FVM Interpolation and Gradient Schemes literature review

AIM:-

To describe what is finite volume method and the need of interpolation schemes and flux limiters in FVM in our own owrds witha an example.

GIVEN:-

Use the 1D heat equation

`d/dx*alpha.(dT)/dx`

EXPLANATION:-

Finite Volume Method (FVM):- This is similar to FDM (Finite differencing method), we use points in the FDM but in FVM we use volume\'s. We ibntergrate the equation so that the equation becomes volume. Now in FVM there is not need of using aligned grids in a particular direction as we are solving for whole volume by using or implementing some math techniques.

Interpolation Schemes:-

The points or locations that need approximations at a particular point o surface or volume, where there are no computatioal points are there or located we use this interpolation schemes. Basically this is the initial point of boundary layer where we can locate and start the approximation at that particular point or location. There are different types of interpolation schemes that are mentioned below.

1. Upwind Interpolation Scheme (UDS):- This is similar to forward finite difference and backward finite difference (depending on flow direction), the name upwind referes to that we are goping to make use of the node which is upwind or upstream of the given node. This is first order accurate.

2. Linear interpolation:- This is very similar to the Central differencing scheme (CDS), so we have to approximate the value of variable at control volume face center by linear interpolationof the values at two nearest computational nodes. This is simplest and most widely used interpolation. This is second order accurate.

3.Quadratic upwind interpolation (QUICK):-We use this interpolation scheme for the second order accurate to decrease the dqamping more compared to linear interpolation. This is used to reduce the oscillations.

There is one more interpolation method called Hybrid interpolation which is blend of Linear interpolation and upward interpolation.

Flux Limiters:-

Flux limiters are used in high resolution schemes to solve problems in fluid dynamics by partial differencial equations(PDE\'s). they are used to avoid the huge oscillations (wiggles) that would occur with high order spatial discretization schemes due to shocks, discontinuities or sharp changes in the solution domain.

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 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 realisable and meaningful values. They are used in high resolution schemes for solving problems described by PDEs and only come into operation when sharp wave fronts 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,

`(du)/dt+1/(Deltaunderset(i)x)[F(underset(i+1/2)u)-F(underset(i-1/2)u)]=0`

{\\displaystyle {\\frac {du_{i}}{dt}}+{\\frac {1}{\\Delta x_{i}}}\\left[F\\left(u_{i+{\\frac {1}{2}}}\\right)-F\\left(u_{i-{\\frac {1}{2}}}\\right)\\right]=0,

where, `F(underset(i+1/2)u)` {\\displaystyle F\\left(u_{i+{\\frac {1}{2}}}\\right)\\ } and `F(underset(i-1/2)u)`{\\displaystyle F\\left(u_{i-{\\frac {1}{2}}}\\right)\\ } 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,

`F(underset(i+1/2)u)=overset(low)underset(i+1/2)f-phi(underset(i)t)(overset(low)underset(i+1/2)f-overset(high)underset(i+1/2)f)`

`F(underset(i-1/2)u)=overset(low)underset(i-1/2)f-phi(underset(i-1)t)(overset(low)underset(i-1/2)f-overset(high)underset(i-1/2)f)`

{\\displaystyle F\\left(u_{i+{\\frac {1}{2}}}\\right)=f_{i+{\\frac {1}{2}}}^{low}-\\phi \\left(r_{i}\\right)\\left(f_{i+{\\frac {1}{2}}}^{low}-f_{i+{\\frac {1}{2}}}^{high}\\right)}

{\\displaystyle F\\left(u_{i-{\\frac {1}{2}}}\\right)=f_{i-{\\frac {1}{2}}}^{low}-\\phi \\left(r_{i-1}\\right)\\left(f_{i-{\\frac {1}{2}}}^{low}-f_{i-{\\frac {1}{2}}}^{high}\\right

where

{\\displaystyle f^{low}=\\ }F `overset(low)f`= low precision, high resolution flux,

{\\displaystyle f^{high}=\\ } `overset(high)f` = high precision, low resolution flux,

{\\displaystyle \\phi \\ (r)=\\ } `underset(t)phi`= flux limiter function,

and {\\displaystyle r\\ }t represents the ratio of successive gradients on the solution mesh, i.e.,

`underset(i)t=(underset(i)u-underset(i-1)u)/(underset(i+1)u-underset(i)u)`

{\\displaystyle r_{i}={\\frac {u_{i}-u_{i-1}}{u_{i+1}-u_{i}}}

The limiter function is constrained to be greater than or equal to zero, i.e., {\\displaystyle \\phi \\ (r)\\geq 0}`phi(t)ge0` `phi(t)ge0`. Therefore, when the limiter is equal to zero (sharp gradient, opposite slopes or zero gradient), 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.