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8.1 Finite difference methods

7 min readaugust 20, 2024

Finite difference methods are powerful numerical techniques for solving partial differential equations in fluid dynamics. By discretizing the domain into a grid and approximating derivatives with finite differences, these methods enable the solution of complex flow problems on computers.

These methods form the foundation of many computational fluid dynamics simulations. Understanding their principles, including , stability, and accuracy, is crucial for effectively applying them to solve real-world fluid flow problems and interpreting the results.

Finite difference methods

  • Numerical techniques used to solve partial differential equations by discretizing the domain into a grid of points
  • Derivatives are approximated using finite differences between neighboring grid points
  • Widely used in computational fluid dynamics to solve governing equations like Navier-Stokes

Discretization of derivatives

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  • Process of approximating continuous derivatives with discrete differences
  • Domain is divided into a grid with finite spacing between points
  • Derivatives at each point are estimated using neighboring point values
  • Enables solving PDEs numerically on a computer

Finite difference approximations

  • Mathematical formulas used to estimate derivatives based on discrete function values
  • Derived from Taylor series expansions truncated to a certain order
  • Common types include forward, backward, and central differences
  • Accuracy depends on the order of approximation and grid spacing

Taylor series expansions

  • Mathematical tool used to approximate functions as polynomial series
  • Enables expressing function values at neighboring points in terms of the function and its derivatives at a reference point
  • Truncating the series introduces an error term called the
  • Forms the basis for deriving finite difference approximations

Truncation errors

  • Errors introduced by truncating Taylor series expansions in finite difference approximations
  • Depend on the order of the approximation and the grid spacing
  • Higher-order approximations and finer grids lead to smaller truncation errors
  • requires truncation errors to vanish as grid spacing approaches zero

Consistency of finite differences

  • Property ensuring that the finite difference approximation converges to the exact derivative as grid spacing approaches zero
  • Requires the truncation error to vanish in the limit of zero grid spacing
  • Necessary condition for the numerical solution to converge to the exact solution
  • Achieved by using appropriate order of approximations based on Taylor series

Stability of finite differences

  • Property related to the growth or decay of errors in the numerical solution over time
  • Unstable schemes lead to unbounded growth of errors, causing the solution to diverge
  • Stability depends on factors like grid spacing, time step, and the specific scheme used
  • Necessary condition for the numerical solution to remain bounded and physically meaningful

Explicit vs implicit methods

  • Two broad categories of finite difference schemes based on how the unknowns are computed
  • Explicit methods calculate the unknown values directly from known values at the current time step
    • Straightforward to implement but may require small time steps for stability
  • Implicit methods solve a system of equations involving both known and unknown values
    • More complex to implement but allow larger time steps while maintaining stability

Forward vs backward differences

  • Two basic types of finite difference approximations for first-order derivatives
  • uses the current and next grid points to estimate the derivative
    • f(xi)f(xi+1)f(xi)Δxf'(x_i) \approx \frac{f(x_{i+1}) - f(x_i)}{\Delta x}
  • uses the current and previous grid points
    • f(xi)f(xi)f(xi1)Δxf'(x_i) \approx \frac{f(x_i) - f(x_{i-1})}{\Delta x}
  • Have first-order accuracy and introduce or dispersion

Central difference scheme

  • Finite difference approximation that uses both forward and backward differences
  • Second-order accurate approximation for first derivatives
    • f(xi)f(xi+1)f(xi1)2Δxf'(x_i) \approx \frac{f(x_{i+1}) - f(x_{i-1})}{2\Delta x}
  • Fourth-order accurate approximation for second derivatives
    • f(xi)f(xi+1)2f(xi)+f(xi1)(Δx)2f''(x_i) \approx \frac{f(x_{i+1}) - 2f(x_i) + f(x_{i-1})}{(\Delta x)^2}
  • Reduces numerical diffusion compared to one-sided differences

Upwind difference scheme

  • Finite difference approximation that considers the direction of information propagation
  • Uses backward differences for positive wave speeds and forward differences for negative wave speeds
  • Helps to stabilize the solution by reducing numerical oscillations
  • Commonly used for advection-dominated problems like supersonic flows

Courant–Friedrichs–Lewy condition

  • Stability criterion for explicit finite difference schemes
  • Relates the time step, grid spacing, and maximum wave speed
  • States that the numerical domain of dependence must contain the physical domain of dependence
  • Mathematically expressed as C=uΔtΔxCmaxC = \frac{u\Delta t}{\Delta x} \leq C_{max}, where CC is the Courant number and CmaxC_{max} is a scheme-dependent constant

Numerical diffusion

  • Artificial diffusion introduced by finite difference approximations
  • Causes smearing or broadening of sharp gradients in the solution
  • Arises from truncation errors in the approximations
  • Can be reduced by using higher-order schemes or finer grids

Numerical dispersion

  • Artificial dispersion introduced by finite difference approximations
  • Causes phase errors and spurious oscillations in the solution
  • Arises from odd-order derivatives in the truncation error
  • Can be reduced by using higher-order schemes or dispersion-relation-preserving methods

Boundary conditions in FDM

  • Specify the behavior of the solution at the boundaries of the computational domain
  • Essential for well-posedness and uniqueness of the numerical solution
  • Common types include Dirichlet (fixed value), Neumann (fixed gradient), and periodic boundary conditions
  • Implemented by modifying the near the boundaries

Initial conditions in FDM

  • Specify the solution at the initial time step
  • Provide the starting point for marching the solution forward in time
  • Must be consistent with the boundary conditions and the governing equations
  • Can be prescribed analytically or interpolated from experimental data

Finite difference grids

  • Discretization of the continuous domain into a set of discrete points
  • Form the basis for applying finite difference approximations
  • Can be structured (regular spacing) or unstructured (irregular spacing)
  • Grid resolution affects the accuracy and computational cost of the solution

Grid resolution effects

  • Finer grids generally lead to more accurate solutions but increase computational cost
  • Coarser grids are computationally cheaper but may introduce larger errors
  • studies are used to assess the sensitivity of the solution to grid resolution
  • refinement techniques can optimize the grid based on solution features

Structured vs unstructured grids

  • Structured grids have regular connectivity and can be mapped to a Cartesian coordinate system
    • Simpler to implement and computationally efficient
    • May have difficulty conforming to complex geometries
  • Unstructured grids have irregular connectivity and can adapt to complex geometries
    • More flexible but computationally expensive
    • Require more complex data structures and algorithms

Staggered grids

  • Arrangement where different variables are stored at different grid locations
  • Commonly used in fluid dynamics to avoid checkerboard instabilities
  • Pressure and velocity components are typically stored at cell centers and cell faces, respectively
  • Improves the coupling between pressure and velocity in incompressible flows

Finite difference operators

  • Mathematical operators that represent the finite difference approximations of derivatives
  • Defined based on the stencil and the order of accuracy
  • Can be combined to form the discrete versions of the governing equations
  • Examples include gradient, divergence, and Laplacian operators

Finite difference stencils

  • Set of grid points used to approximate derivatives at a given point
  • Determine the order of accuracy and the computational molecule of the scheme
  • Can be one-sided (forward or backward) or central
  • Wider stencils generally lead to higher-order accuracy but increased computational cost

Higher-order finite differences

  • Finite difference approximations with accuracy higher than second-order
  • Obtained by including more terms in the Taylor series expansion
  • Require wider stencils and more grid points
  • Offer improved accuracy and reduced numerical errors compared to lower-order schemes

Compact finite differences

  • High-order finite difference schemes with smaller stencils
  • Achieve higher accuracy by implicitly solving for the derivatives
  • Require the solution of a tridiagonal system of equations
  • Offer spectral-like resolution with reduced computational cost compared to explicit schemes

Finite volume methods vs FDM

  • Alternative approach to discretizing partial differential equations
  • Based on integral formulation of the conservation laws over control volumes
  • Store the average values of the solution over the control volumes
  • Can handle complex geometries and discontinuities more naturally than FDM
  • May require flux reconstruction and limiting to maintain stability and accuracy

Advantages of finite differences

  • Conceptually simple and easy to implement
  • Computationally efficient, especially on structured grids
  • Can achieve high-order accuracy with appropriate schemes
  • Well-suited for problems with simple geometries and smooth solutions

Limitations of finite differences

  • Difficulty in handling complex geometries and irregular domains
  • May introduce numerical artifacts like diffusion and dispersion
  • Require careful treatment of boundary conditions and grid interfaces
  • May suffer from stability issues for certain problems and schemes

Applications in fluid dynamics

  • Widely used in computational fluid dynamics to solve the governing equations
  • Can simulate a variety of flow phenomena, including turbulence, heat transfer, and combustion
  • Examples include simulating airflow over aircraft, blood flow in arteries, and ocean currents
  • Often combined with other techniques like finite volume methods and spectral methods

Solving advection equations

  • Advection equations describe the transport of a quantity by a velocity field
  • Finite difference methods can solve advection equations using upwind or central schemes
  • Upwind schemes are stable but introduce numerical diffusion
  • Central schemes are less diffusive but may require stabilization techniques

Solving diffusion equations

  • Diffusion equations describe the spread of a quantity due to random motion
  • Finite difference methods can solve diffusion equations using explicit or implicit schemes
  • Explicit schemes are simple but require small time steps for stability
  • Implicit schemes are stable for larger time steps but require solving a system of equations

Solving Navier-Stokes equations

  • govern the motion of viscous fluids
  • Finite difference methods can solve Navier-Stokes equations using various schemes
  • are commonly used to avoid checkerboard instabilities
  • Fractional step methods can decouple the pressure and velocity computations
  • Turbulence modeling is often required to capture the effects of small-scale motions
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© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.

© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
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