8.2 Discretization methods

Discretization methods are crucial in aerodynamics for solving complex fluid flow equations. These techniques divide the computational domain into smaller elements, allowing for numerical approximations of partial differential equations governing fluid dynamics.

Various methods like finite difference, finite volume, finite element, and spectral approaches offer different trade-offs between accuracy, computational cost, and geometric flexibility. Understanding these methods helps engineers choose the best approach for specific aerodynamic problems.

Discretization methods overview

• Discretization methods are numerical techniques used to approximate the solution of partial differential equations (PDEs) that govern fluid flow and heat transfer in aerodynamics
• These methods involve dividing the computational domain into smaller, discrete elements or volumes and solving the governing equations at these discrete points
• The choice of discretization method depends on factors such as the complexity of the geometry, the desired accuracy, and the computational resources available

Finite difference method

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• Approximates derivatives in the governing PDEs using Taylor series expansions
• Replaces continuous derivatives with finite differences at discrete grid points
• Relatively simple to implement and computationally efficient for structured grids
• Limited accuracy for complex geometries and discontinuities

Finite volume method

• Divides the computational domain into a set of control volumes
• Integrates the governing equations over each control volume, ensuring conservation of mass, momentum, and energy
• Handles complex geometries and discontinuities effectively
• Widely used in computational fluid dynamics (CFD) for aerodynamic simulations

Finite element method

• Discretizes the domain into a set of finite elements (e.g., triangles or tetrahedra)
• Approximates the solution using a weighted sum of basis functions defined on each element
• Handles complex geometries and allows for adaptive mesh refinement
• Computationally expensive compared to finite difference and finite volume methods

Spectral methods

• Approximates the solution using a linear combination of basis functions (e.g., Fourier or Chebyshev polynomials)
• Provides high accuracy for smooth solutions and simple geometries
• Requires fewer grid points compared to other methods for the same level of accuracy
• Limited applicability for complex geometries and discontinuities

Finite difference formulations

Forward, backward, and central differences

• Forward difference: $f'(x) \approx \frac{f(x+h) - f(x)}{h}$, first-order accurate
• Backward difference: $f'(x) \approx \frac{f(x) - f(x-h)}{h}$, first-order accurate
• Central difference: $f'(x) \approx \frac{f(x+h) - f(x-h)}{2h}$, second-order accurate
• Higher-order differences can be derived using Taylor series expansions

Explicit vs implicit schemes

• Explicit schemes calculate the solution at the next time step using only the known values from the previous time step
  • Conditionally stable, requiring small time steps to maintain stability
  • Computationally efficient per time step, but may require many time steps
• Implicit schemes involve solving a system of equations that includes both known and unknown values at the next time step
  • Unconditionally stable, allowing for larger time steps
  • Computationally more expensive per time step, but may require fewer time steps overall

Accuracy and stability considerations

• Spatial accuracy depends on the order of the finite difference approximation and the grid resolution
• Temporal accuracy depends on the order of the time integration scheme and the time step size
• Stability is influenced by the choice of scheme (explicit or implicit), grid resolution, and time step size
• The Courant-Friedrichs-Lewy (CFL) condition relates the time step size to the grid spacing and the characteristic velocity to ensure stability in explicit schemes

Finite volume formulations

Control volume approach

• Divides the computational domain into a set of non-overlapping control volumes
• Integral form of the conservation equations is applied to each control volume
• Flux balance is maintained across control volume faces, ensuring conservation

Conservation laws

• Mass conservation: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0$
• Momentum conservation: $\frac{\partial (\rho \vec{u})}{\partial t} + \nabla \cdot (\rho \vec{u} \otimes \vec{u}) = -\nabla p + \nabla \cdot \overline{\overline{\tau}}$
• Energy conservation: $\frac{\partial (\rho E)}{\partial t} + \nabla \cdot (\rho H \vec{u}) = \nabla \cdot (k \nabla T) + \nabla \cdot (\overline{\overline{\tau}} \cdot \vec{u})$

Flux evaluation and interpolation

• Fluxes across control volume faces are evaluated using interpolation schemes
  • Upwind schemes (first-order, second-order, etc.) consider the direction of information propagation
  • Central schemes (second-order, fourth-order, etc.) use a symmetric stencil around the face
• Gradient reconstruction methods (Green-Gauss, least-squares) are used to compute gradients at cell centers

Boundary conditions treatment

• Boundary conditions are enforced by modifying the fluxes at the domain boundaries
• Common boundary conditions in aerodynamics include:
  • No-slip wall: $\vec{u} = 0$
  • Inflow: specified velocity or pressure
  • Outflow: zero-gradient or specified pressure
  • Symmetry: zero-gradient for scalar quantities, zero normal velocity
• Ghost cells or extrapolation techniques are used to implement boundary conditions

Finite element formulations

Weak form of governing equations

• Governing PDEs are multiplied by a test function and integrated over the domain
• Integration by parts is applied to reduce the order of the derivatives
• Boundary conditions are incorporated into the weak form

Element types and shape functions

• Domain is discretized into a set of finite elements (triangles, quadrilaterals, tetrahedra, hexahedra)
• Shape functions are defined on each element to approximate the solution
  • Linear, quadratic, or higher-order polynomials
  • Lagrange or hierarchical basis functions
• Continuity of the solution across element boundaries is ensured

Galerkin method

• Test functions are chosen to be the same as the shape functions
• Leads to a symmetric system of equations
• Provides optimal convergence properties for self-adjoint problems

Assembly and solution procedures

• Element equations are assembled into a global system of equations
• Sparse matrix storage techniques are used to efficiently store the global matrix
• Linear system of equations is solved using direct or iterative methods
  • Direct methods (LU decomposition, Cholesky factorization) are accurate but computationally expensive
  • Iterative methods (Conjugate Gradient, GMRES) are more efficient for large systems but may require preconditioning

Spectral method formulations

Fourier and Chebyshev polynomials

• Fourier series are used for periodic domains
  • $f(x) = \sum_{k=-\infty}^{\infty} \hat{f}_k e^{ikx}$
  • Efficient computation using Fast Fourier Transform (FFT)
• Chebyshev polynomials are used for non-periodic domains
  • $T_n(x) = \cos(n \arccos(x))$
  • Clustered grid points near the boundaries for better resolution

Collocation and Galerkin approaches

• Collocation approach: governing equations are satisfied at a set of collocation points
  • Easier to implement and computationally efficient
  • Less stable and may require filtering for nonlinear problems
• Galerkin approach: governing equations are projected onto a set of basis functions
  • More stable and accurate, especially for nonlinear problems
  • Computationally more expensive due to the need for numerical integration

Advantages and limitations

• Spectral methods provide exponential convergence for smooth solutions
• Require fewer grid points compared to finite difference and finite volume methods for the same level of accuracy
• Limited applicability for complex geometries and discontinuities
• Efficient for problems with periodic boundary conditions or simple geometries

Discretization error analysis

Truncation and round-off errors

• Truncation error: difference between the exact solution of the PDE and the exact solution of the discretized equation
  • Arises from the approximation of derivatives using finite differences or truncated series expansions
  • Depends on the order of the discretization scheme and the grid resolution
• Round-off error: difference between the exact solution of the discretized equation and the numerical solution obtained using finite-precision arithmetic
  • Accumulates over the course of the simulation due to the limited precision of floating-point numbers
  • Can be mitigated by using higher-precision arithmetic or error-compensated summation algorithms

Convergence and order of accuracy

• Convergence: as the grid resolution increases (h → 0) or the polynomial order increases (p → ∞), the numerical solution approaches the exact solution
• Order of accuracy: rate at which the discretization error decreases with increasing grid resolution or polynomial order
  • For finite difference methods: error = O(h^n), where n is the order of the scheme
  • For spectral methods: error = O(e^(-αN)), where N is the number of modes or polynomials
• Higher-order schemes provide faster convergence but may be more computationally expensive and less stable

Grid refinement studies

• Systematic refinement of the grid or polynomial order to assess the convergence and order of accuracy of the discretization scheme
• Richardson extrapolation can be used to estimate the exact solution and the order of accuracy from a sequence of grid refinements
• Grid convergence index (GCI) provides a standardized way to report the uncertainty associated with the discretization error

Discretization scheme selection

Problem-specific considerations

• Geometry complexity: finite volume and finite element methods are better suited for complex geometries, while finite difference and spectral methods are more efficient for simple geometries
• Solution smoothness: spectral methods are optimal for smooth solutions, while finite volume and finite element methods can handle discontinuities more effectively
• Boundary conditions: spectral methods are efficient for periodic boundary conditions, while finite volume and finite element methods can handle a wider range of boundary conditions

Computational cost vs accuracy trade-offs

• Higher-order schemes provide better accuracy but are computationally more expensive
• For a given level of accuracy, spectral methods require fewer grid points compared to finite difference and finite volume methods
• Adaptive mesh refinement can be used to locally refine the grid in regions of high gradients or complex geometry, reducing the overall computational cost

Hybrid and adaptive methods

• Hybrid methods combine different discretization schemes to exploit their respective strengths
  • Spectral element method: combines the geometric flexibility of finite elements with the accuracy of spectral methods
  • Discontinuous Galerkin method: uses a finite element discretization with discontinuous basis functions, allowing for better handling of discontinuities and adaptive mesh refinement
• Adaptive methods dynamically adjust the grid resolution or polynomial order based on error estimates or solution features
  • h-adaptivity: refines or coarsens the grid while keeping the polynomial order fixed
  • p-adaptivity: increases or decreases the polynomial order while keeping the grid fixed
  • hp-adaptivity: combines h-adaptivity and p-adaptivity for optimal performance