FVM Interpolation and Gradient Schemes literature review

             FVM:: Interpolation and Gradient Schemes literature review

FVM-Flux limiters and interpolation schemes

What is FVM?

FVM is a finite volume method for representing and evaluating PDE's in the form of algebraic equations which are similar to the FDM or FEM. Values are calculate at discrete places on a meshed geometry. Finite volume refers to the small volume surrounding each node point on a mesh. In this method, volume integrals in a partial differential equation that contains a divergence term are converted to surface integrals,using the divergence theorem. Those terms are then evaluated as fluxes at the surface of each finite volume, these methods are conservative.

This method is helpful if the mesh is not even or unstructured meshes, like a mesh we give in openFoam using simple grading which is not possible in matlab. In short FVM computes the solution at each finite volume where FDM computes the solution at each finite points. Also FVM seems to conserve the properties in a better way than the FDM because of the fact that the governing equations solved in an integral form.

So while writing the governing equation in integral form we come up with suitable methods to calculate the flux terms. In this process we use interpolation and gradient scheme. We use the flux limiters to bound the values.

 

1. Interpolation scheme:-

The approximation of surface and volume integrals may require values of the variable at locations other than the computational nodes of CV. Values at these nodes are obtained using the interpolation formaule.

For example in 1D linear heat conduction equation in steady state, the source and the alpha  values are functions of temperature, We need to calculate them seperately of temperature to that particular faces. To calcualte the temperature at that face, we use the interpolation scheme which takes the temperature fro the adjoining cells.

We mention few of the possibilities:-

a. Upwind interpolation:-

It is used for calculating approximate value of the variable at the east face of the control volume, it is first order accurate and is numerically diffusive with coefficient of numerical diffusion. 

`f_e={(f_P if (v*n)_e>0 @ f_E if (v*n)_e<0)`

It is equivalant to using a forward or backward finite difference method . The Upwind relation is given below:-

`γ_e=(ρ*u)_e*((∆x)/2)`

b. Linear interpolation:-

To approximate the value of variable at control volume face centre by two nearest computational nodes location e.

It is equivalant to central differencing scheme of first order derivative. It is second order accurate and produce oscillatory solutions. The rekation for linear interpolation is given below:-

`f_e=f_E*lambda_c+f_P*(1-lambda_c)`,`lambda_c=(x_e-x_P)/(x_E-x_P)`

c. Quadratic upwind interpolation:-

It is third order accurate but prone to oscillations it is derived from polinomial fittings. The QUICK interpolation on the uniform cartesian grid is given by

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

Where D,U,UU denotes the downstream, first upstream and second upstream node respectively (E,P and W or P,E and EE depending on the flow direction)

A quadratic curve is fitted with two upstream nodes and one downstram node but in regions with strong gradients, overshoots and undershoots can occure. This can lead to stability problem in the calculation.

The other interpolation schemes are:-

d. Hybrid interpolation schemes:-(CDS and UDS)

e. Total variable diminishing (TVD) scheme

 

2. Flux limiters:-

Flux limiters are used in high resolution schemes-Numerical schemes used to solve the problems in science and engineering, particularly fluid dyanamics, described by PDE's. They are used in high resolution schemes, such as the MUSCL scheme, to avoid the spurious oscillations that would otherwise occure with high order spatial discritization schemes due to shocks, discountinuities or sharp changes in the solution domain. Use of flux limiters, together with an appropriate high resolution schemes, make the solutions total variable diminishing(TVD).

The main idea behind the flux limiter schemes is to limit the spatial derivatives to realistic values- for scientific and engineering problems this usually means physically realisable and meaningfull values. They are used in hifh resolution schemes for solving problems described ny PDE's and only come into operation when sharp wave fronts are present. For smoothly changing waves, the flux limiters do not operate and and the spatial derivatives can be represented by higher order approximations without introducing spurious oscillations. Consider the 1D semidiscrete scheme below,

`(du_i)/(dt)+(1/(/_\x_i))*[F(u_(i+1/2))-F(u_(i-1/2))]=0`

Where `F(u_(i+1/2)) and F(u_(i-1/2))` represents egde fluxes for the ith cell. These edge fluxes can be represented by low and high resolution schemes, the a flux limiter can switch between these schemes depending upon the gradients close to the particular cell, as follows:-

`F(u_(i+1/2))=f_(i+1/2)^(low)-phi(r_i)*(f_(i+1/2)^(low)-f_(i+1/2)^(high))`

`F(u_(i-1/2))=f_(i-1/2)^(low)-phi(r_(i-1))*(f_(i-1/2)^(low)-f_(i-1/2)^(high))`

Where `f^(low)`= low precision, high resoluation flux 

`f^(high)`=High precision low resolution flux

`phi(r)`=flux limiter function

and r represents the ratio of successive gradients on the solution mesh.

`r_i=(u_i-u_(i-1))/(u_(i+1)-u_i)`

The limiter function is constrained to be grater than or equal to zero, `phi(r)>=0`. Therefore when the limiter is equal to zero(sharp gradient, opposite slopes and zero gradient), the flux is represented by a low resolulation scheme. similarly when the limiter is equal to 1(smooth solution), it is represented by a high resolutiton schemes. The various limiters have differing swithing  characterstics and are selected according to the particular proble and solution scheme. No particular limiter has been found to work well foe all problems, and a particular choice is usually made on a trial and error basis.

 


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