Abstract
One of the bottlenecks for high-order numerical methods on unstructured grids is the design of effective and robust limiters for the simulation of flow with discontinuities. The challenge is to maintain the high-order accuracy in smooth regions while capturing the discontinuities in high resolutions without oscillations. This chapter will provide several high order and high resolution nonoscillatory limiters for the high-order FVM. Before introducing the unstructured FVM limiter, we review the historical research for the nonoscillatory schemes in the development of computational fluid methods.
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Godunov SK (1959) A finite-difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat Sbornik 47:271–306
Harten A, Engquist B, Osher S et al (1987) Uniformly high order accurate essentially non-oscillatory schemes (III). J Comput Phys 71:231–303
Harten A (1991) Recent development in shock-capturing schemes. ICASE report No. 91–8
Zhang XX, Shu CW (2010) On maximum-principle-satisfying high order schemes for scalar conservation laws. J Comput Phys 229:3091–3120
Barth TJ, Jespersen D (1989) The design and application of upwind schemes on unstructured meshes. In: Proceedings of the 27th AIAA aerospace sciences meeting, Reno, NV, Paper AIAA 89–0366
Venkatakrishnan V (1995) Convergence to steady state solutions of the Euler equations on unstructured grids with limiters. J Comput Phys 118:120–130
Park JS, Yoon SH, Kim C (2010) Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids. J Comput Phys 229:788–812
Park JS, Kim C (2011) Higher-order discontinuous Galerkin-MLP methods on triangular and tetrahedral grids. In: AIAA 2011–3059, 20th AIAA computational fluid dynamics conference, Honolulu, Hawaii, 27–30 June 2011
Cockburn B, Shu CW (2001) Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J Sci Comput 16:173–261
Luo H, Baum JD, Löhner R (2007) A hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids. J Comput Phys 225:686–713
Yang M, Wang ZJ (2009) A parameter-free generalized moment limiter for high-order methods on unstructured grids. Adv Appl Math Mech 1(4):451–480
Li WA, Ren YX. High order k-exact WENO finite volume schemes for solving gas dynamic Euler equations on unstructured grids. Int J Numer Meth Fluids DOI:10.1002/fld.2710
Friedrich O (1998) Weighted essentially non-oscillatory schemes for the interpolation of mean values on unstructured grids. J Comput Phys 144:194–212
Dumbser M, Käser M (2007) Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J Comput Phys 221:693–723
Dumbser M, Käser M, Titarev VA et al (2007) Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys 226:204–243
Hu C, Shu CW (1999) Weighted essentially non-oscillatory schemes on triangular meshes. J Comput Phys 150:97–127
Sweby PK (1984) High-resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J Numer Anal 21(5):995–1011
Choi H, Liu JG (1998) The reconstruction of upwind fluxes for conservation laws: its behavior in dynamic and steady state calculations. J Comput Phys 144:237–256
Zoppou C, Roberts S (2003) Explicit schemes for dam-break simulations. J Hydraul Eng 129:11–34
Ĉada M, Torrilhon M (2009) Compact third-order limiter functions for finite volume methods. J. Comput. Phys. 228:4118–4145
Li WA, Ren YX, Lei GD et al (2011) The multi-dimensional limiters for solving hyperbolic conservation laws on unstructured grids. J Comput Phys 230(21):7775–7795
Li WA, Ren YX (2012) The multi-dimensional limiters for solving hyperbolic conservation laws on unstructured grids II: extension to high order finite volume schemes. J Comput Phys 231(11):4053–4077
Wang ZJ (2002) Spectral (finite) volume method for conservation laws on unstructured grids, basic formulation. J Comput Phys 178:210–251
Krivodonova L et al (2004) Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl Numer Math 48:323–338
LeVeque RJ (1996) High-resolution conservative algorithms for advection in incompressible flow. Siam J Numer Anal 33:627–665
Sod G (1978) A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J Comput Phys 27:1–31
Lax PD (1954) Weak solutions of nonlinear hyperbolic equations and their numerical computation. Commun Pure Appl Math 7:159–193
Shu CW, Osher S (1988) Efficient implementation of essentially non-oscillatory shock capturing schemes. J Comput Phys 77:439–471
Jiang GS, Shu CW (1996) Efficient implementation of weighted ENO schemes. J Comput Phys 126:202–228
Woodward P, Colella P (1984) The numerical simulation of two-dimensional fluid flow with strong shocks. J Comput Phys 54:115–173
Yee HC, Sandham ND, Djomehri MJ (1999) Low-dissipative high-order shock-capturing methods using characteristic-based filters. J Comput Phys 150:199–238
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Li, W. (2014). Accuracy Preserving Limiters for High-Order Finite Volume Methods. In: Efficient Implementation of High-Order Accurate Numerical Methods on Unstructured Grids. Springer Theses. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43432-1_3
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DOI: https://doi.org/10.1007/978-3-662-43432-1_3
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