Week 9 - FVM Literature Review
Aim:
Describe What is Finite Volume Method(FVM)
write down the major differences between FDM & FVM.
Describe the need for interpolation schemes and flux limiters in FVM using the 1D linear heat conduction equation as an example.
Finite Volume Method:
The Finite Volume Method (FVM) is a numerical method used to solve partial differential equations (PDEs) by dividing the domain into a set of small control volumes, or "finite volumes." The method involves integrating the PDE over each control volume and then solving the resulting system of algebraic equations.
Difference between FVM and FDM
The major difference between FDM and FVM is that FDM divides the domain into small, fixed-size discrete points, whereas FVM divides the domain into small control volumes. FDM approximates the solution at the discrete points, whereas FVM integrates the PDE over the control volumes to obtain an average value for the solution.
In FVM, interpolation schemes are used to estimate the solution values at the cell interfaces from the known solution values at the cell centers. This is necessary because the PDE is typically solved at the cell centers, but the solution values at the cell interfaces are needed to calculate the fluxes across the interfaces.
Flux LImiters
Flux limiters are used in FVM to prevent the solution from becoming overly diffusive or oscillatory. They are particularly important when solving hyperbolic PDEs, such as the 1D linear heat conduction equation, as they help to ensure that the solution remains physically realistic.
For example, in 1D linear heat conduction equation, the heat flux is determined by the temperature gradient. So, the temperature on the left side of a cell interface is used to calculate the heat flux into that cell and the temperature on the right side of the interface is used to calculate the heat flux out of the cell. The interpolation scheme is used to find the temperature at the cell interfaces from the known temperature values at the cell centers.
1D Linear Heat conduction Equation
The 1D linear heat conduction equation is a PDE that describes how heat is conducted through a material. It can be written as:
`(∂T)/(∂t) = α( (∂²T)/(∂x²))`
where T is the temperature, t is time, x is position, and α is the thermal diffusivity of the material.
To solve this equation using the Finite Volume Method (FVM), we first divide the domain of the equation into a set of small control volumes, or "finite volumes." The size of these control volumes is determined by the desired level of accuracy and the spatial variation of the solution.
Next, we integrate the PDE over each control volume and then solve the resulting system of algebraic equations. This process is commonly referred to as "discretizing" the PDE. The integration of the PDE over the control volume can be expressed mathematically as:
`∫(∂T)/(∂t) dx = ∫α ((∂²T)/(∂x²)) dx`
The result is an equation that describes how the temperature changes within a control volume over time. The equation takes the form of:
`∫(∂T)/(∂t) dx = ∫α ((∂²T)/(∂x²)) dx``(T^((n+1)_i) - T^(n_i) )/ (Δt) = α (T^(n_(i+1) - 2T^(n_i) + T^(n_(i-1))) / (Δx²))`
Where `T^(n_i)` is the temperature of the control volume i at time n, `T^((n+1)_i )` is the temperature of the control volume i at time n+1, Δt is the time step size and Δx is the size of the control volume.
To find the temperature at the cell interfaces from the known temperature values at the cell centers, interpolation scheme is used. The interpolation scheme can be a simple linear interpolation or more complex schemes like higher order polynomials. The flux limiter is used to ensure that the solution remains physically realistic and prevent from becoming overly diffusive or oscillatory.
Once we have the temperature value at all the cell center and interfaces for all the time steps, we can use these values to calculate the heat flux, heat rate and temperature distribution in the material.