Week 9 - FVM Literature Review

Aim: To write a detailed literature review on FVM ad differences between fvm and fdm ,and explain the need for flux limits and interpolation schemes.

FVM: Finite volume method is the most popular method in Cfd and a humongous number of analysis softwares using these methods.

• finite volume refers to a small volume surrounding each node point on a mesh.
• 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.divergence theorem represents any number of dimensions in 1D
• These volume integral terms are evaluated as flixes at a surface of each finite volume method.
• The advantage of Fvm's are they are conservative.
• It can be easily formulated to allow unstructured meshes.
• It evaluates exact expressions for average value of solutions over some volume and uses this data to construct an approximate solution within cells.

FDM:

In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences. Both the spatial domain and time interval (if applicable) are discretized, or broken into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points.

`(delu)/(delt) + C(delu)/(delx) =0`

Using first-order forward differencing for the time derivative

`(U_i^(n+1)-U_i^(n))/(Deltat)`

Using first-order backward/rearward differencing for the space term

`(U_(i)^(n)-U_(i-1)^(n))/(Deltax)`

`(U_i^(n+1)-U_i^(n))/(Deltat)+C*(U_(i)^(n)-U_(i-1)^(n))/(Deltax)`

Major differences between FVM and FDM:

FVM

• The Finite Volume method is an integral scheme. Since you are integrating across the area, chance of error is minimized.
• It is best to apply when the mesh is regular.
• It is difficult to form the mesh and also difficult to form the solution but it is accurate yet slow.

FDM:

• FDM is a differential scheme, that is approximation of a taylor series expansion. You try to write the first order, second order differential in terms of difference between two point values. This may bring in some approximation error.
• It is best suitable when the matches are regular i.e square or rectangular.
• It is easy and fast.

Flux limiters:

• flux limiters are used in high resolution schemes to avoid oscillations or higher order spatial discretisation schemes due to shocks ,discontinuities or sharp changes in solution domain.
• Flux limiters are also called slope limiters because both have the same mathematical form and both have the effect of limiting the solution gradient near shocks or discontinuities.

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

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

`f^(low)=`low precision flux

`f^(high)=`high precision flux

`phi(r)=`Flux limiter function

Interpolation Schemes:

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

Upwind Interpolation:

upwind schemes denote a class of numerical discretization methods for solving hyperbolic partial differential equations. Upwind schemes use an adaptive or solution-sensitive finite difference stencil to numerically simulate the direction of propagation of information in a flow field.

`(U_i^(n+1)-U_i^(n))/(Deltat)`

Linear Upwind Difference Second order Scheme:
We can approximate the value of the variable at CV face centre by linear interpolation of the values at two nearest computational nodes. LUDS is second-order accurate, equivalent to the one-sided 3-point finite differece the variable value is approximated by

`Iapprox (3 u_i − 4 u_(i−1) + u_(i− 2))/(2Deltax)`

Linear Interpolation:
The linear interpolation is equivalent to the use of central difference formula of the first order derivative, and hence, this scheme is also termed as central difference scheme (CDS). This scheme is second order accurate, and may produce oscillatory solutions .

`I=(U_r+U_l)/2`

Quadratic Interpolation:
Approximated by quadratic profile between two nearest nodes.A quadratic upwind interpolation scheme can be derived using polynomial fitting.
The QUICK interpolation on a uniform Cartesian grid is given by

`I = (3 u_(i+1) + 3 u_i − 7 u_i−1 + u_(i− 2))/(8Deltax)`