Total variation diminishing

  1. Numerical differential equations
  2. Computational fluid dynamics


In numerical methods, total variation diminishing (TVD) is a property of certain discretization schemes used to solve hyperbolic partial differential equations. The concept of TVD was introduced by Ami Harten.[1]

In systems described by partial differential equations, such as the following hyperbolic advection equation,

\[\frac{\part u}{\part t} + a\frac{\part u}{\part x} = 0, \]

the total variation (TV) is given by,

\[TV = \int \left| \frac{\part u}{\part x} \right| dx ,\]

and the total variation for the discrete case is,

\[TV = \sum_j \left| u_{j+1} - u_j \right| .\]

A numerical method is said to be total variation diminishing (TVD) if,

\[TV \left( u^{n+1}\right) \leq TV \left( u^{n}\right) .\]

A system is said to be monotonicity preserving if the following properties are maintained as a function of t:

  • No new local extrema can be created within the solution spatial domain,
  • The value of a local minimum is non-decreasing, and the value of a local maximum is non-increasing.

Harten 1983 proved the following properties for a numerical scheme,

Monotone schemes are attractive for solving engineering and scientific problems because they do not provide non-physical solutions.

Godunov's theorem proves that only first order linear schemes preserve monotonicity and are therefore TVD. Higher order linear schemes, although more accurate for smooth solutions, are not TVD and tend to introduce spurious oscillations (wiggles) where discontinuities or shocks arise. To overcome these drawbacks, various high-resolution, non-linear techniques have been developed, often using flux/slope limiters.

See also

References

  1. ↑ Harten, Ami (1983), "High resolution schemes for hyperbolic conservation laws", J. Comput. Phys. 49: 357–393, doi:10.1006/jcph.1997.5713

Further reading

  • Hirsch, C. (1990), Numerical Computation of Internal and External Flows, Vol 2, Wiley.
  • Laney, C. B. (1998), Computational Gas Dynamics, Cambridge University Press.
  • Toro, E. F. (1999), Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag.
  • Tannehill, J. C., Anderson, D. A., and Pletcher, R. H. (1997), Computational Fluid Mechanics and Heat Transfer, 2nd Ed., Taylor & Francis.
  • Wesseling, P. (2001), Principles of Computational Fluid Dynamics, Springer-Verlag.


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