Week 9 - FVM Literature Review

AIM:-

Describe the need for interpolation schemes and flux limiters in FVM(Finite Volume Method).

Objective:-

1) What is FVM(Finite Volume Method).

2) Differences between FDM(Finite Difference Method) VS FVM(Finite Volume Method).

3) Describe the need for interpolation schemes and flux limiters in FVM(Finite Volume Method).

Introduction:-

We know that Fluid Dynamics there are three-numerical method based solvers, which is used to solve for the governing equation of Navier-stokes.

1) Finite Volume Method.

2) Finite Difference Method.

3)Finite Element Method.

Theory:-

The Finite Volume Method (FVM) is a numerical technique used to solve problems involving fluid flow, heat transfer, and other physical phenomena described by partial differential equations (PDEs). Here's a simplified explanation:

Basic Concept

1. Discretization of the Domain:

   • The physical domain (e.g., the space where fluid is flowing) is divided into small, non-overlapping control volumes or cells.
   • These cells together cover the entire domain without any gaps or overlaps.

2. Integral Form of the Governing Equations:

   • Instead of solving the PDEs directly, FVM transforms them into integral forms over each control volume.
   • This means we look at the balance of fluxes (like mass, momentum, or energy) across the boundaries of each cell.

3. Conservation Principle:

   • FVM ensures that the quantities like mass, momentum, and energy are conserved within each control volume.
   • This is achieved by calculating the fluxes entering and leaving the cell boundaries.

Steps Involved in FVM

1. Divide the Domain:

   • Split the domain into a grid of control volumes (cells).

2. Integral Equations:

   • Convert the PDEs into integral form over each control volume.

3. Calculate Fluxes:

   • Determine the fluxes across the boundaries of each control volume using interpolation methods.

4. Solve Equations:

   • Set up algebraic equations for each control volume based on the flux balances and solve them simultaneously.

Advantages of FVM

• Conservation: Naturally conserves quantities like mass, momentum, and energy.
• Flexibility: Can handle complex geometries and boundary conditions.
• Stability: Generally provides stable solutions for a wide range of problems.

Example

Imagine you want to simulate the temperature distribution in a metal rod being heated at one end.

1. Divide the Rod: Split the rod into several small segments (control volumes).
2. Heat Equation: Use the heat equation in its integral form for each segment.
3. Calculate Heat Flux: Determine how much heat flows into and out of each segment.
4. Temperature Distribution: Solve the resulting equations to find the temperature at different points along the rod.

In summary, the Finite Volume Method is like dividing a big problem into smaller, manageable pieces (control volumes), ensuring that what goes into each piece comes out correctly, and solving for the unknowns in a way that respects conservation laws.

The Finite Difference Method (FDM) is a numerical technique used to solve problems involving differential equations, such as those describing heat transfer, fluid flow, and other physical processes. Here's a simple explanation:

Basic Concept

1. Discretization of the Domain:

   • The physical domain (e.g., a line, area, or volume) is divided into a grid of points.
   • These grid points represent specific locations where the solution will be approximated.

2. Approximating Derivatives:

   • The derivatives in the differential equations are approximated using differences between the values at these grid points.
   • This involves replacing continuous derivatives with discrete differences.

Steps Involved in FDM

1. Divide the Domain:

   • Create a grid of points over the domain where you want to find the solution.

2. Finite Difference Approximation:

   • Replace the derivatives in the differential equations with finite difference approximations. For example:
     • A first derivative $\frac{d u}{dx}$dxdu​ at a point can be approximated by $\frac{u(x + \Delta x) - u(x)}{\Delta x}$Δxu(x+Δx)−u(x)​.
     • A second derivative $\frac{d^2 u}{dx^2}$dx2d2u​ can be approximated by $\frac{u(x + \Delta x) - 2u(x) + u(x - \Delta x)}{\Delta x^2}$Δx2u(x+Δx)−2u(x)+u(x−Δx)​.

3. Set Up Equations:

   • Use these approximations to write equations for each grid point.
   • These equations relate the value at each point to its neighbors.

4. Solve the System:

   • Solve the resulting system of algebraic equations to find the approximate values at each grid point.

Advantages of FDM

• Simplicity: Easy to understand and implement.
• Direct Approach: Directly approximates the derivatives of the differential equations.
• Widely Used: Suitable for a wide range of problems with well-established techniques.

Example

Imagine you want to find the temperature distribution along a metal rod heated at one end.

1. Divide the Rod: Split the rod into a grid of points along its length.
2. Heat Equation: The heat equation involves a second derivative with respect to position.
3. Finite Difference Approximation: Approximate this second derivative at each grid point using the temperatures at neighboring points.
4. Set Up Equations: Write an equation for each grid point based on these approximations.
5. Solve for Temperatures: Solve the system of equations to get the temperature at each point along the rod.

The Finite Element Method (FEM) is a powerful numerical technique used to solve complex problems in engineering and physics, such as structural analysis, heat transfer, and fluid dynamics. Here's a simplified explanation:

Basic Concept

1. Discretization of the Domain:

   • The physical domain (e.g., a structure, a fluid region) is divided into smaller, simple-shaped regions called elements (e.g., triangles, quadrilaterals in 2D; tetrahedra, hexahedra in 3D).
   • The collection of these elements is known as a mesh.

2. Element Equations:

   • Each element is described by its own set of equations derived from the governing differential equations (like the equations of motion, heat equations).
   • The solution within an element is typically approximated by simple functions (often polynomials).

3. Assembly:

   • The equations from all the elements are assembled into a global system of equations that represents the entire domain.

4. Boundary Conditions and Solving:

   • Apply boundary conditions (constraints at the edges or surfaces of the domain).
   • Solve the global system of equations to find the approximate solution over the entire domain.

Steps Involved in FEM

1. Create the Mesh:

   • Divide the domain into small elements.

2. Choose Approximation Functions:

   • Select simple functions (often polynomials) to approximate the solution within each element.

3. Derive Element Equations:

   • Formulate the equations for each element based on the governing differential equations and the chosen approximation functions.

4. Assemble the Global System:

   • Combine the element equations to form a large system of equations that covers the whole domain.

5. Apply Boundary Conditions:

   • Incorporate any known values or constraints at the domain boundaries into the global system.

6. Solve the System:

   • Solve the resulting system of equations to get the approximate values at the nodes of the elements.

Advantages of FEM

• Flexibility: Can handle complex geometries and boundary conditions very well.
• Accuracy: Provides high accuracy for complicated problems.
• Versatility: Applicable to a wide range of physical problems.

Example

Imagine you want to analyze the stress distribution in a bridge component under a load.

1. Create the Mesh: Divide the component into small triangular or quadrilateral elements.
2. Choose Approximation Functions: Use simple polynomial functions to approximate the displacement within each element.
3. Derive Element Equations: Formulate equations for each element based on the mechanical properties and the load.
4. Assemble the Global System: Combine these equations into a global system that represents the entire component.
5. Apply Boundary Conditions: Include constraints like fixed supports or applied forces.
6. Solve the System: Solve for the displacements at the nodes, then compute stresses and strains.

Differences between FDM(Finite Difference Method) VS FVM(Finite Volume Method).

The Finite Difference Method (FDM) and the Finite Volume Method (FVM) are both numerical techniques used to solve partial differential equations (PDEs), but they have distinct approaches and characteristics. Here's a comparison to help understand their differences:

Discretization Approach

Finite Difference Method (FDM):

• Grid Points: FDM divides the domain into a grid of points.
• Derivatives Approximation: It approximates derivatives directly at these grid points using finite differences. For example, the first derivative $\frac{du}{dx}$dxdu​ might be approximated by $\frac{u(x + \Delta x) - u(x)}{\Delta x}$Δxu(x+Δx)−u(x)​.
• Equation Formulation: The differential equations are converted into algebraic equations based on the differences between values at grid points.

Finite Volume Method (FVM):

• Control Volumes: FVM divides the domain into a series of small control volumes (cells).
• Flux Balance: It focuses on conserving fluxes (like mass, momentum, energy) through the surfaces of these control volumes.
• Integral Form: The differential equations are integrated over each control volume, converting them into algebraic equations based on the fluxes across the volume boundaries.

Conservation Properties

FDM:

• Local Accuracy: Emphasizes the accuracy of the differential approximation at specific points.
• Conservation: Does not inherently ensure global conservation of quantities (e.g., mass, energy) unless special techniques are applied.

FVM:

• Global Conservation: Naturally ensures the conservation of quantities across the entire domain because it is based on flux balances.
• Physical Intuition: Aligns well with physical principles of conservation, making it particularly suitable for problems in fluid dynamics and heat transfer.

Flexibility and Applicability

FDM:

• Simplicity: Easier to implement for simple geometries and structured grids.
• Complex Geometries: Less flexible for handling complex geometries and unstructured grids.

FVM:

• Flexibility: More versatile in handling complex geometries and unstructured grids.
• Applications: Widely used in computational fluid dynamics (CFD) due to its strong conservation properties.

Accuracy and Stability

FDM:

• Local Error: The accuracy depends on the order of the finite difference scheme used.
• Stability: Requires careful consideration of stability criteria (e.g., Courant-Friedrichs-Lewy (CFL) condition) for time-dependent problems.

FVM:

• Flux Calculation: Accuracy depends on the methods used to approximate the fluxes at the control volume boundaries.
• Stability: Generally stable for a wide range of problems due to its conservation approach.

Example of Use

FDM Example:

• Heat Equation in a Rod: Divide the rod into grid points and approximate the temperature gradient using finite differences to solve the heat equation.

FVM Example:

• Fluid Flow in a Pipe: Divide the pipe into small control volumes and calculate the fluxes of fluid entering and leaving each volume to solve the Navier-Stokes equations.

Summary

• FDM: Uses differences between grid points to approximate derivatives, simple to implement, suitable for structured grids, but less flexible with complex geometries and doesn't inherently conserve quantities.
• FVM: Uses control volumes and flux balances to ensure conservation of quantities, flexible with complex geometries and unstructured grids, widely used in fluid dynamics and heat transfer problems.

Understanding these differences helps in choosing the appropriate method based on the specific requirements and nature of the problem to be solved.

Describe the need for interpolation schemes and flux limiters in FVM(Finite Volume Method).

In the Finite Volume Method (FVM), interpolation schemes and flux limiters play crucial roles in accurately calculating the fluxes across the boundaries of control volumes. Here's an easy-to-understand explanation of why they are needed and how they work:

Interpolation Schemes

Need for Interpolation:

• Non-uniform Grids: In many practical problems, the grid points (or control volumes) may not be evenly spaced. To compute fluxes (like heat, mass, or momentum) at the boundaries of these control volumes, we need to estimate values at points where we don’t have direct data.
• Boundary Fluxes: The flux at the boundary between two control volumes is often needed to solve the equations. Since we have values at the center of the control volumes, we must interpolate to estimate the values at the boundaries.

How Interpolation Schemes Work:

• Linear Interpolation: One of the simplest methods, where the value at the boundary is estimated by assuming a straight line between the values at the centers of the adjacent control volumes.

  • Example: If we have values $u_i$ui​ and $u_{i+1}$ui+1​ at centers of control volumes $i$i and $i+1$i+1, the boundary value $u_{i+1/2}$ui+1/2​ might be approximated as the average: $u_{i+1/2} = \frac{u_i + u_{i+1}}{2}$ui+1/2​=2ui​+ui+1​​.

• Higher-Order Interpolation: More complex methods can use polynomial or spline functions to achieve greater accuracy, but they are also more computationally intensive.

Flux Limiters

Need for Flux Limiters:

• Avoiding Oscillations: Higher-order interpolation schemes can sometimes introduce unrealistic oscillations, especially near sharp changes or discontinuities (like shock waves in fluid flow). These oscillations can make the numerical solution physically incorrect.
• Ensuring Stability: Flux limiters help in controlling these oscillations, ensuring the numerical solution remains stable and physically realistic.

How Flux Limiters Work:

• Detecting Discontinuities: Flux limiters identify regions with sharp changes or discontinuities in the solution.
• Adjusting Interpolation: In smooth regions, higher-order interpolation is used for accuracy. Near discontinuities, the interpolation is adjusted (limited) to prevent oscillations.
  • Example: If the solution is smooth, a higher-order scheme might be used. Near a discontinuity, the scheme might switch to a lower-order (more diffusive) scheme to maintain stability.

Putting It All Together

1. Interpolation Schemes:

   • They help estimate values at control volume boundaries by interpolating data from the center of control volumes.
   • Different schemes offer a trade-off between simplicity and accuracy.

2. Flux Limiters:

   • They adjust the interpolation to prevent unrealistic oscillations and ensure stability, particularly near sharp gradients or discontinuities.
   • They make the solution more robust by switching between higher-order accuracy and lower-order stability as needed.

Example to Illustrate

Imagine you're simulating how a pollutant spreads in a river. The concentration of the pollutant is measured at specific points (grid points):

1. Interpolation:

   • You need to know the concentration at the boundaries between these points to calculate how much pollutant flows from one section of the river to the next.
   • Interpolation schemes estimate these boundary values based on the known concentrations at the grid points.

2. Flux Limiters:

   • If there's a sudden increase in pollutant concentration (like a spill), simple interpolation might overshoot, predicting negative concentrations or oscillations.
   • Flux limiters detect this sudden change and adjust the interpolation to prevent these non-physical results, ensuring the simulation stays realistic.

In summary, interpolation schemes in FVM help estimate boundary values necessary for calculating fluxes, while flux limiters ensure that these estimates remain stable and physically realistic, especially near discontinuities or sharp changes in the solution.