Interpolation and Gradient Schemes literature review

Objective: To describe the need for interpolation schemes and flux limiters in the Finite Volume Method.

Finite Volume Method (FVM):
The finite volume method (FVM) is a method for representing and evaluating PDE in the form of algebraic equations. Similar to the finite difference method or finite-element method, values are calculated at discrete places on a meshed geometry. \"Finite volume\" refers to the small volume surrounding each node point on a mesh. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many CFD packages.

Interpolation schemes:

Why do we need interpolation schemes in FVM?
As we all know we need to evaluate surface integrals and volume integrals in our finite volume formulation but surface integrals as well as volume integrals, required the value of the unknown variable at a location other than the computational node of the control volume. And the values at these locations are obtained by using the interpolation formula. That\'s why we required interpolation schemes in FVM.

Therefore there are various possibilities for interpolations:
1. Upwind Interpolation
2. Linear Interpolation
3. Quadratic Upwind Interpolation(QUICK)
4. Other Interpolation schemes

1. Upwind Interpolation (UDS):
It is essentially used for convection dominant problems. Here, the velocity at a given point direction of that velocity that determines the values of the variables to be used in the interpolation process. Name \"Upwind\" refers to we are going to make use of a node that is upwind or upstream of the given node.
UDS for approximating the value of a variable `f` at the east face of the control volume is given by,

`f_e={(f_P if(V.n)_e>0),(f_E if (V.n)_e<0))}`

where `f_P` is the point to the left of face e
           `f_E` is the point to the right of face e
            (V.n) is Velocity vector

Numerical stencil:

`--P---e---E--`

`***`UDS is equivalent to Backward or forward finite difference approximation (depending on the flow direction)

Advantage:
The only approximation that satisfies boundness criterion unconditionally i.e. irrespective of what velocity is and irrespective of grid spacing the solution will always remain bounded. The use of UDS will never yield an oscillatory solution.
Disadvantage:
1. UDS is numerically diffusive i.e. if we\'ve stopped peaks or oscillatory values of which are actually present in solution those will be damped out because of numerical diffusion.
2. Numerical diffusion is magnified in multi-dimensional problems if the flow is oblique to the grid.
3. Peaks or rapid variation in variables will be smeared out because of the introduction of the diffusion coefficient that we call the numerical diffusion coefficient by the use of UDS.
4. Only first-order accurate, hence, very fine grids are required to obtain an accurate solution.

2. Linear Interpolation (CDS):
Approximation of the value of the variable at CV face center by linear interpolation of the value at two nearest computational nodes.

`f_e= f_Elambda_e+f_P(1-lambda_e)`

Therefore, `lambda_e=(x_e-x_P)/(x_E-x_P)`

where, `lambda`is waiting function, depends on the relative location of computational nodes E & P with respect to the eastern face.

`***`The linear interpolation is equivalent to the use of central difference formula of the first-order derivative & hence, this scheme is also termed as Central Difference Schemes(CDS).

Numerical stencil:

`--P---e---E--`

`***`CDS schemes is second-order accurate. It may produce an oscillatory solution.
`***`Leads to a very simple approximation for evaluation of gradient required for evaluation of diffusive fluxes.
`***`Approximation of the gradient has the second-order accuracy on uniform grids.

3. Quadratic Upwind Interpolation schemes (QUICK):
Quadratic Upwind Interpolation is second-order accurate derived for the purpose which has better damping property than CDS & got better accuracy than CDS.
`***`Approximate the value of the variable at CV face center by quadratic interpolation of the values at three nearest computational nodes (one downstream node D and two upstream nodes U, UU.

`f_e= f_U+g_1(f_D-f_U)+g_2(f_(UU)-f_U)`

Therefore, `g_1=((x_e-x_U)(x_e-x_(UU)))/((x_D-x_U)(x_D-x_(UU))`
                `g_2=((x_e-x_U)(x_e-x_(D)))/((x_(UU)-x_U)(x_(UU)-x_(D)))`

Numerical stencil:

`--(UU)/(W)---(U)/(P)--e--D/E---\"EE\"`

if `U_e>0`
Downstream node,
`D~=E`
`U-=P`
`UU-=W`

if `U_e<0`
Downstream node,
`D-=P`
`U-=E`
`\"UU\"-=\"EE\"`

Function f can be approximated by a quadratic interpolation as

`f(x)= a_0+a_1(x-x_0)+a_2(x-x_U)^2`
where, `a_0,a_1,a_2`is coefficients.

`***`Leonard (1979) named this scheme as QUICK (Quadratic Upwind Interpolation for convective kinematics).
`***`It is Third-order accurate on both uniform and non-uniform grids.

`***`On uniform grids, its simplified form is

`phi_e=6/8*phi_U+3/8*phi_D-1/8*phi_(UU)`

4. Other Interpolation methods:
    `->`Hybrid Interpolation schemes:
            Blend of CDS and UDS scheme based on local peclet number

`f_e=gamma*f_(CDS)+(1-gamma)f_(UDS)`
where, `gamma` depends on Peclet number.

     `->`Total Variation Diminishing (TVD) schemes:
             Total variation diminishing scheme is given by

`f_e= f_P+1/2*psir(f_E-f_P)`
where, `psi`is flux limiter function.

Flux limiters:

Flux limiters are used in high-resolution schemes– numerical schemes used to solve problems in science and engineering, particularly fluid dynamics, described by Partial Differential Equations (PDE\'s). They are used in high-resolution schemes, such as the MUSCL schemes to avoid the spurious oscillations (wiggles) that would otherwise occur with high order spatial discretization schemes due to shocks, discontinuities or sharp changes in the solution domain. The use of flux limiters, together with an appropriate high-resolution scheme, make the solutions total variation diminishing(TVD).
Note that flux limiters are also referred to as slope limiters because they both have the same mathematical form, and both have the effect of limiting the solution gradient near shocks or discontinuities. In general, the term flux limiter is used when the limiter acts on system fluxes, and the slope limiter is used when the limiter acts on system states (like pressure, velocity, etc.).

The main idea behind the construction of flux limiter schemes is to limit the spatial derivatives to realistic values – for scientific and engineering problems this usually means physically realizable and meaningful values. They are used in high-resolution schemes for solving problems described by PDEs and only come into operation when sharp wavefronts are present. For smoothly changing waves, the flux limiters do not operate and the spatial derivatives can be represented by higher-order approximations without introducing spurious oscillations.

Consider the 1D semi-discrete scheme below,

where, and represent edge fluxes for the ith cell. If these edge fluxes can be represented by low and high-resolution schemes, then a flux limiter can switch between these schemes depending upon the gradients close to the particular cell, as follows,

,,

where, low precision, high-resolution flux,

 high precision, low-resolution flux,

  flux limiter function,

and  represents the ratio of successive gradients on the solution mesh, i.e.,

.

The limiter function is constrained to be greater than or equal to zero, i.e. . Therefore, when the limiter is equal to zero (sharp gradient, opposite slopes or zero gradients), the flux is represented by a low-resolution scheme. Similarly, when the limiter is equal to 1 (smooth solution), it is represented by a high-resolution scheme. The various limiters have differing switching characteristics and are selected according to the particular problem and solution scheme. No particular limiter has been found to work well for all problems, and a particular choice is usually made on a trial and error basis.