Week 9 - FVM Literature Review
Finite Difference Method(FDM): This method solves the equation after dividing it into single points or nodes. This method involves solving the discretizations using the differencing equations. This method uses 3 different discretization schemes. They are:
1. Forward Differencing Scheme(FDS)
2. Backward Differencing Scheme(BDS)
3. Central Differencing Scheme(CDS)
Finite Volume Method(FVM): In the FVM Method, the system is divided into an unstructured mesh which is conservative. In the algebraic form, the partial equation represented in the volume integral which contains a divergence term is converted into the surface integral by the Gauss Divergence Theorem. These terms are solved as a flux in each surface. Because, the flux entering the surface is identical to the adjacent surface, this is a conservative principle. The FVM Method is a method for representing and evaluating partial differential equations in the form of algebraic equations. An advantage of FVM is that it is easily formulated to allow unstructured meshes. This method is used in many CFD packages.
Study of Various Interpolation Schemes and Flux Limiters in FVM:
Interpolation Schemes In FVM: The interpolation schemes used in FVM are as follows:
1. Upwind Interpolation Scheme(UDS)
2. Central Differencing Approximation(CDS)
3. Quadratic Upwind Differencing Scheme(QUICK)
4. Hybrid Interpolation Scheme(UDS & CDS)
5. Total Variation Diminishing Scheme(TVD)
Upwind Interpolation Scheme(UDS): In this scheme, we use the node in the upwind or upstream of the given node. It is generally used for convection dominated problems(i.e. `P_e>1`)
`f_e`= `f_p`if `(v.n)_e>0`
`f_E`if `(v.n)_e<0`
The values for this at the interface is given by:
For `v>0; u_(i-1/2)approxu_(i-1);u_(i+1/2)approxu_i`
`I_c approxv.(u_i-u_(i-1))/(trianglex)`- Backward Differencing Scheme(BDS)
For `v>0; u_(i-1/2)approxu_(i-1);u_(i+1/2)approxu_i`
`I_c approxv.(u_(i+1)-u_i)/(trianglex)`- Forward Differencing Scheme(FDS)
Taylor series expansion is
`v.u_(i+1/2)=v.u_i-((vtrianglex)/2).((delu)/(delx))_(i+1/2)`where the second term has the truncation error which resembles a diffusive flux term.
Upwind schemes are the approximations that satisfy the boundedness unconditionally and is first order accurate in space.
Central Differencing Approximation(CDS): Approximate values of the variable at the control volume(CV) face centre by the linear interpolation of the values at the two nearest computational nodes.
`f_e=fE.lamda_E+f_p(1-lamda_e)`where `lamda_e=(x_e-x_p)/(x_E-x_P)`
The linear interpolation is equivalent to that of using central difference formula of the first order derivative and hence this scheme is also termed as Central Differencing Scheme(CDS).
Interpolation Polynomial:
`p1(x)=u_L.(x_R-x)/(x_r+x)+u_R.(x-x_L)/(x_R-x_L)`
The interface values are given by:
`u_(i-1/2) approx (u_(i-1)+u_i)/2`and `u_(i+1/2) approx (u_i+u_(i+1))/2`
`I_c approx v.(u_(i+1)-u_(i-1))/(2trianglex)`- Central Differencing Scheme(CDS)
Taylor Series Approximation Polynomial:
`u_(i+1/2)=(u_i+u_(i+1))/2-((trianglex)^2/8).((del^2u)/(delx^2))_(i+1/2)+....`
This approximation is second order precise. It is the most simplest and most widely used interpolation technique for the evaluation of gradients required to measure diffusive fluxes.
Flux Limiters: Flux limiters are used in the numerical schemes to solve problems of fluid dynamics, described by highly coupled non-linear partial differential equations. If we tend to use low order numerical schemes for the solving of the governing equations, then we might get a highly oscillatory and unstable solution near the discontinuity if present. Whereas, if we tend to use higher-order numerical schemes to capture the phenomenon more accurately, we tend to exploit the computational power unnecessarily. The main idea behind the use of flux limiters is to limit the spatial derivative terms near the discontinuities to physical rear values so as to maintain physical consistency. There are several types of flux/slope limiter functions. They come in handy when sharp wave-fronts/peaks are present. For smoothly changing functions/waves, the flux limiters do not operate and the spatial derivatives can be represented normally. It is imperative to note that flux limiters are also referred to as slope limiters because they 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^(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))`
Here, `f^(low)`= Low Precision Flux(1st order precision).
`f^(high)`= High Precision Flux(highest order precision).
`phi(r)`= Flux Limiter Function and `r=(u_(i)-u_(i-1))/(u_(i+1)-u_i)`.