Total variation diminishing

In numerical methods, total variation diminishing (TVD) is a property of certain discretization schemes used to solve hyperbolic partial differential equations. The most notable application of this method is in computational fluid dynamics. The concept of TVD was introduced by Ami Harten.[1]

Model equation

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

 

the total variation (TV) is given by

 

and the total variation for the discrete case is,

 

where  .

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

 

Characteristics

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A numerical scheme is said to be monotonicity preserving if the following properties are maintained:

  • If   is monotonically increasing (or decreasing) in space, then so is  .

Harten 1983 proved the following properties for a numerical scheme,

Application in CFD

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In Computational Fluid Dynamics, TVD scheme is employed to capture sharper shock predictions without any misleading oscillations when variation of field variable “ ” is discontinuous. To capture the variation fine grids (  very small) are needed and the computation becomes heavy and therefore uneconomic. The use of coarse grids with central difference scheme, upwind scheme, hybrid difference scheme, and power law scheme gives false shock predictions. TVD scheme enables sharper shock predictions on coarse grids saving computation time and as the scheme preserves monotonicity there are no spurious oscillations in the solution.

Discretisation

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Consider the steady state one-dimensional convection diffusion equation,

 ,

where   is the density,   is the velocity vector,   is the property being transported,   is the coefficient of diffusion and   is the source term responsible for generation of the property  .

Making the flux balance of this property about a control volume we get,

   

Here   is the normal to the surface of control volume.

Ignoring the source term, the equation further reduces to:

 
 
A picture showing the control volume with velocities at the faces,nodes and the distance between them, where 'P' is the node at the center.

Assuming

  and  

The equation reduces to

 

Say,

 
 

From the figure:

 
 

The equation becomes:  The continuity equation also has to be satisfied in one of its equivalent forms for this problem:

 

Assuming diffusivity is a homogeneous property and equal grid spacing we can say

 

we get The equation further reduces to The equation above can be written as where   is the Péclet number

 

TVD scheme

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Total variation diminishing scheme[2][3] makes an assumption for the values of   and   to be substituted in the discretized equation as follows:

 
 

Where   is the Péclet number and   is the weighing function to be determined from,

 

where   refers to upstream,   refers to upstream of   and   refers to downstream.

Note that   is the weighing function when the flow is in positive direction (i.e., from left to right) and   is the weighing function when the flow is in the negative direction from right to left. So,

 

If the flow is in positive direction then, Péclet number   is positive and the term  , so the function   won't play any role in the assumption of   and  . Likewise when the flow is in negative direction,   is negative and the term  , so the function   won't play any role in the assumption of   and  .

It therefore takes into account the values of property depending on the direction of flow and using the weighted functions tries to achieve monotonicity in the solution thereby producing results with no spurious shocks.

Limitations

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Monotone schemes are attractive for solving engineering and scientific problems because they do not produce non-physical solutions. Godunov's theorem proves that linear schemes which preserve monotonicity are, at most, only first order accurate. 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

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References

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  1. ^ Harten, Ami (1983), "High resolution schemes for hyperbolic conservation laws", J. Comput. Phys., 49 (2): 357–393, Bibcode:1983JCoPh..49..357H, doi:10.1016/0021-9991(83)90136-5, hdl:2060/19830002586
  2. ^ Versteeg, H.K.; Malalasekera, W. (2007). An introduction to computational fluid dynamics : the finite volume method (2nd ed.). Harlow: Prentice Hall. ISBN 9780131274983.
  3. ^ Blazek, Jiri (2001). Computational fluid dynamics : Principles and Applications (1st ed.). London: Elsevier. ISBN 9780080430096.

Further reading

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  • 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.
  • Anil W. Date Introduction to Computational Fluid Dynamics, Cambridge University Press.