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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). 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. ==Compact finite difference formulation== ===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 (error (numerical integration)|truncation error )] (TE). To obtain a third-order scheme, the coefficients of ''f''''i'' , ''f'' 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 () (+ 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 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「compact finite difference」の詳細全文を読む スポンサード リンク
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