is to extend the regularized relations on the face-neighboring cells to current cell as additional reconstruction relations. In the implementation of MSR, one does not need to search and store the possible large stencil encountered in the high order FV schemes. Instead, during the implementation of MSR, the cells involved in the reconstruction will increase accordingly. Furthermore, this procedure is operationally compact in the sense that in every step of this multi-step procedure, only the information of current and face-neighboring cells is used. As being discussed in [40], this property is sufficient to ease the data transfer between different sub-domains in the parallel computing based on the domain decomposition approach and is also beneficial to reduce the cache missing encountered by traditional high order FV schemes using a very large stencil. The computational cost is only slightly larger than the traditional k-exact reconstruction using large stencil.

This paper is organized as follows. Section 2 and Section 3 detail the MSR algorithms on 1D and 2D unstructured meshes respectively. In Section 4, numerical results of 1D tests and 2D tests are presented. Finally, conclusions are given in Section 5.

Section snippets

Notations and the basic ideas of the multi-step reconstruction

In this section, the 1D MSR is introduced and the spectral property for corresponding high order FV schemes is analyzed. To facilitate the derivation, some notations are introduced first. In 1D FV methods, the physical domain Ω is decomposed into N non-overlapping control volumes (cells). Ωi[xi1/2,xi+1/2] is the ith cell. hi=xi+1/2xi1/2 is the length of Ωi and xi=(xi1/2+xi+1/2)/2 is the center of Ωi. We denote the cell average of variable u on Ωi asu¯i=1hiΩiu(x)dx. The reconstruction

Notations

In the 2D case, we are interested in the FV method on the unstructured triangular grids. The computational domain Ω is composed of a collection of N non-overlapping space filling triangles, i.e.Ω=i=1NΩi, where Ωi is the i-th cell. For a specific triangular cell Ωi, the boundary is composed of 3 edges or interfacesΩi=m=13Im. Supposing 3 nodes (vertices) of the specific cell Ωi is point 1, 2, 3 and coordinates of those points are (X1,Y1),(X2,Y2),(X3,Y3) respectively (capital letters are used

Numerical results

Numerical tests are presented in this section. These tests demonstrate the property of MSR scheme on 1-D and 2-D unstructured grids. 1-D tests are presented in Sections 4.1–4.3; 2-D tests are presented in Sections 4.4–4.8.

Conclusion

In this paper, MSR procedure is presented for finite volume method on both 1D grids and 2D unstructured grids. MSR is a recursive algorithm which can eventually provide sufficient relations for high order reconstruction in a multi-step procedure. In each step, the PIT is used to regularize the existing reconstruction relations which are usually not sufficient to perform the high order reconstruction but are sufficient to obtain the RRR of the next step. And in the next step, the RRR on the

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 91752114 and 11672160). The 2D finite volume method framework is based on Qian Wang's code.

References (59)

  • Y. Liu et al.

    Spectral difference method for unstructured grids I: basic formulation

    J. Comput. Phys.

    (2006)
  • M. Dumbser et al.

    A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes

    J. Comput. Phys.

    (2008)
  • M. Dumbser

    Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations

    Comput. Fluids

    (2010)
  • M. Dumbser et al.

    Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations

    J. Comput. Phys.

    (2009)
  • L. Zhang et al.

    A class of hybrid DG/FV methods for conservation laws I: basic formulation and one-dimensional systems

    J. Comput. Phys.

    (2012)
  • L. Zhang et al.

    A class of hybrid DG/FV methods for conservation laws II: two-dimensional cases

    J. Comput. Phys.

    (2012)
  • A. Harten et al.

    Uniformly high order accurate essentially non-oscillatory schemes III

    J. Comput. Phys.

    (1987)
  • C.W. Shu et al.

    Efficient implementation of essentially non-oscillatory shock-capturing schemes

    J. Comput. Phys.

    (1988)
  • C.W. Shu et al.

    Efficient implementation of essentially non-oscillatory shock-capturing schemes, II

    J. Comput. Phys.

    (1989)
  • G.S. Jiang et al.

    Efficient implementation of weighted ENO schemes

    J. Comput. Phys.

    (1996)
  • D.S. Balsara et al.

    Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy

    J. Comput. Phys.

    (2000)
  • A.K. Henrick et al.

    Simulations of pulsating one-dimensional detonations with true fifth order accuracy

    J. Comput. Phys.

    (2006)
  • R. Borges et al.

    An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws

    J. Comput. Phys.

    (2008)
  • D.S. Balsara et al.

    Efficient high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydro-dynamics

    J. Comput. Phys.

    (2009)
  • D.S. Balsara

    Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics

    J. Comput. Phys.

    (2009)
  • R. Abgrall

    On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation

    J. Comput. Phys.

    (1994)
  • R. Abgrall et al.

    Construction of very high order residual distribution schemes for steady inviscid flow problems on hybrid unstructured meshes

    J. Comput. Phys.

    (2011)
  • Q. Wang et al.

    Compact high order finite volume method on unstructured grids I: basic formulations and one-dimensional schemes

    J. Comput. Phys.

    (2016)
  • Q. Wang et al.

    Compact high order finite volume method on unstructured grids III: variational reconstruction

    J. Comput. Phys.

    (2017)
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