Week 9 - FVM Literature Review

Finite Volume Method-

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.

These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages. \"Finite volume\" refers to the small volume surrounding each node point on a mesh.

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. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative.

The advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes, and this method is used in many computational fluid dynamic packages.

Difference between Finite Volume Method(FVM), Finite Difference Method(FDM)

Finite Difference Method(FDM)

The finite-difference method is defined dimension per dimension; this makes it easy to increase the “element order” to get higher-order accuracy. If you can fit the simulation in a rectangular or box-shaped geometry using a regular grid, efficient implementations are much easier than for finite-element and finite-volume methods. Regular grids are useful for very-large-scale simulations on supercomputers often used in, as mentioned before, meteorological, seismological, and astrophysical simulations.

With the finite-difference method, you may easily run into problems handling curved boundaries for the purpose of defining the boundary conditions. Boundary conditions are needed to truncate the computational domain. They represent communication with the surrounding world, which is the part that you do not want included in your simulation. If one can overcome the boundary-condition problem on curved boundaries, the method gives very efficient and high quality results.

Finite element Method(FEM)

The finite-volume method is a natural choice for CFD problems, since the partial differential equations you have to solve for CFD are conservation laws. However, both finite differences and finite elements can also be used for CFD. Efficient technology for CFD with the finite-element method has become increasingly popular over the last 10 to 15 years. Techniques for CFD with the finite-difference and finite-volume method have been known and used much longer.

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.

Interpolation schemes in Finite Volume Method

Interpolation is a process to estimate the values at unknown points by using the points with known value and sample points on either sides of the unknown point. In FVM, interpolation schemes are used to find values of volume integrals required at the points other than nodes.

Different types of Interpolation schemes

1. Upwind Interpolation Scheme (UDS)

f_e= {(f_(p ) if (v.n)_e>0),(f_E if (v.n)_e<0):}

2. Linear Interpolation Scheme

To approximate the values of variables at CV face centre by linear interpolation of the values at two nearest computational nodes. This is equivalent to the use of the central difference scheme of first order derivative, and hence it is also known as central difference scheme. This also leads to very simple approximation of evaluation of gradients required for evaluation of diffusive fluxes.

u_(i+1/2) = {u_i + u_(i+1)}/2 - (Deltax)^2/8.({del^2u}/{delx^2})_(i+1/2)+...

3. Quadratic Upwind Interpolation Scheme

This approximates the value of variable at CV face centre by quadratic interpolation of the values at three nearest computational nodes (one downstream node, D and two upstream nodes, U). It is third order accurate for both uniform and nonuniform grids.

phi_e = 6/8phi_c + 3/8phi_D - 1/8phi_(UU)

4. Hybrid Interpolation Scheme

The Hybrid difference scheme is a method used in the numerical solution for convection and diffusion problems. It is the blend of CDS and UDS scheme, based on Peclet number.

f_e = gammaf_(CDS) + (1-gamma)f_(UDS)

5. Total variation Diminishing Scheme

In numerical methods , TVD is a property of certain discretization schemes used to solve hyperbolic partial differential equations.

f_e = f_p + 1/2 psi(r)(f_E - f_p)

Flux Limiters-

For convective fluid flow, it is observed that the 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.

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.

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

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

Where f^low = Low precision flux

F^high = High precision flux

phi(r) = flux limiter function where, r ={u_i-u_(i-1)}/{u_(i+1) - u_i