page_title: Compact finite difference
section_title: First order derivatives
hierarchical_section_title: Compact finite difference / Compact finite difference formulation / First order derivatives
caption_attribution_description: English: The above mentioned is a full formula of a three point Hermitian formula
context_page_description: The compact finite difference formulation, or Hermitian formulation, is a numerical method to solve the compressible Navier–Stokes equation. This method is both accurate and numerically very stable.
  The expression for partial derivatives is developed and expressed mainly on dependent variables. An approach to increase accuracy of the estimates of the derivatives, in particular a problem involving shorter length scales or equivalently high frequencies, is to include the influence of the neighboring points in the calculations. This approach is analogous to the solution of a partial differential equation by an Implicit scheme to an explicit scheme. The resulting approximation is called a compact finite difference formulation or a Hermitian formulation.
  Forward difference formulae and backward difference formulae are first order accurate, and central difference formula are second order accurate; compact finite difference formulae provide a more accurate method to solve equations.
context_section_description: Consider a three-point Hermitian formula involving the first derivative:
  Substituting the Taylor series expansion of the terms fᵢ₊₁ and fᵢ₋₁ results in:
  In the above expression, only the first few terms are to be considered zero, and the rest of the higher order terms will be considered as the truncation error (TE). To obtain a third-order scheme, the coefficients of fᵢ , f'ᵢ, f''ᵢ , f'''ᵢ will be zero. If it is a fourth-order scheme, the coefficient of f’’’’ᵢ will also be zero.
  From the above equations, one can solve for a₁ , a₀ , a−1 and b₀ in the form of b₁ and b−1
  The Truncation error will be:
  Substituting (4) in the above equation results in:
  The standard compact finite difference formula for first order derivatives of f(x) has a 3-point formulation
  
  [Δx/3] [f’ᵢ₋₁ + 4f’ᵢ + f’ᵢ₊₁] = -fᵢ₋₁ + fᵢ₊₁
  
  where f’ᵢ = df(xᵢ)/dx
  
  Similarly, for a fifth point formulation, take m=2 in the beginning.
  One-parameter family of compact finite difference schemes for first order derivatives Scheme
caption: Extended formula of a three-point Hermitian formula
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