Week 9 - FVM Literature Review

Aim : To discuss about different types of Interpolation schemes and Flux Limiters used in finite volume method.

Finite volume method (FVM) :

• Finite volume method (FVM) can be used on all differential equations, which can be written in the divergence form. This effectively writes the equation using divergence operators. The equation is then integrated over the volume. We can then apply Gauss’s theorem converting the volume integral over the divergence into a surface integral across the boundaries. The integral is therefore turned from integrating the differential of the dependent variable inside of the cells into surface integrals of the fluxes of the dependent variable across the boundary of the cells. These integrals can usually be calculated using suitable numerical approximation methods. This simplifies the differential equation substantially.
• Finite volume method (FVM), like FEM, is based on an unstructured (e.g. triangular) mesh. Therefore, it is suitable for irregular and complex geometries. FVM has another advantage over FEM for fluid mechanic problems. So far, the numerical methods that we presented have been based on PDEs. By contrast, FVM is based on the integral form of the conservation laws, rather than their differential form.
• The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. Similar to the finite difference method (FDM) or finite element method (FEM), values are calculated at discrete places on a meshed geometry.

Differences bewteen FVM and FDM :

• A finite difference method (FDM) discretezation is based upon the differential form of the PDE to be solved. Each derivative is replaced with an approximate difference formula that can generallly be derived from a Taylor series expansion. The computational Domain is usually divided into grids and the solution will be obtained at each nodal point.
• A Finite volume method (FVM) discretezation is based upon the integral form of PDE to be solved (eg. conservation of mass, momentum or energy). The PDE is written in a form that can be solved for a given finite volume for cells. The coputational domain is discretized into finite volumes and then for every volume, the governing equations are solved. The resulting system of the equation usually involves fluxes of the conserved variables, and thus the calculation of fluxes is very important in FVM.
• The basic advantage of this method FVM is it doesnot require the use of structured grids, and the effort to convert the given mesh into a structured numerical grid internally is completely avoided.
• Finite Difference Method : It is difficult to satisfy conservation and to apply for irregular geometries.
• Finite Volume Method : It tends to be biased toward edges and one-dimensional physics.

Interpolation Schemes in Finite Volume Method :

The approximation surface & volume integral may require values of the variable at locations other then the computational nodes of the control volume.Values at this locations are obtained using interploation schemes which are as follows:-

1. Upwind Interploation Scheme(UDS)
2. Central Differencing Schemes(CDS)
3. Quadratic Upwind Differencing Scheme(QUICK)
4. Hybrid Interpolation Scheme(HIS)
5. Total Variation Diminishing scheme(TVD)

Below are the some Interpolation Scheme which explain in detail :

1. Upwind Interpolation Scheme(UDS) :

In computational physics, upwind schemes basically a numerical discretization method which use to solve hyperbolic partial differential equations. Upwind schemes use an adaptive or solution-sensitive finite difference stencil to numerically simulate the direction of propagation of information in a flow field.

To understand this scheme lets take an example of 1D linear convection equation which is

∂u∂t+a∂u∂x=0∂u∂t+a∂u∂x=0

The simplest upwind scheme possible in this case is the first-order upwind scheme which is given by

For Backward differencing,

.................(1)

For Forward differencing,

.................(2)

which can written as follows,

a+=max(a,0)a+=max(a,0), a−=min(a,0)a-=min(a,0)

..................(3)

Above two equations (1) and (2) can be combined and written as

...................(4)

Above equation(4) is a general way of writing any upwind-type schemes.

2. Quadratic Upwind Differencing Scheme(QUICK) :

In computational Fluid dynamics QUICK, which stands for Quadratic Upstream Interpolation for Convective Kinematics, is a higher-order differencing scheme that considers a three-point upstream weighted quadratic interpolation for the cell face values this method can be use to solve steady convection-diffusion equation.

Fig:-Qudractic profile use in QUICK scheme

QUICK scheme for 1-D convection-diffusion problems can be written as follows :

where,

Flux Limiters :

For convective fluid flow it has been observed that low order scheme are usually stable but quite scatter in nature around points of discontinuity while the higher order schemes are unstable in nature & show oscillations in the around points of discontinuity.Flux limters functions are used to tune high order & low order schemes in such a way that the resulting scheme gives a high order accuracy in the smooth region of the flow & maintain first order accuracy around points of discontinuity.

For this purpose Total Variation Diminishing scheme is used which can be wriiten as follows:-