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A Novel Multi-Dimensional Limiter for High-Order Finite Volume Methods on Unstructured Grids

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Waves Hyperbolic equations and systems

Published online by Cambridge University Press:  31 October 2017

Yilang Liu ,
Weiwei Zhang  and
Chunna Li
Yilang Liu*
Affiliation:
School of Aeronautics, Northwestern Polytechnical University, No. 127 Youyi West Road, Xi'an 710072, China
Weiwei Zhang*
Affiliation:
School of Aeronautics, Northwestern Polytechnical University, No. 127 Youyi West Road, Xi'an 710072, China
Chunna Li*
Affiliation:
National Key Laboratory of Aerospace Flight Dynamics, School of Astronautics, Northwestern Polytechnical University, No. 127 Youyi West Road, Xi'an 710072, China
*
*Corresponding author. Email addresses:[email protected](W.W. Zhang), [email protected](Y. L. Liu), [email protected](C. N. Li)
*Corresponding author. Email addresses:[email protected](W.W. Zhang), [email protected](Y. L. Liu), [email protected](C. N. Li)
*Corresponding author. Email addresses:[email protected](W.W. Zhang), [email protected](Y. L. Liu), [email protected](C. N. Li)
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Abstract

This paper proposes a novel distance derivative weighted ENO (DDWENO) limiter based on fixed reconstruction stencil and applies it to the second- and highorder finite volume method on unstructured grids. We choose the standard deviation coefficients of the flow variables as the smooth indicators by using the k-exact reconstruction method, and obtain the limited derivatives of the flow variables by weighting all derivatives of each cell according to smoothness. Furthermore, an additional weighting coefficient related to distance is introduced to emphasize the contribution of the central cell in smooth regions. The developed limiter, combining the advantages of the slope limiters and WENO-type limiters, can achieve the similar effect of WENO schemes in the fixed stencil with high computational efficiency. The numerical cases demonstrate that the DDWENO limiter can preserve the numerical accuracy in smooth regions, and capture the shock waves clearly and steeply as well.

Keywords

High-order finite volume methodunstructured gridsDDWENO limitershock waves

MSC classification

Secondary: 35L25: Higher-order hyperbolic equations 65M08: Finite volume methods 65M12: Stability and convergence of numerical methods 74J40: Shocks and related discontinuities
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 5 , November 2017 , pp. 1385 - 1412
Copyright
Copyright © Global-Science Press 2017 

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Footnotes

Communicated by Chi-Wang Shu

References

[1] Gao, C Q, Zhang, W W, Li, X T, et al. Mechanism of frequency lock-in in transonic buffeting flow. Journal of Fluid Mechanics, 2017, 818: 528–561.CrossRefGoogle Scholar
[2] Kou, J Q, Zhang, W W. An improved criterion to select dominant modes from dynamic mode decomposition. European Journal of Mechanics B/Fluids, 2017, 62: 109–129.CrossRefGoogle Scholar
[3] Kou, J Q, Zhang, W W, Liu, Y L, et al. The lowest Reynolds number of vortex-induced vibrations. Physics of Fluids, 2017, 29: 041701.CrossRefGoogle Scholar
[4] Li, J, Zhong, C W, Wang, Y, et al. Implementation of dual time-stepping strategy of the gaskinetic scheme for unsteady flow simulations. Physical Review E, 2017, 5(95): 053307.Google Scholar
[5] Wang, Z J. High-order methods for the Euler and Navier-Stokes equations on unstructured grids. Progress in Aerospace Sciences, 2007, 43: 1–41.CrossRefGoogle Scholar
[6] Ollivier-Gooch, C, Nejat, A, Michalak, K. Obtaining and verifying high-order unstructured finite volume solutions on the Euler equations. AIAA Journal, 2009, 47(9): 2105–2120.CrossRefGoogle Scholar
[7] Liu, Y L, Zhang, W W, Jiang, Y W, et al. A high-order finite volume method on unstructured grids using RBF reconstruction. Computers and Mathematics with Applications, 2016, 72: 1096–1117.CrossRefGoogle Scholar
[8] Jameson, A. Analysis and design of numerical schemes for gas dynamics, 1: artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence. International Journal of Computational Fluid Dynamics, 1995, 4(3-4): 171–218.CrossRefGoogle Scholar
[9] Jameson, A, Schmidt, W, Turkel, E. Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. AIAA paper, 1981, 1259: 1981.CrossRefGoogle Scholar
[10] Swanson, R C, Radespiel, R, Turkel, E. On some numerical dissipation schemes. Journal of Computational Physics, 1998, 147(2): 518–544.CrossRefGoogle Scholar
[11] Harten, A. High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics, 1983, 49(3): 357–393 CrossRefGoogle Scholar
[12] Harten, A, Engquist, B, Osher, S, et al. Uniformly high-order essentially non-oscillatory schemes, III. Journal of Computational Physics, 1997, 131: 3–47.CrossRefGoogle Scholar
[13] Friedrich, O. Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. Journal of Computational Physics, 1998, 144: 194–212.CrossRefGoogle Scholar
[14] Hu, C, Shu, C W. Weighted essentially non-oscillatory schemes on triangular meshes. Journal of Computational Physics, 1999, 150:97–127 CrossRefGoogle Scholar
[15] Liu, Y, Zhang, Y T. A robust reconstruction for unstructured WENO schemes. Journal of Scientific Computing, 2013, 54: 603–621.CrossRefGoogle Scholar
[16] Jiang, G S, Shu, C W. Efficient implementation of Weighted ENO schemes. Journal of Computational Physics, 1996, 126: 202–228.CrossRefGoogle Scholar
[17] Shen, Y Q, Zha, G C, Wang, B Y. Improvement of stability and accuracy of implicit WENO scheme, AIAA Journal, 2009, 47: 331–344 CrossRefGoogle Scholar
[18] Qiu, J X, Shu, C W. Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method Π : Two dimensional case. Computers & Fluids, 2005, 34: 642–663.CrossRefGoogle Scholar
[19] Luo, H, Baum, J D, L?hner, R. A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids. Journal of Computational Physics, 2007, 225: 686–713.CrossRefGoogle Scholar
[20] Ivan, L, Groth, C P T. High-order central ENO finite-volume scheme with adaptive mesh refinement. AIAA 2007-4323, 2007.CrossRefGoogle Scholar
[21] Susanto, A, Ivan, L, Sterck, H D, et al. High-order central ENO finite-volume scheme for ideal MHD. Journal of Computational Physics, 2013, 250: 141–164.CrossRefGoogle Scholar
[22] Charest, M R J, Groth, C P T, Gauthier, P Q. A high-order central ENO finite-volume scheme for three-dimensional low-speed viscous flows on unstructured mesh. Communications in Computational Physics, 2015, 3(17): 615–656.CrossRefGoogle Scholar
[23] Ivan, L, Sterck, H D, Susanto, A, et al. High-order central ENO finite-volume scheme for hyperbolic conservation laws on three-dimensional cubed-sphere grids. Journal of Computational Physics, 2015, 282: 157–182.CrossRefGoogle Scholar
[24] Charest, M R J, Canfield, T R, Morgan, N R, et al. A high-order vertex-based central ENO finite-volume scheme for three-dimensional compressible flows. Computers & Fluids, 2015, 114: 172–192.CrossRefGoogle Scholar
[25] Barth, T J, Jespersen, D C. The design and application of upwind schemes on unstructured meshes. AIAA 89-0366, 1989.CrossRefGoogle Scholar
[26] Venkatakrishnan, V.: Convergence to Steady-State Solutions of the Euler Equations on Unstructured Grids with Limiters. Journal of Computational Physics, 1995, 118: 120–130.CrossRefGoogle Scholar
[27] Michalak, C, Ollivier-Gooch, C. Accuracy preserving limiter for the high-order accurate solution of the Euler equations. Journal of Computational Physics, 2009, 228: 8693–8711.CrossRefGoogle Scholar
[28] Hou, J M, Simons, F, Mahgoub, M, et al. A robust well-balanced model on unstructured grids for shallow water flows with wetting and drying over complex topography. Computer Methods in Applied Mechanics and Engineering, 2013, 257: 126–149.CrossRefGoogle Scholar
[29] Kim, K H, Kim, C. Accurate, efficient and monotonic numerical methods for multidimensional compressible flows Part Π: Multi-dimensional limiting process. Journal of Computational Physics, 2005, 208: 570–615.CrossRefGoogle Scholar
[30] Kang, H M, Kim, K H, Lee, D H. A new approach of a limiting process for multi-dimensional flow. Journal of Computational Physics, 2010, 229: 7102–7128.CrossRefGoogle Scholar
[31] Park, J S, Yoon, S H, Kim, C. Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids. Journal of Computational Physics, 2010, 229: 788–812.CrossRefGoogle Scholar
[32] Park, J S, Kim, C. High-order multi-dimensional limiting strategy for discontinuous Galerkin methods in compressible inviscid and viscous flows. Computers & Fluids, 2014, 96: 377–396.CrossRefGoogle Scholar
[33] Li, W A, Ren, Y X. The multi-dimensional limiters for discontinuous Galerkin method on unstructured grids. Computers & Fluids, 2014, 96(13): 368–376.CrossRefGoogle Scholar
[34] Jawahar, P, Kamath, H. A high-resolution procedure for Euler and Navier-Stokes computations on unstructured grids. Journal of Computational Physics, 2000, 164(1): 165–203.CrossRefGoogle Scholar
[35] Yoon, T H, Asce, F, Kang, S K. Finite volume model for two-dimensional shallow water flows on unstructured grids. Journal of Hydraulic Engineering, 2004, 130(7): 678–688.CrossRefGoogle Scholar
[36] Zhang, T S, Shahrouz, A. A slope limiting procedure in discontinuous Galerkin finite element method for gas dynamics applications. International Journal of Numerical Analysis and Modeling, 2005, 2: 163–178.Google Scholar
[37] Cueto-Felgueroso, L, Colominas, I, Fe, J, et al. High-order finite volume schemes on unstructured grids using moving least-squares reconstruction. Application to shallow water dynamics. International Journal for Numerical Methods in Engineering, 2006, 65(3): 295–331.CrossRefGoogle Scholar
[38] Choi, H, Liu, J G. The reconstruction of upwind fluxes for conservation laws: its behavior in dynamic and steady state calculation. Journal of Computational Physics, 1998, 144: 237–256.CrossRefGoogle Scholar
[39] Li, W A, Ren, Y X. The multi-dimensional limiters for solving hyperbolic conservation laws on unstructured grids. Journal of Computational Physics, 2011, 230: 7775–7795.CrossRefGoogle Scholar
[40] Li, W A, Ren, Y X. The multidimensional limiters for solving hyperbolic conservation laws on unstructured grids Π: Extension to high order finite volume schemes. Journal of Computational Physics. 2012, 231: 4053–4077.CrossRefGoogle Scholar
[41] Liu, Y L, Zhang, W W. Accuracy preserving limiter for the high-order finite volume method on unstructured grids. Computers & Fluids, 2017, 149: 88–99.CrossRefGoogle Scholar
[42] Roe, P L. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 1981, 43: 357–372.CrossRefGoogle Scholar
[43] Jameson, A. The Evolution of Computational Methods in Aerodynamics. Journal of Applied Mechanics, 1983, 50(4):1052–1070.CrossRefGoogle Scholar
[44] AGARD AR-211, Test Cases for Inviscid Flow Field, AGARD, 1985.Google Scholar
[45] Vassberg, C, Jameson, A. In pursuit of grid convergence, Part I: Two-dimensional Euler solutions. AIAA paper 2009-4114, 2009.CrossRefGoogle Scholar
[46] Woodward, P, Colella, P. The numerical simulation of two-dimensional fluid flow with strong shocks. Journal of Computational Physics, 1984, 54(1): 115–173.CrossRefGoogle Scholar