Week 9 - FVM Literature Review
Objective: To study and understand
1. The difference between FDM and FVM.
2. Interpolation schemes in FVM
3. Flux limiters in FVM
Description:
There are total 4 Numerical Methods:
1. Finite Element Method (FEM)
2. Boundary Element Method (BEM)
3. Finite Volume Method (FVM)
4. Finite Difference Method (FDM)
Efficient technology for CFD with the finite-difference and finite-volume method has become increasingly popular and have
been using for much longer periods.
FDM
Boundary conditions are needed to truncate the computational domain. They represent communication with the surrounding
world, which is the part that we do not want to include in our simulation. With the finite-difference method, we may easily run
into problems handling boundaries for the purpose of defining the boundary conditions.
The finite-difference method is the most direct approach to discretizing based upon the differential form of the PDE to be
solved. We consider a point in space where we take the continuum representation of the equations and replace it with a set of
discrete equations, called finite-difference equations.
The finite-difference method is typically defined on a regular/structured grid (i.e. the computational domain is usually divided
into hexahedral cells (the grid), and the solution will be obtained at each nodal point.) and this fact can be used for a very efficient
solution methods. Regular grids are useful for very-large-scale simulations on supercomputers.
The method is therefore not usually used for irregular CAD geometries, but more often for rectangular or block-shaped models.
Examples include Formula 1 car computational-fluid-dynamic (CFD) simulations and space-shuttle CFD simulations. The finite-difference method is more difficult to use for handling material discontinuities. In addition, it does not lend itself to local grid
refinement or anything similar to “adaptive mesh refinement.” This may be needed to resolve local rapid variations in solutions
such as around a corner of a complex shape, as described earlier.
But, if we can fit the simulation in a rectangular or box-shaped geometry using a regular grid, efficient implementations are
much easier than for finite-volume methods.
If one can overcome the boundary-condition problem on curved boundaries, the method gives very efficiently and high-quality
results.
The finite-difference method is defined dimension per dimension; this makes it easy to increase the “element order” to get
higher-order accuracy.
FVM
The finite-volume method is similar to the finite-differencing method in that the CAD model is first divided into very small but
finite-sized elements of geometrically simple shapes. Apart from this, the finite-volume method is very different from the
finite-differencing method, beginning with the concept of elements, which are instead referred to as cells
A finite volume method (FVM) discretization is based upon an integral form of the PDE to be solved (e.g. conservation of mass,
momentum, or energy) and can handle discontinuities in solutions. The PDE is written in a form which can be solved for a given
finite volume (or cell). The computational domain is discretized into finite volumes and then for every volume the governing
equations are solved. The resulting system of equations usually involves fluxes of the conserved variable, and thus the
calculation of fluxes is very important in FVM.
The finite-volume method’s strength is that it only needs to do flux evaluation for the cell boundaries. This also holds for
nonlinear problems, which makes it extra powerful for robust handling of (nonlinear) conservation laws appearing in transport
problems.
The local accuracy of the finite-volume method, such as close to a corner of interest, can be increased by refining the mesh
around that corner. However, the functions that approximate the solution when using the finite-volume method cannot be
easily made of higher-order. This is a disadvantage of the finite-volume method compared to the finite-difference method.
The basic advantage of this method over FDM is it does not require the use of structured grids, and the effort to convert the
given mesh into a structured numerical grid internally is completely avoided. As with FDM, the resulting approximate solution is a
discrete, but the variables are typically placed at cell centers rather than at nodal points. This is not always true, as there are also
face-centered finite volume methods. In any case, the values of field variables at non-storage locations (e.g. vertices) are
obtained using interpolation.
FVM can guarantee the local and global conservation of the system.
Interpolation schemes:
The approximations of surface and volume integrals require values of the variable at a location other than the computational
nodes of the control volume. Values at these locations are obtained using interpolation formulae.
Interpolation methods:
Upwind interpolation
Linear interpolation
Quadratic upwind interpolation
Hybrid, TVD, and ENO interpolation schemes
1. Upwind interpolation (UDS)
As the name indicates that we use the node in the upwind or upstream of the given node.
It is used in the case of convection-dominated problems.
The upwind interpolation for approximating the value of a variable f at the east face of a control volume is given by
`fe = {fp, if (v. n)e >0 fE if (v. n)e <0`
This interpolation scheme is equivalent to using the Forward difference method or backward difference method depending on the
flow direction.
If the flow is along -X direction, then the values for this at the interface is given by
For v>0;ui− ≈ ui−1; ui+ ≈ ui
`c ≈ v(ui − ui-1)/(Δx)`Which is equivalent to Backward Difference a Scheme
If the flow is along -X direction, then the values for this at the interface is given by
For v> 0; ui− ≈ ui−1; ui+ ≈ ui
`c ≈ v(ui+1 − ui)/(Δx)` which is equivalent to forwarding Difference Scheme
It is said to be equivalent because all these three methods are of the order of accuracy 1.
Advantages:
UDS is the only approximation that satisfies the boundedness criterion unconditionally irrespective of velocity and grid
spacing i.e. never yield oscillatory solutions.
Disadvantages:
UDS is numerically diffusive
Numerical diffusion is magnified in multi-dimensional problems if the flow is oblique to the grid.
Peaks or rapid variations in the variables will be smeared out.
Only first-order accurate. Hence very fine grids are required to obtain accurate solutions.
2. Linear interpolation (CDS)
Approximate values of the variable at the control volume face center by the linear interpolation of the values at the two
nearest computational nodes.
fe = fEλe + fp(1 − λe)
Where `λe = (xe − xp) / (xE − xP)`
t is simple and most widely used.
The linear interpolation is equivalent to the use of the central difference formula of the first-order derivative and hence this scheme
is also termed as Central Differencing Scheme(CDS).
Average interface values are given by
`ui− 1/2 ≈(ui−1 + ui)/2 and ui+1/2 ≈ (ui + ui+1)/2`
`Ic ≈ v.(ui+1 − ui−1)/(2Δx)`
It is said to be equivalent because both these methods are of the order of accuracy 2.
Advantages:
Approximation of gradients has second order of accuracy on uniform grids
It's formally of first order on non-uniform grids.
Disadvantages:
It may produce oscillatory solutions in the case of large grid space.
3. Hybrid interpolation scheme
It is a blend of UDS and CDS. The hybrid difference scheme exploits the favorable properties of the
upwind and central differencing schemes. It switches to upwind differencing when central differencing produces inaccurate
results at high Pe numbers.
The scheme is fully conservative and since the coefficients are always positive it is unconditionally bounded. It satisfies the
transportiveness requirement by using an upwind formulation for large values of Peclet number.
`fe = γfCDS + (1 − γ)fUDS`
Advantages:
Produces physically realistic solutions and highly stable compared to QUICK
Disadvantages:
Accuracy is of the first order.
4. Quadratic Upwind Interpolation for convective kinematics (QUICK)
It approximates the values of the variable at the CV face center by Quadratic Interpolation of the values at the three nearest
computational nodes (1 Downstream node D & 2 Upstream nodes U, UU).
Quadratic - As it uses 3 nodes
Upwind - As it uses two upstreams
Convective Kinematics - As it is used for convective domain problems
fe = fE + g1(fD − fU) + g2(fUU − fU)
Where,
`g1 = ((xe − xU)(xe − xUU)) /((xUU − xU)(xUU − xD))`
`g2 = ((xe − xU)(xe − xD)) /((xUU − xU)(xUU − xD))`
Advantages:
It is the better version of UDS and CDS because it is more accurate than UDS and more damping in nature than CDS as it
uses 3 nodes to solve.
It is of 3rd order accurate.
Disadvantages:
The use of QUICK can lead to subtle problems caused by such unbounded results.
Implementation of boundary conditions can be problematic with such higher-order schemes.
5. Total variation diminishing scheme (TVD)
The fact that the QUICK scheme and other higher-order schemes can give undershoots and overshoots has led to the
development of second-order schemes that avoid these problems. The class of TVD (total variation diminishing) schemes has
been specially formulated to achieve oscillation-free solutions and has proved to be useful in CFD calculations.
TVD schemes are designed to address this undesirable oscillatory behaviour of higher-order schemes. In TVD schemes the
tendency towards oscillation is counteracted by adding an artificial diffusion fragment or by adding a weighting towards
upstream contribution.
It is 2nd order accurate.
`fe = fp + 1/2ψ(r)(fE − fP )`
where,
ψ(r) is called a flux limiter function
where
`r = (ui − ui−1)/( ui+1 − ui)`
Flux limiter:
For convective fluid flow, it is observed that low order schemes are usually stable but quite dissipative in nature around the
points of discontinuity or shocks while the higher-order schemes are unstable in nature and show oscillations in the vicinity of
discontinuity or shocks. Highly accurate and oscillation-free schemes are known as High-Resolution Schemes.
In order to make the scheme TVD, we must constrain or limit the range of possible values of the additional convective flux
fe = fp + 1/2ψ(r)(fE − fP ), which was originally introduced to make the scheme higher-order. Hence, the function ψ(r) is
called a flux limiter function.
Flux Limiter functions are used to fine-tune the higher order and lower order schemes in such a way that the resulting scheme
gives a higher order accuracy in the smooth region of the flow and maintains first-order accuracy in the vicinity of shocks and
discontinuities. For such a scheme TVD scheme is employed.
`(ui+1/2 ) = f^(low)(i +1/2 ) − ψ(ri)(f^(low)(i +1/2 ) − f^(high) (i +1/2 ))`
`(ui+1/2 ) = f^(low)(i +1/2 ) − ψ(ri)(f^(low)(i +1/2 ) − f^(high) (i -1/2 ))`
where,
`f^(low)`is low precision flux (first-order)
f^(high)is high precision flux (higher-order)
Conclusion: Finite Volume Method (FVM) is really a powerful approach to solve CFD problems. With the help of proper interpolation
schemes and the accurate use of flux limiters in case of any discontinuities present, we can solve a wide range of CFD problems.