Flux limiter

The main idea behind the construction of flux limiter schemes is to limit the spatial derivatives to realistic values – for scientific and engineering problems this usually means physically realisable and meaningful values. They are used in high resolution schemes for solving problems described by PDEs and only come into operation when sharp wave fronts are present. For smoothly changing waves, the flux limiters do not operate and the spatial derivatives can be represented by higher order approximations without introducing spurious oscillations. Consider the 1D semi-discrete scheme below,

:$\backslash frac\{d\; u\_i\}\{d\; t\}\; +\; \backslash frac\{1\}\{\backslash Delta\; x\_i\}\; \backslash left[$

F \left( u_{i + {1}/{2}} \right) - F \left( u_{i - {1}/{2}} \right) \right] = 0,

where, $F\; \backslash left(\; u\_\{i\; +\; \{1\}/\{2\}\}\; \backslash right)$ and $F\; \backslash left(\; u\_\{i\; -\; 1/2\}\; \backslash right)$ represent edge fluxes for the i-th cell. If these edge fluxes can be represented by low and high resolution schemes, then a flux limiter can switch between these schemes depending upon the gradients close to the particular cell, as follows,

$$F\; \backslash left(\; u\_\{i\; +\; 1/2\}\; \backslash right)\; =\; f^\backslash text\{low\}\_\{i\; +\; 1/2\}\; -\; \backslash phi\backslash left(\; r\_i\; \backslash right)\; \backslash left(\; f^\backslash text\{low\}\_\{i\; +\; 1/2\}\; -\; f^\backslash text\{high\}\_\{i\; +\; 1/2\}\; \backslash right)\; ,$$

$$F\; \backslash left(\; u\_\{i\; -\; 1/2\}\; \backslash right)\; =\; f^\backslash text\{low\}\_\{i\; -\; 1/2\}\; -\; \backslash phi\backslash left(\; r\_\{i-1\}\; \backslash right)\; \backslash left(\; f^\backslash text\{low\}\_\{i\; -\; 1/2\}\; -\; f^\backslash text\{high\}\_\{i\; -\; 1/2\}\; \backslash right)\; ,$$

where

• $f^\backslash text\{low\}$ is the low resolution flux,
• $f^\backslash text\{high\}$ is the high resolution flux,
• $\backslash phi\backslash \; (r)$ is the flux limiter function, and
• $r$ represents the ratio of successive gradients on the solution mesh, i.e., $$r\_\{i\}\; =\; \backslash frac\{u\_\{i\}\; -\; u\_\{i-1\}\}\{u\_\{i+1\}\; -\; u\_\{i\}\}\; .$$

The limiter function is constrained to be greater than or equal to zero, i.e., $\backslash phi\backslash \; (r)\; \backslash ge\; 0$. Therefore, when the limiter is equal to zero (sharp gradient, opposite slopes or zero gradient), the flux is represented by a low resolution scheme. Similarly, when the limiter is equal to 1 (smooth solution), it is represented by a high resolution scheme. The various limiters have differing switching characteristics and are selected according to the particular problem and solution scheme. No particular limiter has been found to work well for all problems, and a particular choice is usually made on a trial and error basis.

The following are common forms of flux/slope limiter function, $\backslash phi\; (r)$:

• CHARM [not 2nd order TVD] {{citation |last=Zhou |first=G. |year=1995 |title=Numerical simulations of physical discontinuities in single and multi-fluid flows for arbitrary Mach numbers |type=PhD Thesis |publisher=Chalmers Univ. of Tech. |location=Goteborg, Sweden }} $$$$

\phi_{cm}(r) = \begin{cases}

\frac{r\left(3r+1\right)}{\left(r+1\right)^{2}}, & r>0, & \lim_{r\to\infty}\phi_{cm}(r)=3 \\

0 \quad \quad\, , & r\le 0

\end{cases}

• HCUS [not 2nd order TVD] {{citation |last1=Waterson |first1=N.P. |first2=H. |last2=Deconinck |year=1995 |title=A unified approach to the design and application of bounded higher-order convection schemes |type=VKI Preprint 1995-21 }} $$\backslash phi\_\{hc\}(r)\; =\; \backslash frac\{\; 1.5\; \backslash left(r+\backslash left|\; r\; \backslash right|\; \backslash right)\}\{\; \backslash left(r+2\; \backslash right)\}\; ;\; \backslash quad\; \backslash lim\_\{r\; \backslash to\; \backslash infty\}\backslash phi\_\{hc\}(r)\; =\; 3.$$
• HQUICK [not 2nd order TVD] $$\backslash phi\_\{hq\}(r)\; =\; \backslash frac\{2\; \backslash left(r\; +\; \backslash left|r\; \backslash right|\; \backslash right)\}\{\; \backslash left(r+3\; \backslash right)\}\; ;\; \backslash quad\; \backslash lim\_\{r\; \backslash to\; \backslash infty\}\backslash phi\_\{hq\}(r)\; =\; 4.$$
• Koren{{citation |last=Koren |first=B. |year=1993 |contribution=A robust upwind discretisation method for advection, diffusion and source terms |title=Numerical Methods for Advection–Diffusion Problems |editor1-first=C.B. |editor1-last=Vreugdenhil |editor2-first=B. |editor2-last=Koren |publisher=Vieweg |location=Braunschweig |page=117 |isbn=3-528-07645-3 }} – third-order accurate for sufficiently smooth data{{Citation | first=D. |last=Kuzmin |title=On the design of general-purpose flux limiters for implicit FEM with a consistent mass matrix. I. Scalar convection |journal=Journal of Computational Physics |volume=219 |issue=2 |year=2006 |pages=513–531 |doi=10.1016/j.jcp.2006.03.034 |bibcode = 2006JCoPh.219..513K }} $$\backslash phi\_\{kn\}(r)\; =\; \backslash max\; \backslash left[\; 0,\; \backslash min\; \backslash left(2\; r,\; \backslash dfrac\{(1\; +\; 2\; r)\}\{3\},\; 2\; \backslash right)\; \backslash right];\; \backslash quad\; \backslash lim\_\{r\; \backslash to\; \backslash infty\}\backslash phi\_\{kn\}(r)\; =\; 2.$$
• minmod – symmetric {{citation |last=Roe |first=P.L. | author1-link = Philip L. Roe | year=1986 |title=Characteristic-based schemes for the Euler equations |journal=Annu. Rev. Fluid Mech. |volume=18 |pages=337–365 | doi = 10.1146/annurev.fl.18.010186.002005 |bibcode = 1986AnRFM..18..337R }} $$\backslash phi\_\{mm\}\; (r)\; =\; \backslash max\; \backslash left[\; 0\; ,\; \backslash min\; \backslash left(\; 1\; ,\; r\; \backslash right)\; \backslash right]\; ;\; \backslash quad\; \backslash lim\_\{r\; \backslash to\; \backslash infty\}\; \backslash phi\_\{mm\}(r)\; =\; 1.$$
• monotonized central (MC) – symmetric {{citation |last=van Leer |first=B. | author-link = Bram van Leer |year=1977 |title=Towards the ultimate conservative difference scheme III. Upstream-centered finite-difference schemes for ideal compressible flow |journal=J. Comput. Phys. |volume=23 |pages=263–275 |doi=10.1016/0021-9991(77)90094-8 |issue=3 |bibcode = 1977JCoPh..23..263V }} $$\backslash phi\_\{mc\}\; (r)\; =\; \backslash max\; \backslash left[\; 0\; ,\; \backslash min\; \backslash left(\; 2\; r,\; 0.5\; (1+r),\; 2\; \backslash right)\; \backslash right]\; ;\; \backslash quad\; \backslash lim\_\{r\; \backslash to\; \backslash infty\}\backslash phi\_\{mc\}(r)\; =\; 2.$$
• Osher {{citation |last1=Chakravarthy |first1=S.R. |last2=Osher |first2=S. |year=1983 |contribution=High resolution applications of the Osher upwind scheme for the Euler equations |id=AIAA Paper 83-1943 |title=Proc. AIAA 6th Computational Fluid Dynamics Conference |pages=363–373 |url=http://www.aiaa.org/content.cfm?pageid=406&gTable=mtgpaper&gID=55466 |access-date=2008-03-31 |archive-url=https://web.archive.org/web/20110517180238/http://www.aiaa.org/content.cfm?pageid=406&gTable=mtgpaper&gID=55466 |archive-date=2011-05-17 |url-status=dead }} $$\backslash phi\_\{os\}\; (r)\; =\; \backslash max\; \backslash left[\; 0\; ,\; \backslash min\; \backslash left(\; r,\; \backslash beta\; \backslash right)\; \backslash right],\; \backslash quad\; \backslash left(1\; \backslash leq\; \backslash beta\; \backslash leq\; 2\; \backslash right)\; ;\; \backslash quad\; \backslash lim\_\{r\; \backslash to\; \backslash infty\}\backslash phi\_\{os\}\; (r)\; =\; \backslash beta.$$
• ospre – symmetric $$\backslash phi\_\{op\}\; (r)\; =\; \backslash frac\{1.5\; \backslash left(r^2\; +\; r\; \backslash right)\; \}\{\backslash left(r^2\; +\; r\; +1\; \backslash right)\}\; ;\; \backslash quad\; \backslash lim\_\{r\; \backslash to\; \backslash infty\}\backslash phi\_\{op\}\; (r)\; =\; 1.5\; \backslash ,\; .$$
• smart [not 2nd order TVD] {{citation |last1=Gaskell |first1=P.H. |first2=A.K.C. |last2=Lau |year=1988 |title=Curvature-compensated convective transport: SMART, a new boundedness-preserving transport algorithm |journal=Int. J. Numer. Methods Fluids |volume=8 |pages=617–641 | doi = 10.1002/fld.1650080602 | issue = 6 | bibcode = 1988IJNMF...8..617G }} $$\backslash phi\_\{sm\}(r)\; =\; \backslash max\; \backslash left[\; 0,\; \backslash min\; \backslash left(2\; r,\; \backslash left(0.25\; +\; 0.75\; r\; \backslash right),\; 4\; \backslash right)\; \backslash right]\; ;\; \backslash quad\; \backslash lim\_\{r\; \backslash to\; \backslash infty\}\backslash phi\_\{sm\}(r)\; =\; 4.$$
• superbee – symmetric $$\backslash phi\_\{sb\}\; (r)\; =\; \backslash max\; \backslash left[\; 0,\; \backslash min\; \backslash left(\; 2\; r\; ,\; 1\; \backslash right),\; \backslash min\; \backslash left(\; r,\; 2\; \backslash right)\; \backslash right]\; ;\; \backslash quad\; \backslash lim\_\{r\; \backslash to\; \backslash infty\}\backslash phi\_\{sb\}\; (r)\; =\; 2.$$
• Sweby – symmetric {{citation |last=Sweby |first=P.K. |year=1984 |title=High resolution schemes using flux-limiters for hyperbolic conservation laws |journal=SIAM J. Numer. Anal.| volume=21 |pages=995–1011 |doi=10.1137/0721062 |issue=5 |bibcode = 1984SJNA...21..995S }} $$\backslash phi\_\{sw\}\; (r)\; =\; \backslash max\; \backslash left[\; 0\; ,\; \backslash min\; \backslash left(\; \backslash beta\; r,\; 1\; \backslash right),\; \backslash min\; \backslash left(\; r,\; \backslash beta\; \backslash right)\; \backslash right],\; \backslash quad\; \backslash left(1\; \backslash leq\; \backslash beta\; \backslash leq\; 2\; \backslash right)\; ;\; \backslash quad\; \backslash lim\_\{r\; \backslash to\; \backslash infty\}\backslash phi\_\{sw\}\; (r)\; =\; \backslash beta.$$
• UMIST – symmetric {{citation |last1=Lien |first1=F.S. |first2=M.A. |last2=Leschziner |year=1994 |title=Upstream monotonic interpolation for scalar transport with application to complex turbulent flows |journal=Int. J. Numer. Methods Fluids |volume=19 |pages=527–548 |doi=10.1002/fld.1650190606 |issue=6 |bibcode = 1994IJNMF..19..527L }} $$\backslash phi\_\{um\}(r)\; =\; \backslash max\; \backslash left[\; 0,\; \backslash min\; \backslash left(2\; r,\; \backslash left(0.25\; +\; 0.75\; r\; \backslash right),\; \backslash left(0.75\; +\; 0.25\; r\; \backslash right),\; 2\; \backslash right)\; \backslash right]\; ;\; \backslash quad\; \backslash lim\_\{r\; \backslash to\; \backslash infty\}\backslash phi\_\{um\}(r)\; =\; 2.$$
• van Albada 1 – symmetric {{citation |last1=Van Albada |first1=G.D. |first2=B. |last2=Van Leer | author2-link = Bram van Leer |first3=W.W. |last3=Roberts |year=1982 |title=A comparative study of computational methods in cosmic gas dynamics |journal=Astronomy and Astrophysics |volume=108 |issue=1 |pages=76–84 |bibcode=1982A&A...108...76V }} $$\backslash phi\_\{va1\}\; (r)\; =\; \backslash frac\{r^2\; +\; r\}\{r^2\; +\; 1\; \}\; ;\; \backslash quad\; \backslash lim\_\{r\; \backslash to\; \backslash infty\}\backslash phi\_\{va1\}\; (r)\; =\; 1.$$
• van Albada 2 – alternative form [not 2nd order TVD] used on high spatial order schemes {{citation |last1=Kermani |first1=M.J. |last2=Gerber |first2=A.G. |last3=Stockie |first3=J.M. |year=2003 |contribution = Thermodynamically Based Moisture Prediction Using Roe’s Scheme |title=4th Conference of Iranian AeroSpace Society |location=Amir Kabir University of Technology, Tehran, Iran, January 27–29 }} $$\backslash phi\_\{va2\}\; (r)\; =\; \backslash frac\{2\; r\}\{r^2\; +\; 1\}\; ;\; \backslash quad\; \backslash lim\_\{r\; \backslash to\; \backslash infty\}\backslash phi\_\{va2\}\; (r)\; =\; 0.$$
• van Leer – symmetric {{citation |last=van Leer |first=B. | author-link = Bram van Leer |year=1974 |title=Towards the ultimate conservative difference scheme II. Monotonicity and conservation combined in a second order scheme |journal=J. Comput. Phys. |volume=14 |pages=361–370 |doi=10.1016/0021-9991(74)90019-9 |issue=4 |bibcode = 1974JCoPh..14..361V }} $$\backslash phi\_\{vl\}\; (r)\; =\; \backslash frac\{r\; +\; \backslash left|\; r\; \backslash right|\; \}\{1\; +\; \backslash left|\; r\; \backslash right|\; \}\; ;\; \backslash quad\; \backslash lim\_\{r\; \backslash to\; \backslash infty\}\backslash phi\_\{vl\}\; (r)\; =\; 2.$$
• All the above limiters indicated as being symmetric, exhibit the following symmetry property, $$\backslash frac\{\; \backslash phi\; \backslash left(\; r\; \backslash right)\}\{r\}\; =\; \backslash phi\; \backslash left(\; \backslash frac\{1\}\{r\}\; \backslash right)\; .$$

This is a desirable property as it ensures that the limiting actions for forward and backward gradients operate in the same way.

File:LimiterRegion.png

Unless indicated to the contrary, the above limiter functions are second order TVD. This means that they are designed such that they pass through a certain region of the solution, known as the TVD region, in order to guarantee stability of the scheme. Second-order, TVD limiters satisfy at least the following criteria:

• $r\; \backslash le\; \backslash phi(r)\; \backslash le\; 2r,\; \backslash left(\; 0\; \backslash le\; r\; \backslash le\; 1/2\; \backslash right)\; \backslash$,
• $r\; \backslash le\; \backslash phi(r)\; \backslash le\; 1,\; \backslash left(\; 1/2\; \backslash le\; r\; \backslash le\; 1\; \backslash right)\; \backslash$,
• $\backslash phi(1)\; =\; 1\; \backslash$,
• $1\; \backslash le\; \backslash phi(r)\; \backslash le\; r,\; \backslash left(\; 1\; \backslash le\; r\; \backslash le\; 2\; \backslash right)\; \backslash$,
• $1\; \backslash le\; \backslash phi(r)\; \backslash le\; 2,\; \backslash left(\; r\; >\; 2\; \backslash right)\; \backslash$,

The admissible limiter region for second-order TVD schemes is shown in the Sweby Diagram opposite, and plots showing limiter functions overlaid onto the TVD region are shown below. In this image, plots for the Osher and Sweby limiters have been generated using $\backslash beta\; =\; 1.5$.

File:LimiterPlots1.png

An additional limiter that has an interesting form is the van-Leer's one-parameter family of minmod limiters.{{citation |last=Van Leer |first=B. | author-link = Bram van Leer |year=1979 |title=Towards the ultimate conservative difference scheme V. A second order sequel to Godunov's method |journal=J. Comput. Phys. |volume=32 |issue=1 |pages=101–136 |doi=10.1016/0021-9991(79)90145-1 |bibcode = 1979JCoPh..32..101V }}{{citation |last1=Harten |first1=A. |first2=S. |last2=Osher |year=1987 |title=Uniformly high-order accurate nonoscillatory schemes. I |journal=SIAM J. Numer. Anal. |volume=24 |pages=279–309 |doi=10.1137/0724022 |issue=2 |bibcode = 1987SJNA...24..279H |s2cid=15957238 |url=http://www.dtic.mil/get-tr-doc/pdf?AD=ADA158177 |archive-url=https://web.archive.org/web/20170923124820/http://www.dtic.mil/get-tr-doc/pdf?AD=ADA158177 |url-status=dead |archive-date=September 23, 2017 }}{{citation |last1=Kurganov |first1=A. |first2=E. |last2=Tadmor|author2-link=Eitan Tadmor |year=2000 |title=Solution of Two-Dimensional Riemann problems for Gas Dynamics without Riemann Problem Solvers |publisher=Report by Dept. of Mathematics, Univ. Michigan }}. Available on-line at: [http://citeseer.ist.psu.edu/410715.html CiteSeer]. It is defined as follows

$$\backslash phi\_\{mg\}(r,\backslash theta)\; =\; \backslash max\backslash left(0,\backslash min\backslash left(\backslash theta\; r,\backslash frac\{1+r\}\{2\},\backslash theta\backslash right)\backslash right),\backslash quad\backslash theta\backslash in\backslash left[1,2\backslash right].$$

Note: $\backslash phi\_\{mg\}$ is most dissipative for $\backslash theta=1,$ when it reduces to $\backslash phi\_\{mm\},$ and is least dissipative for $\backslash theta\; =\; 2$.

• Godunov's theorem
• High resolution scheme
• MUSCL scheme
• Sergei K. Godunov
• Total variation diminishing

{{reflist}}

• {{citation |last=Hirsch |first=C. |year=1990 |title=Numerical Computation of Internal and External Flows, Volume 2: Computational Methods for Inviscid and Viscous Flows |publisher=Wiley |isbn=978-0-471-92452-4 }}
• {{citation |last1=Leonard |first1=B.P. |first2=M.A. |last2=Leschziner |first3=J. |last3=McGuirk |year=1978 |contribution=The QUICK algorithm: a uniformly 3rd-order finite-difference method for highly convective flows |title=Proc. 1st Conf. on Numerical Methods in Laminar & Turbulent Flow |location=Swansea |page=807 }}

• {{citation |last=Laney |first=Culbert B. |year=1998 |title=Computational Gasdynamics |publisher=Cambridge University Press |isbn=978-0-521-57069-5 }}
• {{citation |last=LeVeque |first=Randall |year=1990 |title=Numerical Methods for Conservation Laws |series=ETH Lectures in Mathematics Series |publisher=Birkhauser-Verlag |isbn=3-7643-2464-3 }}
• {{citation |last=LeVeque |first=Randall |year=2002 |title=Finite Volume Methods for Hyperbolic Problems |publisher=Cambridge University Press |isbn=0-521-00924-3 }}
• {{citation |last=Toro |first=E.F. |year=1999 |title=Riemann Solvers and Numerical Methods for Fluid Dynamics |publisher=Springer-Verlag |edition=2nd |isbn=3-540-65966-8 }}
• {{citation |last1=Tannehill |first1=John C. |first2=Dale Arden |last2=Anderson |first3=Richard H. |last3=Pletcher |year=1997 |title=Computational Fluid Mechanics and Heat Transfer |edition=2nd |publisher=Taylor and Francis |isbn=1-56032-046-X |url-access=registration |url=https://archive.org/details/computationalflu0000tann }}
• {{citation |last=Wesseling |first=Pieter |year=2001 |title=Principles of Computational Fluid Dynamics |publisher=Springer-Verlag |isbn=3-540-67853-0 }}

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