Week 9 - FVM Literature Review

Finite Volume Method (FVM) is a numerical method for solving partial differential equations (PDEs) that involve the conservation laws, such as mass, momentum, and energy. The method involves dividing the physical domain into a finite number of small control volumes or cells, where the conservation laws are applied to the volume-averaged quantities. The fluxes of conserved quantities across the cell interfaces are computed using numerical schemes and updated in time until the solution converges to a steady state.

The major differences between FDM and FVM are:

1. In FDM, the governing equations are discretized at the grid points, while in FVM, the equations are discretized at the cell centers.

2. FDM approximates the derivatives of the solution at the grid points using finite differences, while FVM approximates the fluxes of the solution across the cell interfaces using numerical schemes.

3. FDM is a point-wise method, while FVM is a volume-wise method. This means that FDM focuses on the solution at the grid points, while FVM focuses on the solution in the control volumes.

The need for interpolation schemes in FVM arises when computing the fluxes at the cell interfaces. The fluxes are computed as the product of the average values of the conserved quantities in the adjacent cells and the velocity of the flow across the interface. Since the flow velocity is not constant across the interface, an interpolation scheme is needed to estimate the velocity at the interface.

Flux limiters are used in FVM to prevent the numerical solution from becoming unphysical, such as generating negative concentrations or producing oscillations in the solution. The flux limiters control the amount of flux that is allowed to pass through the cell interfaces, based on the local gradient of the solution. The most commonly used flux limiters are the minmod limiter, the superbee limiter, and the van Leer limiter.

For example, let's consider the 1D linear heat conduction equation with constant thermal conductivity k:

∂T/∂t = k ∂^2T/∂x^2

where T is the temperature, t is the time, and x is the spatial coordinate. We can discretize this equation using FVM as follows:

1. Divide the domain [0, L] into a finite number of cells of equal size Δx.

2. Define the cell centers as x_i = (i-1/2)Δx, where i is the cell index.

3. Integrate the governing equation over the cell volume V_i = Δx, and apply the divergence theorem to obtain:

d/dt ( ∫_Vi T dV ) = k ( ∫_Ai ( ∂T/∂x ) dA )

where A_i is the area of the cell interface between cells i and i+1.

4. Approximate the cell-averaged temperature T_i as T_i = (1/Δx) ∫_Vi T dV.

5. Approximate the flux of temperature at the cell interface A_i as F_i+1/2 = -k ( ∂T/∂x )_i+1/2.

6. Apply a numerical scheme, such as the upwind scheme or the central difference scheme, to compute the temperature gradient ( ∂T/∂x )_i+1/2.

7. Update the temperature in each cell using an explicit time-stepping scheme, such as the forward Euler method:

T_i^(n+1) = T_i^n + (Δt/Δx) ( F_i-1/2 - F_i+1/2 )

where n is the time step index and Δt is the time step size.