Week 9 - FVM Literature Review
Finite volume method: This method is for representing and evaluating the partial differential equation in form of algebraic equation. 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 finite volume refers to the small volume surrounding each node point in the mesh. This volume is the control volume and the partial equation is integrated for this volume. The solution is found by grid generation, discretization and by solving discretising equation at each of nodal points.
Finite element method: This method subdivides a large system into smaller parts that are called finite elements. This is done by the construction of a mesh of the object.The FEM the uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function. This method creates local approximations of a solution using local data, and construct a global approximation by sitching them together.
The 1D heat conduction equation:
The one dimensional heat equation for steady state is
`d((Gamma(dphi)/dx))/dx+S=0`
where `Gamma` is thermal diffusion coefficient and S is the source term.
When integrating the above equation over the control volume, we get
`(GammaA(dphi)/dx)_e-(GammaA(dphi)/dx)_w+(StriangleV)=0`
The value of `Gamma_w=(Gamma_W+Gamma_P)/2`
`Gamma_e=(Gamma_P+Gamma_E)/2`
`(GammaA(dphi/dx))_e=Gamma_eA_e(phi_E-phi_P)/(deltax_(PE))`
`(GammaA(dphi/dx))_w=Gamma_wA_w(phi_W-phi_P)/(deltax_(PW))`
`StriangleV=S_u+S_pphi_p`
on subsituting the values , we get
`(Gamma_e/(deltax_(PE))A_e+Gamma_w/(deltax_(WP))A_w-S_p)=(Gamma_w/(deltax_(WP))A_w)phi_W+(Gamma_e/(deltax_(PE))A_e)phi_E+S_u`
which can be written as
`a_pphi_p=a_Wphi_W+a_Ephi_E+S_u`
To calculate the value of `phi_w` and `phi_e`, we have the value at `phi_P`, `phi_E` and `phi_W` by interpolation the value can be calculated.
Some major interpolation schemes used are
1) Upwind interpolation scheme
2) Linear interpolation scheme
3) Quadratic interpolation scheme
4) Hybrid interpolation scheme
Flux limiters: Flux limiters are used in high resolution numerical schemes used to solve partial differential equations. They are used in high resolution schemes, such as the MUSCL scheme, to avoid the spurious oscillations (wiggles) that would otherwise occur with high order spatial discretization schemes due to shocks, discontinuities or sharp changes in the solution domain. Use of flux limiters, together with an appropriate high resolution scheme, make the solutions total variation diminishing (TVD).