Flux limiters

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(Difference between revisions)

Latest revision as of 10:19, 22 February 2007

Most of us might have seen the behaviour of numerical schemes in order to capture shocks and discontinuity that arises in hyperbolic equations. Physically, these equations model the convective fluid flow. It has been observed that low-order schemes are usually stable but quite dissipative in nature around the points of discontinuity/shocks. On the other hand higher-order numerical schemes are unstable in nature and show oscillations in the vicinity of discontinuity. One can have a better understanding of such behaviour by analysing the modified equation of these schemes.

Now as said above one can not have high order accuracy without oscillations. The objective is to have highly accurate and stable oscillation free schemes. This kind of schemes are known as high resolution schemes. In 1984 Harten proposed the constuction and gave TVD criteria for such scheme. Here we talk only about using flux limiters function to construct high resolution schemes. The idea is to tune the numerical flux of high order and low order scheme using the flux limiter function in such a way that the resulting scheme gives a high order accuracy in the smooth region of flow and sticks with first order of accuracy in the vicinity of socks/discontinuities as follows:

Define the numerical flux fuction of high resolution conservative scheme as

$F_{j+ \frac{1}{2}} = L_{j+\frac{1}{2}} + \phi(H_{j+ \frac{1}{2}} - L_{j+ \frac{1}{2}})$

where $L, H$ are the numerical flux of conservative low order and high order schemes respectively. and $\phi$ is a function of smoothness parameter $\theta$ usually defined as

$\theta_j = \frac{U_j -U_{j-1}}{U_{j+1} - U{j}}$

What remains next is to define the limiter fuction $\phi(\theta)$ in such a way that it satisfies at least the following properties:

remains positive $\forall \theta$ ,
satisfies $\phi(1) = 1$
passes through a perticular region known as TVD region associted with the underlying scheme in order to guarantee stability of the scheme.

References

C. B. Laney, "Computational Gas Dynamics"
E. F. Toro, "Riemann Solvers and Numerical Methods for Fluid Dynamics"
R. J. Leveque, "Numerical Methods for Conservation Laws"

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-Most of us might have seen the behaviour of numerical schemes in order to capture shocks and discontinuity that arises in hyperbolic equations. Physically, these equations model the convective fluid flow. It has been observed that low-order schemes are usually stable but quite dissipative in nature around the points of discontinuity/shocks. On the other hand higher-order numerical schemes are unstable in nature and show oscillations in the vicinity of discontinuity. One can have a batter understanding of this behaviour by analysing the modified equation of these schemes.
+Most of us might have seen the behaviour of numerical schemes in order to capture shocks and discontinuity that arises in hyperbolic equations. Physically, these equations model the convective fluid flow. It has been observed that low-order schemes are usually stable but quite dissipative in nature around the points of discontinuity/shocks. On the other hand higher-order numerical schemes are unstable in nature and show oscillations in the vicinity of discontinuity. One can have a better understanding of such  behaviour by analysing the modified equation of these schemes.
-The problem is that one can not have high order accuracy without oscillations and without oscillations one has to compromise for accuracy. One needs to optimize these two extremes in order to have highly accurate and stable oscillation free methods. In order to do so, an idea of flux limiters came into the picture. According to the idea, one tunes the numerical flux in such a way that the resulting scheme gives a second order accuracy in the smooth region of flow and sticks with first order of accuracy in the vicinity of shocks/discontinuities. This kind of schemes are known as high resolution schemes and in 1984 Harten gave a paper on one such scheme in Math. Comp. The idea goes like this:
+Now as said above one can not have high order accuracy without oscillations. The objective is to have highly accurate and stable oscillation free schemes. This kind of schemes are known as high resolution schemes. In 1984 Harten proposed the constuction and gave TVD criteria for such scheme. Here we talk only about using flux limiters function to construct high resolution schemes. The idea is to tune the numerical flux of high order and low order scheme using the flux limiter function in such a way that the resulting scheme gives a high order accuracy in the smooth region of flow and sticks with first order of accuracy in the vicinity of socks/discontinuities as follows:
 Define the numerical flux fuction of high resolution conservative scheme as
-<math>F_{j+ \frac{1}{2}} = L_{j+\frac{1}{2}} + \phi(H_{j+ \frac{1}{2}} - L_{j+ \frac{1}{2}})</math>
+:<math>F_{j+ \frac{1}{2}} = L_{j+\frac{1}{2}} + \phi(H_{j+ \frac{1}{2}} - L_{j+ \frac{1}{2}})</math>
 where <math>L, H</math> are the numerical flux of conservative low order and high order schemes respectively.
-and <math>\phi</math> is a function of smoothness parameter <math>\theta</math> defined by <math>\theta_j = \frac{U_j -U_{j-1}}{U_{j+1} - U{j}}</math>.
+and <math>\phi</math> is a function of smoothness parameter <math>\theta</math> usually defined as
-What remains next is to define the limiter fuction <math>\phi(\theta)</math> in such a way that it satisfies at least the following:
-. remains positive <math>\forall \theta</math>,
+:<math>\theta_j = \frac{U_j -U_{j-1}}{U_{j+1} - U{j}}</math>
-. satisfies <math> \phi(1) = 1</math>
-. passes through a perticular region known as TVD region associted with the underlying scheme in order to guarantee
+What remains next is to define the limiter fuction <math>\phi(\theta)</math> in such a way that it satisfies at least the following properties:
-stability of the scheme.
-There are other properties too which it should satisfy for better results.
+* remains positive <math>\forall \theta</math>,
+* satisfies <math> \phi(1) = 1</math>
+* passes through a perticular region known as TVD region associted with the underlying scheme in order to guarantee stability of the scheme.
-In 70's Vanleer came up with his conservative schemes and published 5 papers in a row in Journal of Computational Phyics. Then in 1984 (in JCP) P. K. Sweby gave a scheme. There are many other methods which use more or less the same idea i. e. the idea of adding antidiffusive term in low order scheme. This gave various schemes like Flux Corrected Transport (FCT) by Book and Borris, Piecewise Parabolic Method (PPM) by Colella and Woodward (JCP) etc.
+==References==
-Look for further references in above mentioned papers. One should consult the following books:
+* C. B. Laney, "Computational Gas Dynamics"
-C. B. Laney's "Computational Gas Dynamics", E. F. Toro's "Reimann Solvers and Numerical Methods for Fluid Dynamics", R. J. Leveque's "Numerical Methods for Conservation Laws".
+* E. F. Toro, "Riemann Solvers and Numerical Methods for Fluid Dynamics"
+* R. J. Leveque, "Numerical Methods for Conservation Laws"

Flux limiters

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Latest revision as of 10:19, 22 February 2007

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