Week 9 - FVM Literature Review

Aim:

To describe the need for the interpolation schemes in the FVM and flux limiter.

METHOD OF THE CFO:

To solve the real-world problem, the people discovery three simples schemes for the numerical solution are

• Finite Difference Method
• Finite Element Method
• Finite volume Method

Finite Difference Method:

This method solves the equation, that is divide into single points or nodes. This method involves solving the discretizations using the differencing equation.

Finite Element Method:

This method involves dividing the domain into elements of the desire. By solving the element in the system ,The finite element analysis is a particular method for solving two or three-dimensional problems.

Finite Volume Method:

In the FVM Method the system to divide into an unstructured mesh which is conservative. In the algebraic form, the partial equation represents in the volume integral, which converted into the surface integral by the Gauss divergence theorem. These terms are solved as a flu x in each surface. Because the flux entering the surface is identical to the adjacent surface , this is a conservative principle.

The finite volume method (FVM) is a method for representing and evaluating partial diffrential 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 theorm. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to thot 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.

Interpolation Schemes in Finite Volume Method (FVM):

The interpolation schemes used in FVM method are as follows

1. Upwind Interpolation Scheme.
2. Central Differencing
3. Quadratic Upwind Differencing
4. Hybrid Interpolation
5. Total variation Diminishing

Let us consider the Control volume P and the centroid at the Pas shown in the above figure. The ab volume with the centroids located at P, E(EAST) , W(W ES T), N(NORT H), S(SOUTH POINT), and the further p The faces center is denoted as n, e, s, and w. Come back to the Heat conduction equation, because of and south control volume. Consider the small volume 'dv' in the Control volume P, the integral form of volume is given by,

Need for interpolation schemes:

To calculate the Tw a nd T" Temperature at the west and east faces, we have the value at the Tp, T E, a nd Tw by interpolating the values, we can get the temperature at the east and west faces. By using them we can calculate the thermal diffusivity of the faces are

a, and aw .The re are different ways of Interpolation. Some of the ways are given by

• Upwind interpolation
• Linear Interpolation
• Quadratic Interpolation
• Hybrid Interpolation

Flux Limiters: