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Study of Flux Limiters to Minimize the Numerical Dissipation Based on Entropy-Consistent Scheme

Published online by Cambridge University Press:  24 July 2017

J. Ren ,
G. Wang ,
J. H. Feng  and
M. S. Ma
J. Ren
Affiliation:
Department of Fluid DynamicsSchool of AeronauticsNorthwestern Polytechnical UniversityXi'an, China
G. Wang*
Affiliation:
Department of Fluid DynamicsSchool of AeronauticsNorthwestern Polytechnical UniversityXi'an, China
J. H. Feng
Affiliation:
College of ScienceChang'an UniversityXi'an, China
M. S. Ma
Affiliation:
Computational Aerodynamics InstituteChina Aerodynamics Research and Development CenterMianyang, China
*
*Corresponding author ([email protected])
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Abstract

The use of limiters may impact both accuracy and resolution of a solution. And the accuracy and resolution are highly dependent on the amount of numerical dissipation in a scheme, so the ability of limiters to control numerical dissipation should be improved. In this view, based on the examination of several classical limiters to control dissipation, a class of general piecewise-linear flux limiters (termed GPL limiters) are presented in this paper for Multi-step time-space-separated unsteady schemes. The GPL limiters can satisfy the second-order TVD criterion and contain some existing limiters such as Superbee and Minmod. Using the decrement of discrete total entropy to represent the amount of numerical dissipation, an entropy dissipation function of GPL limiters is defined with three parameter variables. By proving the monotonicity of this function, a new GPL type limiter (named MDF individually), which can minimize the numerical dissipation and improve the calculation accuracy, is proposed. A high resolution entropy-consistent scheme is obtained by MDF limiter, which will be proved to satisfy entropy stability and entropy consistency. Computational results of this scheme for several 1-D and 2-D Euler test cases are presented, demonstrating the accuracy, monotonicity and robustness of MDF limiter.

Keywords

Flux-limiterHigh resolutionSecond order TVD criterionNumerical dissipation
Type
Research Article
Information
Journal of Mechanics , Volume 34 , Issue 2 , April 2018 , pp. 135 - 149
Copyright
Copyright © The Society of Theoretical and Applied Mechanics 2018 

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References

1. Zhang, D., Jiang, C., Liang, D. and Cheng, L., “A Review on TVD Schemes and a Refined Flux-limiter for Steady-state Calculations,” Journal of Computational Physics, 302, pp. 114–154 (2015).CrossRefGoogle Scholar
2. Sweby, P. K., “High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws,” SIAM Journal on Numerical Analysis, 21, pp. 995–1011 (1984).CrossRefGoogle Scholar
3. Yee, H. C., “Construction of Explicit and Implicit Symmetric TVD Schemes and Their Applications,” Journal of Computational Physics, 68, pp. 151–179 (1987).Google Scholar
4. Daru, V. and Tenaud, C., “High Order One-Step Monotonicity-Preserving Schemes for Unsteady Compressible Flow Calculations,” Journal of Computational Physics, 193, pp. 563–594 (2004).Google Scholar
5. Waterson, N. P. and Deconinck, H., “Design Principles for Bounded Higher-Order Convection Schemes – A Unified Approach,” Journal of Computational Physics, 224, pp. 182–207 (2007).Google Scholar
6. LeVeque, R. J., “High-Resolution Conservative Algorithms for Advection in Incompressible Flow,” SIAM Journal on Numerical Analysis, 33, pp. 627–665 (1996).Google Scholar
7. Kuan, K. B. and Lin, C. A., “Adaptive QUICK-Based Scheme to Approximate Convective Transport,” AIAA Journal, 38, pp. 2233–2237 (2000).Google Scholar
8. Lin, C. H. and Lin, C. A., “Simple High Order Bounded Convection Scheme to Model Discontinuities,” AIAA Journal, 35, pp. 563–565 (1997).CrossRefGoogle Scholar
9. Thomas, J. M., Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations, Springer-Verlag, New York, pp. 204–209 (1999).Google Scholar
10. Harten, A., “High Resolution Schemes for Hyperbolic Conservation Laws,” Journal of Computational Physics, 49, pp. 357–393 (1983).CrossRefGoogle Scholar
11. Venkatakrishnan, V., “Convergence to Steady State Solutions of the Euler Equations on Unstructured Grids with Limiters,” Journal of Computational Physics, 118, pp. 120–130 (1995).Google Scholar
12. Hubbard, M. E., “Multidimensional Slope Limiters For MUSCL-Type Finite Volume Schemes on Unstructured Grids,” Journal of Computational Physics, 155, pp. 54–74 (1999).Google Scholar
13. Buffard, T. and Clain, S., “Monoslope and Multislope MUSCL Methods for Unstructured Meshes,” Journal of Computational Physics, 229, pp. 3745–3776 (2010).Google Scholar
14. Michalak, C. and Ollivier-Gooch, C., “Accuracy Preserving Limiter for High-Order Accurate Solution of the Euler Equations,” Journal of Computational Physics, 228, pp. 8693–8711 (2009).Google Scholar
15. Park, J. S., Yoon, S.-H. and Kim, C., “Multi-Dimensional Limiting Process for Hyperbolic Conservation Laws on Unstructured Grids,” Journal of Computational Physics, 229, pp. 788–812 (2010).CrossRefGoogle Scholar
16. Li, W., Ren, Y.-X., Lei, G. and Luo, H., “The Multi-Dimensional Limiters for Solving Hyperbolic Conservation Laws on Unstructured Grids,” Journal of Computational Physics, 230, pp. 7775–7795 (2011).Google Scholar
17. Li, W. and Ren, Y.-X., “The Multi-Dimensional Limiters for Solving Hyperbolic Conservation Laws on Unstructured Grids II: Extension to High Order Finite Volume Schemes,” Journal of Computational Physics, 231, pp. 4053–4077 (2012).CrossRefGoogle Scholar
18. Hundsdorfer, W. and Trompert, R. A., “Method of Lines and Direct Discretization: A Comparison for Linear Advection,” Applied Numerical Mathematics, 13, pp. 469–490 (1994).Google Scholar
19. Kemm, F., “A Comparative Study of TVD-Limiters – Well-Known Limiters and an Introduction of New Ones,” International Journal for Numerical Methods in Fluids, 67, pp. 404–440 (2011).Google Scholar
20. Jeng, Y. N. and Payne, U. J., “An Adaptive TVD Limiter,” Journal of Computational Physics, 118, pp. 229–241 (1995).Google Scholar
21. Kadalbajoo, M. K. and Kumar, R., “A High Resolution Total Variation Diminishing Scheme for Hyperbolic Conservation Law and Related Problems,” Applied Mathematics and Computation, 175, pp. 1556–1573 (2006).CrossRefGoogle Scholar
22. Dubey, R. K., “Flux Limited Schemes: Their Classification and Accuracy Based on Total Variation Stability Regions,” Applied Mathematics and Computation, 224, pp. 325–336 (2013).Google Scholar
23. Harten, A., Engquist, B., Osher, S. and Chakravarthy, S.R., “Uniformly High Order Accurate Essentially Non-Oscillatory Schemes, III,” Journal of Computational Physics, 71, pp. 231–303 (1987).CrossRefGoogle Scholar
24. Abgrall, R., “On Essentially Non-Oscillatory Schemes on Unstructured Meshes: Analysis and Implementation,” Journal of Computational Physics, 144, pp. 45–58 (1994).Google Scholar
25. Friedrich, O., “Weighted Essentially Non-Oscillatory Schemes for the Interpolation of Mean Values on Unstructured Grids,” Journal of Computational Physics, 144, pp. 194–121 (1998).Google Scholar
26. Hu, C. and Shu, C. W., “Weighted Essentially Non-Oscillatory Schemes on Triangular Meshes,” Journal of Computational Physics, 150, pp. 97–127 (1999).Google Scholar
27. Dumbser, M. and Käser, M., “Arbitrary High Order Non-Oscillatory Finite Volume Schemes on Unstructured Meshes for Linear Hyperbolic Systems,” Journal of Computational Physics, 221, pp. 693–723 (2007).Google Scholar
28. Hou, J., Simons, F. and Hinkelmann, R., “A New TVD Method for Advection Simulation on 2D Unstructured Grids,” International Journal for Numerical Methods in Fluids, 71, pp. 1260–1281 (2013).Google Scholar
29. Zhang, D., Jiang, C., Yang, C. and Yang, Y., “Assessment of Different Reconstruction Techniques for Implementing the NVSF Schemes on Unstructured Meshes,” International Journal for Numerical Methods in Fluids, 74, pp. 189–221 (2014).CrossRefGoogle Scholar
30. Arora, M. and Roe, P. L., “A Well-Behaved TVD Limiter for High-Resolution Calculations of Unsteady Flow,” Journal of Computational Physics, 132, pp. 3–11 (1997).Google Scholar
31. Cockburn, B. and Shu, C. W., “Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems,” Journal of Scientific Computing, 16, pp. 173–261 (2001).Google Scholar
32. Tadmor, E., “The Numerical Viscosity of Entropy Stable Schemes for Systems of Conservation Laws. I,” Mathematics of Computation, 49, pp. 91–103 (1987).Google Scholar
33. Ismail, F. and Roe, P. L., “Affordable, Entropy-Consistent Euler Flux Functions II: Entropy Production at Shocks,” Journal of Computational Physics, 228, pp. 5410–5436 (2009).CrossRefGoogle Scholar
34. Dafermos, C., Hyperbolic Conservation Laws In Continuum Physics, Springer, Berlin (2000).Google Scholar
35. Van Leer, B., “Towards the Ultimate Conservative Difference Scheme, ii. Monotonicity and Conservation Combined in a Second Order Scheme,” Journal of Computational Physics, 14, pp. 361–370 (1974).CrossRefGoogle Scholar
36. Gottlieb, S., Shu, C.W. and Tadmor, E., “Strong Stability-Preserving High-Order Time Discretization Methods,” SIAM Review, 43, pp. 89–112 (2001).Google Scholar
37. Jiang, G. S. and Shu, C. W., “Efficient Implementation of Weighted ENO Schemes,” Journal of Computational Physics, 126, pp. 202–228 (1996).Google Scholar
38. Woodward, P. and Colella, P., “The Numerical Simulation of Two-Dimensional Fluid Flow with Strong Shocks,” Journal of Computational Physics, 54, pp. 115–173 (1984).Google Scholar
39. Liou, M.-S., “A Sequel to AUSM: AUSM +,” Journal of Computational Physics, 129, pp. 364–382 (1996).CrossRefGoogle Scholar
40. Kim, S., Kim, C., Rho, O.-H. and Hong, S. K., “Cures for the Shock Instability: Development of A Shock-Stable Roe Scheme,” Journal of Computational Physics, 185, pp. 342–374 (2003).Google Scholar
41. Liou, M.-S., “A Sequel to AUSM, Part II: AUSM +-up for All Speeds,” Journal of Computational Physics, 214, pp. 137–170 (2006).Google Scholar
42. Nishikawa, H. and Kitamura, K. “Very Simple, Carbuncle-Free, Boundary-Layer-Resolving, Rotated-Hybrid Riemann Solvers,” Journal of Computational Physics, 227, pp. 2560–2581 (2008).CrossRefGoogle Scholar