The compact finite difference (CTFD) formulation, or Hermitian formulation, is a numerical method to solve the compressible Navier–Stokes equation. This method is both accurate and numerically very stable (especially for high-order derivatives).
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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 (CTFD) 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.
First order derivatives
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
Second order derivatives
Similar to the first order derivatives equation, i.e., eq. (5) the equation for a 2nd order derivative will be:
Applications
Compact finite difference formulation is very frequently used to solve problems based on the Navier–Stokes equation, and is used extensively in solving hyperbolic equations.