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MUSCL scheme: having trouble understanding what total variation diminishing means

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Old   January 15, 2015, 14:04
Default MUSCL scheme: having trouble understanding what total variation diminishing means
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Josh Dawson
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Hello everyone, I hope everyone is having a good day! I was hoping someone, in laymen terms, could help me understand what total variation diminishing means. Thinking aloud...It seems like the scheme targets a property of the convective terms and their interpolation from cell center to cell face. This "property" eliminates the scheme's sensitivity to spurious oscillations common to many central-diff schemes. This inherent characteristic is what makes an algorithm such as MUSCL appropriate for flows with large gradients and flow discontinuities, correct? Are any of those thoughts in the ballpark? Any help or literature reference would be greatly appreciated! Thanks in advance and have a great day!
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Old   January 16, 2015, 08:04
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You can find some information on the Wiki:

Probably you can find some literature there as well.

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Old   January 16, 2015, 08:10
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You are definitely on the right track.

A simple and general explanation from Versteeg (2007) chapter 5.10.2:
"It has been established that the desirable property for a stable,
non-oscillatory, higher-order scheme is monotonicity preserving. For a
scheme to preserve monotonicity, (i) it must not create local extrema and
(ii) the value of an existing local minimum must be non-decreasing and that
of a local maximum must be non-increasing. In simple terms,
monotonicity-preserving schemes do not create new undershoots and overshoots
in the solution or accentuate existing extremes.
Monotonicity-preserving schemes have the property that the total variation
of the discrete solution should diminish with time."
I can generally recommend you familiarize yourself with the Sweby-diagram, which illustrates the monotone and 2nd order TVD regions. A very good introduction is given in Hirsch's book, chapter 8.3.

Hope this helps.


Versteeg, H. K. An Introduction to Computational Fluid Dynamics: The Finite
Volume Method. 2nd ed. Harlow, England: Pearson Education Ltd, 2007.

Sweby, P. K. “High Resolution Schemes Using Flux Limiters for Hyperbolic
Conservation Laws.” SIAM Journal on Numerical Analysis 21, no. 5 (October 1,
1984): 995–1011.

Hirsch, Charles. Numerical Computation of Internal and External Flows: The
Fundamentals of Computational Fluid Dynamics. 2nd ed.
Oxford; Burlington, MA: Butterworth-Heinemann, 2007.
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discontinuity, large gradient, muscl, second order, tvd

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