FVM Literature Review
Finite volume method (FVM)
The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. 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.
FVM computes solution and volume where FDM computes solution at each point. Here in FVM governing equations are solved in an integral form so it serves to conserve the properties in a better way than FDM.
Interpolation Scheme:
The approximation of the surface and volume integrals may require values of the variable at the location other than the computational nodes of the CV values at these locations are obtained by the interpolation scheme.
The different interpolation schemes
It is used to calculating the approximate value of the variables at the east face of the control volume. It is the first order accurate and is numerically diffusive with a coefficient of numerical diffusion.
`underset(e)f = { (underset(p)f, if underset(e)((v.n))>0),(underset(E)f, if underset(e)((v.n))<0):}`
It is equivalent to using a backward or forward finite difference approximation. The upwind is given below. e stands for east side.
`underset(e)Gamma^(n um) =underset(e)((rho*u))*(Deltax)/2`
Linear Interpolation:
To approximate the value of a variable at control volume face center by two computational nodes locations e.
It is equivalent to the central difference schemes of the first-order derivative. It is second-order accurate and produces oscillatory solutions, the relation of linear interpolation is given below.
`underset(e)lambda= (underset(e)x-underset(p)x)/(underset(E)x-underset(P)x)`
Quadratic upwind interpolation:
It is third-order accurate but is prone to oscillations and is derived from polynomial fittings. The interpolation on a uniform cartesian grid is given by
`underset(e)(phi)= 6/8*underset(U)(phi) +3/8*underset(D)(phi)-1/8*underset(UU)(phi)`
Where D, U, and UU denote the downstream, the first upstream, and the second upstream node respectively.
A Quadratic curve is fitted with two upstream nodes and one downstream node. but in regions with strong gradients, overshoots and undershoots can occur which can lead to stability problems.
Flow limiters:
Flux limiters are used in the high-resolution schemes - numerical schemes used to solve problems especially in fluid dynamics problems described by PDEs. They are used in high-resolution schemes such as the MUSCL scheme, to eliminate the spurious oscillations that would have occurred with high order spatial discretization scheme due to shock, discontinuities, or sharp changes in the solution domain with a point of inflection. The use of flux limiters, together with appropriate high-resolution schemes makes the solution's total variation diminishing.
The main moto of flux limiters schemes is to limit the spatial derivatives to realistic values concerned with certain science and engineering problems that usually means physically realizable and meaningful values. They are used in high-resolution schemes for solving problems described by PDEs and is only considered or taken into account when sharp wave fonts are present. For smoothly changing waves, the flux limiters do not operate and can be represented by higher-order approximations without any introduction of spurious oscillations.
`(du_i)/(dt)= 1/((Deltax)_i)*[F(u_(i+1/2))-F(u_(i-1/2))]=0,`
`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))`
r represents the successive gradients on the solution mesh i.e,
`r_i=(u_i-u_(i-1))/(u_(i+1)-u_i)`
The flux is represented by low resolution schemes when the limiter is equal to zero. When the limiter is equal to one it is represented by high-resolution schemes.