Week 9 - FVM Literature Review

Finite Volume Method :-

The finite volume method (FVM) is a widely used numerical technique for solving partial differential equations (PDEs) that arise in various fields of engineering and science, including fluid dynamics, heat transfer, electromagnetics, and structural mechanics. Here are some key literature reviews on FVM:

1. "A review of finite volume method for partial differential equations" by Kandasamy et al. (2016) provides a comprehensive overview of the FVM, its mathematical foundation, and its application to various types of PDEs. The review covers topics such as discretization schemes, boundary conditions, convergence analysis, and parallelization techniques.

2. "Recent advances in finite volume methods and their applications in fluid dynamics" by Darwish and Moukalled (2013) focuses on the application of FVM to fluid dynamics problems. The review covers topics such as high-resolution schemes, turbulence modeling, multiphase flow, and compressible flow.

3. "A review of finite volume methods for solving partial differential equations with rough coefficients" by Eymard et al. (2016) addresses the challenges of using FVM for PDEs with rough coefficients, which can lead to numerical instability and inaccurate solutions. The review covers topics such as upwinding schemes, domain decomposition, and adaptive mesh refinement.

4. "A review of the finite volume method for non-linear conservation laws" by Castro et al. (2013) focuses on the application of FVM to non-linear conservation laws, which arise in many areas of physics and engineering. The review covers topics such as shock-capturing schemes, well-balanced schemes, and positivity-preserving schemes.

Finite Difference Method (FDM) and Finite Volume Method (FVM) are numerical methods used to solve partial differential equations. The major differences between FDM and FVM are:

1. Approach: In FDM, the solution domain is divided into a grid of points, and the values of the variables are approximated at these points. The derivatives in the partial differential equations are approximated using finite differences between these points. In FVM, the solution domain is divided into a grid of control volumes, and the values of the variables are approximated at the centers of these control volumes. The integral form of the partial differential equations is used to approximate the fluxes between these control volumes.

2. Conservation: FVM is a conservative method, meaning that it ensures the conservation of mass, momentum, and energy at each control volume. FDM does not necessarily conserve these quantities and may require additional techniques to enforce conservation.

3. Fluxes: FVM calculates the fluxes of variables across the boundaries of the control volumes, while FDM approximates the derivatives of variables at the points within the grid.

4. Boundary Conditions: In FVM, boundary conditions are enforced at the control volume boundaries, while in FDM, boundary conditions are enforced at the grid points.

5. Accuracy: FVM is generally more accurate than FDM because it better approximates the fluxes between control volumes. However, FDM can be simpler and less computationally expensive than FVM, making it suitable for certain problems.

Flux limiters are used in Finite Volume Method (FVM) to improve the accuracy and stability of numerical solutions. In this method, the fluxes of conserved quantities, such as mass, momentum, and energy, are calculated at the boundaries of these control volumes.

However, the flux calculation can introduce numerical errors, especially when the solution contains sharp gradients or discontinuities. These errors can lead to numerical instabilities and spurious oscillations in the solution.

To address this issue, flux limiters are introduced to reduce the magnitude of the fluxes at the cell interfaces. These limiters are functions that modify the fluxes in a way that preserves the monotonicity of the solution and prevents oscillations. The flux limiter function is applied to the difference between the interpolated values of the solution at the cell interfaces.

There are various types of flux limiters used in FVM, including:

1. Van Leer limiter: This limiter uses a nonlinear function to reduce the magnitude of the fluxes and preserve the positivity of the solution.

2. Superbee limiter: This limiter uses a piecewise linear function to reduce the magnitude of the fluxes and prevent oscillations.

3. Minmod limiter: This limiter uses a piecewise constant function to reduce the magnitude of the fluxes and preserve the monotonicity of the solution.

4. MC limiter: This limiter uses a quadratic function to reduce the magnitude of the fluxes and prevent oscillations.

The choice of flux limiter depends on the characteristics of the solution and the numerical scheme used for the discretization. The selection of an appropriate flux limiter is essential for obtaining accurate and stable solutions in FVM.

Need of interpolation schemes in FVM-

1. The approximation of surface and volume integrals requires value of variable at locations other than computationals nodes of CV.
2. Value at these locations are obtained using interpolation formula.

Various possibilites of interploation

1. Upwind Interpolation
2. Linear Interpolation
3. Quadric upwind interpolation
4. Hybrid,TVD and ENO Interpolation schemes

Upwind Interpolation: Upwind interpolation is a technique that is used to estimate the value of a variable at the boundary of a control volume based on the direction of flow of the variable. In upwind interpolation, the value of the variable is estimated using the value at the nearest neighboring control volume that is upstream (in the direction of flow) of the current control volume. Upwind interpolation is commonly used in solving problems involving advection-dominated flows, where the direction of flow is an important factor in the solution.

Linear interpolation: Approximate the value of variable at CV face center by Linear interpolation of the values at two nearest computational nodes.

`f_e=f_E.lambda_e+f_p(1-lambda_e)`

`lambda_e=(x_e-x_p)/(x_E-x_P)`

The Linear Interpolation is equvalent to the use of central difference formula of the first oreder derivative, and hence, this scheme is also termed central deiffernce scheme (CDS).

Quadratic Upwind Interpolation: Quadratic upwind interpolation is a more complex technique that is used to estimate the value of a variable at the boundary of a control volume based on the direction of flow of the variable. Quadratic upwind interpolation uses a weighted average of the values at three neighboring control volumes, with the weights determined by the direction of flow of the variable. Quadratic upwind interpolation is more accurate than linear interpolation but is also more computationally expensive.

Hybrid Interpolation Scheme: The Hybrid scheme is a combination of central and upwind interpolation schemes. It switches between the two schemes depending on the direction of flow of the variable at the boundary of the control volume. The Hybrid scheme provides a good compromise between accuracy and stability.

Total Variation Diminishing (TVD) Interpolation Scheme: The TVD scheme is designed to prevent the formation of spurious oscillations that can occur in the solution of hyperbolic partial differential equations. It does this by imposing a condition called total variation diminishing, which ensures that the total variation of the solution does not increase over time. The TVD scheme is widely used in numerical methods for solving problems involving shock waves.

Essentially Non-Oscillatory (ENO) Interpolation Scheme: The ENO scheme is designed to reduce numerical errors by selecting the most accurate interpolation scheme based on the smoothness of the solution. It uses a set of weighted stencil points to determine the most accurate interpolation scheme at each point in the solution domain. The ENO scheme is particularly useful for solving problems involving discontinuous solutions or shocks.