Week 9 - FVM Literature Review

Q.1 What is Finite Volume Method(FVM)?

The Finite Volume Method (FVM) is a numerical method for solving partial differential equations, particularly those that govern fluid flow, heat transfer, and other transport phenomena. It divides the computational domain into a finite number of control volumes or cells, and approximates the integral form of the governing equations over each control volume. This yields a system of algebraic equations that can be solved numerically to obtain a discrete approximation of the solution.

The FVM is based on the conservation laws that govern the behavior of physical systems. These laws state that the total amount of a conserved quantity, such as mass, energy, or momentum, remains constant over time within a closed system. The FVM discretizes these laws by approximating the fluxes of the conserved quantities across the cell faces, and integrating them over each control volume. This yields a discrete equation that relates the values of the conserved quantity at the cell centers to the fluxes across the cell faces.

One of the advantages of the FVM is its ability to handle complex geometries and unstructured meshes. Unlike other numerical methods such as the Finite Difference Method (FDM), which require a structured grid, the FVM can be used with arbitrary meshes, including unstructured and non-uniform meshes. This makes it a powerful tool for simulating real-world systems with complex geometries, such as fluid flows through porous media or around complex obstacles.

The FVM is widely used in various fields of engineering and science, including computational fluid dynamics, heat transfer, and electromagnetic field analysis. It has also been used in the design of energy-efficient buildings, the simulation of air pollution dispersion, and the modeling of blood flow in the human body, among many other applications.

Q.2 Write down the major differences between FDM & FVM.

The Finite Difference Method (FDM) and the Finite Volume Method (FVM) are two numerical techniques for solving partial differential equations. While they share some similarities, there are also several major differences between the two methods:

1. Control volumes vs. nodes: In FVM, the computational domain is divided into a finite number of control volumes, each with a volume and surface area. In contrast, FDM divides the domain into a grid of nodes, with the solution values calculated at each node.

2. Conservation laws: FVM is based on the conservation laws that govern the behavior of physical systems. It approximates the integral form of the governing equations over each control volume, resulting in a set of algebraic equations that can be solved numerically. FDM, on the other hand, approximates the differential form of the equations at each node.

3. Meshes: FDM requires a structured grid, where each node is connected to its nearest neighbors by a fixed pattern of links. FVM, on the other hand, can use an unstructured mesh with irregularly shaped control volumes that are not required to be connected in any particular pattern.

4. Fluxes and gradients: FVM directly calculates the fluxes of conserved quantities (e.g. mass, momentum, energy) across the cell faces and calculates the gradient of the solution within each control volume. FDM, on the other hand, calculates the gradients of the solution at each node and calculates the fluxes across the grid edges.

5. Computational efficiency: FDM is generally faster than FVM for problems with a regular structure and a small number of dimensions. FVM is better suited for problems with complex geometries and irregularly shaped control volumes, but it can be more computationally expensive.

Overall, both FDM and FVM are powerful numerical techniques that can be used to solve a wide range of partial differential equations. The choice between the two methods depends on the specific characteristics of the problem at hand, such as the geometry of the system, the nature of the governing equations, and the desired level of accuracy and computational efficiency.

Q.3 Describe the need for interpolation schemes and flux limiters in FVM. You can use the 1D linear heat conduction equation as an example. 

The finite volume method (FVM) is a numerical method used to solve partial differential equations (PDEs) on a discrete grid. In FVM, the solution is approximated by considering the flux of the conserved quantities (such as mass, momentum, and energy) across the cell boundaries. The fluxes are calculated using an interpolation scheme that approximates the solution within each cell based on the values at the cell centers.

However, the accuracy of the solution obtained through FVM can be affected by numerical errors, such as numerical diffusion and oscillations. These errors can occur due to the approximations made in the interpolation scheme, especially near sharp gradients or discontinuities in the solution.

One way to mitigate these errors is through the use of flux limiters, which are numerical techniques that limit the fluxes across cell boundaries based on the gradients of the solution. Flux limiters ensure that the numerical solution does not generate non-physical oscillations or overshoots near sharp gradients or discontinuities in the solution.

For example, consider the 1D linear heat conduction equation:

$$ \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} $$

where $u(x,t)$ is the temperature distribution, $\alpha$ is the thermal diffusivity, and $t$ and $x$ are time and space variables, respectively. In FVM, the equation is discretized by dividing the domain into cells and approximating the solution within each cell.

To calculate the fluxes across the cell boundaries, an interpolation scheme is used to approximate the solution at the cell faces. However, if the solution has sharp gradients or discontinuities, the interpolation scheme can generate non-physical oscillations or overshoots, which can lead to numerical errors.

To mitigate these errors, flux limiters can be used. For example, the minmod limiter can be used to limit the fluxes as:

$$ F_{i+\frac{1}{2}} = \frac{1}{2}(F_{i}+F_{i+1}) - \frac{1}{2} \text{minmod}(\Delta F_i, \Delta F_{i+1}) $$

where $F_{i+\frac{1}{2}}$ is the flux at the cell boundary between cell $i$ and $i+1$, and $\Delta F_i = F_{i} - F_{i-1}$. The minmod function limits the slope of the solution by selecting the minimum of the two slopes calculated using the neighboring cells. This ensures that the fluxes are limited near sharp gradients or discontinuities, reducing numerical errors and producing a more accurate solution.

The minmod function is defined as: