FVM Literature Review

INTERPOLATION SCHEMES:

In Finite Difference Method, the governing equations were solved at each and every point of the volume. The Finite Volume methods solve only a single point(centre) in the volume and says that if the governing equations are valid at a particular point it is valid in the entire volume.

But when the geometry is complex or the volume provided by meshing is large in number, there is a boundary or face that separate two volumes. So there is a problem encountered to assign the value at this region.

To overcome these Interpolation schemes were introduced.

NOTE: Here we take the east cell as the reference for explaining various schemes.

Various Interpolation Schemes are:

First-order Interpolation Schemes:

The first order differencing schemes are the oldest differencing schemes specifically designed to fix the stability problems of the discrete advection equation.

The First Order Upwind Scheme (FOU):

The method was originally developed from a comparison with boundary layer flow, where the transport equations are parabolic, with scalars being transported downstream by the flow, and so the value of a scalar at a face can be approximated by its value at the first node upstream of the face. Thus for the east face of a cell

Where `m_e`is the mass flux.

The Hybrid Scheme:

In order to combine the accuracy of Central Differencing with the stability of the First Order Upwind scheme, a hybrid of the two schemes was suggested by Spalding.

Using this combination of first and second-order differencing the hybrid scheme approximates the value of the scalar at a cell east face as

The Exponential and Power Schemes:

For one-dimensional flows, the scheme can be shown to give exact solutions to the flow, but as for all the first-order schemes it exhibits numerical diffusion in multidimensional flows. The scheme uses an exponential approximation to the scalar field and is called the exponential scheme, although it is sometimes referred to as the exact scheme.

Second-Order Interpolation Schemes:

Central Difference Scheme (CDS):

For the one-dimensional cell the value of the scalar at the right or eastern face of the cell is estimated using a linear interpolation between the two neighbouring values, so

Second-Order Upwind Scheme (SOU):

For the second-order upwind scheme the value of a scalar at a face is approximated by a second-order upwind extrapolation. For the east face of a finite volume mesh, the scalar is estimated as

Monotonic Second-Order Upwind Differencing Scheme (MSOU):

The Monotonic Second-Order Upwind scheme was developed. A flux limiter is added to the SOU differencing scheme to prevent the formation of oscillations in the scalar field. The method is implemented as a second-order correction to first-order upwind differencing (FOU) and so is numerically stable. The approximation of the value of a scalar at the east face of a volume is given by

Third or higher Order Differencing Scheme :

Third Order Upwind differencing or Quadratic Upstream Interpolation for Convective Kinematics (QUICK):

The third-order upwind scheme was first described by Leonard in 1979. The two schemes in his first paper, QUICK and QUICKEST, were both explicit finite volume methods, but other authors quickly applied them to implicit finite volume discretisations. In what has become typical Leonard style, he named the schemes QUICK standing for “Quadratic Upwind Interpolation for Convective Kinematics” whilst the QUICKEST scheme had “Estimated Streaming Terms” tagged on the end.

For a third-order upwind scheme the value of a scalar at the east face of a volume is approximated by

assuming a regular mesh with constant `Deltax`Leonard changed, the form of a central difference
scheme with a third-order correction

There are various other interpolation schemes

FLUX LIMITERS:

Whilst the deferred correction schemes improve the numerical stability of the linear systems generated by the QUICK differencing, one of the fundamental problems of high order differencing schemes remains, with the solution field often exhibiting spurious (ie: non-physical) oscillations. To overcome this problem Leonard has applied a number of flux–limiter schemes to the basic third-order upwind biased method.

Few examples are:

For example in ULTRA scheme,

The ULTRA flux limiter is a bound checking algorithm which can be applied to any higher-order differencing scheme, placing an upper and lower limit on the values that the interpolating polynomial can take. It is similar to the flux limiters of Roe that are the basis of the MSOU scheme described above

For any high order interpolation on the east face of a cell, the ULTRA limited interpolation for that face is given by,

.