Finite Volume Method (FVM) Literature Review

Finite Volume Method (FVM):-

It is a numerical discretization method for the approximation of a single or a system of partial differential equations (PDEs). These partial differential equations (PDEs) are frequently called as conservation laws. These PDEs may be parabolic, elliptical or hyperbolic in nature. These PDEs express the conservation or balance between one or more physical quantities (physical properties) and describes the relations between partial derivatives of quantities such as temperature, pressure, density with respect to variables within the domain like space and time under consideration.

Considering finite small portion of domain, a mesh is constructed. The elements of the mesh/grid/cell are called control volumes. The integration of the PDE over each control volume results in a balance equation. And the set of such balance equations are then discretized with respect to a set of discrete unknowns.

The discretization procedure used in the finite volume method is distinctive and involves two basic steps. In the first step, the partial differential equations are integrated and transformed into balance equations over an element (or finite volumes) into which the domain has been subdivided, then the Gauss theorem is applied to transform the volume integrals of the convection and diffusion terms into surface integrals. The result is a set of semi-discretized equations. In the second step, interpolation profiles are chosen to approximate the variation of the variables within the element and relate the surface values of the variables to their cell values and thus transform the algebraic relations into algebraic equations.

In order for the FVM to be efficient, the numerical fluxes are generally

• conservative i.e. the flux entering a control volume from its neighbor must be the opposite of the one entering the neighbor from the control volume,
• consistent i.e. the numerical flux of a regular function interpolation tends to the continuous flux as the mesh size vanishes.

It is sometimes possible to discretize the fluxes at the boundaries of the control volume by the finite difference method (FDM). In this case, the method has often been referred to as a finite difference method or conservative finite difference method. The specificity of the FVM with respect to the FDM is that the discretization is performed on the local balance equations, rather than the PDE: the fluxes on boundaries of the control volumes are discretized, rather than the continuous differential operator.

The resulting system of discrete equations depends on a discrete (finite) set of unknowns, and may be either linear or non-linear, depending on the original problem itself; this system is then solved exactly or approximately, using for example direct or iterative solvers in the case of linear equations and fixed point or Newton Raphson method in the case of nonlinear equations.

Finite volume methods can be compared and contrasted with the finite difference method (FDM), which approximate derivatives using nodal values, or finite element method (FEM), which create local approximations of a solution using local data, and construct a global approximation by stitching them together. Where FVM calculate exact expression for average value of solution over volume.

The advantage of FVM is that it allows easily to formulate unstructured mesh, also FVM formulation itself gives physical intuition. In many CFD packages FVM method is used.

Meshing in FVM can be done by following two methods.

1. Cell centered
2. Vertex centered

*Need of interpolation scheme:

While solving approximation off volume and surface integral it is required to have values at variable locations (space) other than computational nodes of control volume. So to obtained the values at various location interpolation scheme is needed.

The interpolation schemes used in FVM are as follows:-

1. Upwind interpolation scheme
2. Linear interpolation scheme
3. Quadratic upwind differencing scheme
4. Hybrid interpolation scheme
5. Total variation diminishing scheme

1. Upwind interpolation scheme:

In this scheme we use the node in upwind (upstream) of known (given) node. This scheme is generally used for convection dominating problems i.e. Pe>1 (Peclete number should be less than 1).

`f_e= {[f_p" if "(v,n)_e >0],[f_E" if "(v,n)_e <0]:}`

The value at interreference,

for `v<0;" " u_(i-1/2)~~u_(i);" "u_(i+1/2)~~u_(i+1)`

By forward differencing, `I_c~~v (u_(i+1)-u_(i))/(Deltax)`

for `v>0;" " u_(i-1/2)~~u_(i-1);" "u_(i+1/2)~~u_(i)`

By backward differencing, `I_c~~v (u_(i)-u_(i-1))/(Deltax)`

Hence Taylor series expansion,

`vu_(i+1/2)=v.u_i - v(Deltax)/2.((delu)/(delx))_(i+1/2)+...`

This scheme is first order accurate.

2. Linear interpolation scheme:

It is simplest and most widely used interpolation scheme. For this scheme we have to approximate values of variable at control volume face centre by linear interpolation of the values at the two nearest nodes. This is similar to first order central differencing about node.

`f_e=f_Elambda_e +(1-lambda_e)f_p`

where `lambda_e=(x_e-x_p)/(x_E -x_P)`

This scheme is second order accurate.

3.Quadratic upwind differencing scheme:

For this scheme we approximate values of variable at control volume face centre by quadratic interpolation of values at three nearest computational nodes.

`f_e= f_E +g1(f_D - f_U)+g2(f_(UU)-f_U)`

where,

`g1=((x_e-x_U)(x_e-x_(UU)))/((x_(UU)-x_U)(x_(UU)-x_D))`

`g2=((x_e-x_U)(x_e-x_D))/((x_(UU)-x_U)(x_(UU)-x_D))`

4. Hybrid interpolation scheme

This scheme is combination of above upwind interpolation and linear interpolation scheme.

5.Total variation diminishing scheme

This scheme is employed to capture shock predictions.

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

Where `psi` is flux limiter.

Flux limiter:

This quantity is used in high resolution schemes , numerical schemes used to solve problems in engineering, particularly fluid dynamics, described by PDEs. This are used in high resolution scheme to avoid the spurious oscillations that would may occur with high order spatial discretization schemes due to shock, or discontinuity or sharp changes in solution domain. Using flux limiter with an appropriate scheme gives solution total variation diminishing scheme.

The flux limiter also referred as slope limiters as they both denotes same mathematical formulation and also both have effect of limiting the solution gradient near shock and discontinuity.

`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))`

where,

`f^(low)=` low resolution flux,

`f^(high)=` high resolution flux,

`phi(r)=` flux limiter function

r = successive gradient on solution mesh

`r_i= (u_(i)-u_(i-1))/(u_(i+1)-u_(i))`

The limiter function must be greater than or equal to zero. `phi(r)>=0