Week 9 - FVM Literature Review

Describe the need for interpolation schemes and flux limiters.

Interpolation schemes:

When integral forms are approximated there will be a requirement of values of variables at locations other than that of computational nodes or control-volume centers. To calculate the value of variables at these locations we need interpolation schemes.

example:

consider 1d - heat conduction equation `d/dx(alpha.(dT)/dx) + s = 0` here, `alpha`and s can be function of temperature. If that is the case then the equation becomes non-linear and needed to be solved iteratively.

In Finite volume method different interpolation schemes can be used for different terms, there are numerous interpolation schemes used and here are few frequently used interpolation schemes:

1. Upwind interpolation:

Upwind interpolation scheme is similar to forward or backward difference for the first derivative and mainly depends on the flow direction of the fluid through the mesh. Let `phi` be any quantity for which generic conservative equation is given then in upwind interpolating scheme:

`phi_p = {(phi_P if (v.n_e)>0),(phi_E if (v.n_e)<0):}`

here the error order of the equation of first order, so it is first order scheme

when flow is from direction is changed values also change, based of control volume centers values. Here variation is constant variation not linear.

Advantages:

Un-conditionally satisfies the boundedness criterion initially to give stable solution, will never yeild oscillating type of solution

Disadvantages:

Very fine grid size is needed to obtain accurate results due to high truncation error.

2. Linear interpolation:

Here linear approximation is done between the two adjacent control volume centers to obtain value at the face, at any location of the cartesian grid 'e'

`phi_e = phi_Elamda_e +(1-lamda_e)phi_P`

where, `lamda_e`is called linear interpolation factor given by `lamda_e = (x_e-x_P)/(x_E-x_P)`

linear interpolation is second order accurate, simplest second order scheme most widely used.

Note: This scheme may produce oscillatory solutions. ie, solution not bounded needs flux limiters

3. Quadratic upwind interpolation for convective kinematics (QUICK scheme)

Here rather than constant and straight line profiles a parabolic profile is used, to construct a parabola three points are required and third point is taken in the upstream side.

`phi_e = phi_U+g_1(phi_D-phi_U)+g_2(phi_U-phi_(UU))`

where, D, U and UU denote the downstream, first upstream and second upstream nodes or centre volume points. The coefficient g1 and g2 are given by:

`g_1 = ((x_e-x_u)(x_e-x_(UU)))/((x_D-x_U)(x_D-x_(UU))` and `g_2 = ((x_e-x_U)(x_D-x_e))/((x_U-x_UU)(x_D-x_(UU)))`

quick scheme is having truncation error of third order

One can always generate higher and higher order schemes by using taylor table method but the solution might not always be accurate.

Few other schemes are:

Linear upwind scheme

Skewed upwind scheme

Hybrid interpolation scheme

Total varying diminishing scheme

Flux limiters:

Higher order interpolation schemes when used may produce unbounded solution with dissipative nature at points of discontinuities. To obtain High Resolution schemes containing high accuracy and bounded solution flux limiters are used.

The main objective of using flux limiters is to limit the spatial derivatives to obtain realistic values for solving realistic problems computationally, they are used to maintain higher order accuracy at normal surfaces and when shock or discontinuities are present first order accuracy is maintained.

Consider 1D semi-discrete scheme:

`(du_i)/dt + 1/(trianglex_i)(F(u_(i,e) -u_(i,w))) = 0`

where, `u_(i,e) and u_(i,w)`represent edge fluxes for a given cell, these flux edges are represented in the form of lower order and higher order schemes, a flux limiter function is also multiplied in such a way that it can switch between these two for a given cell as per requirement to limit dissipation at the discontinuities.

`F(u_(i,e)) = f_(i,e)^(low) - phi(r_i)(f_(i,e)^(low)-f_(i,e)^(high))`

`F(u_(i,w)) = f_(i,w)^(low) - phi(r_(i-1))(f_(i,w)^(low)-f_(i,w)^(high))`

where,

`f^(low) and f^(high)`are lower and higher order scheme flux

`phi(r)` is flux limiter function

`r_i = (u_i - u_W)/(u_E - u_i)`

Limiters are of so many types and can be chosen based on the given problem.

ex: CHARM, HCUS, HQUICK, koren, minmod..etc.