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# compact - 46 CHAPTER 3 FINITE DIFFERENCE APPROXIMATION OF...

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Unformatted text preview: 46 CHAPTER 3. FINITE DIFFERENCE APPROXIMATION OF DERIVATIVES 3.5 Compact Differencing Schemes A major disadvantage of the finite difference approach presented earlier is the widening of the computational stencil as the order of the approximation is increased. These large stencils are cumbersome near the edge of the domain where no data is available to perform the differencing. Fortunately, it is possible to derive high-order finite difference approximation with compact stencils at the expense of a small complication in their evalution: implicit differencing schemes (as opposed to explicit schemes) must be used. Here we show how these schemes can be derived. 3.5.1 Derivation of 3-term compact schemes The Taylor series expansion of u i + m , where u i + m = u ( x i + m Δ x ) about point x i can be written as u i + m = ∞ summationdisplay n =0 ( m Δ x ) n n ! u ( n ) (3.56) where u ( n ) is the n-th derivative of u with respect to x at x i , with m being an arbitrary number. From this expression it is easy to obtain the following sum and difference u i + m ± u i- m = ∞ summationdisplay n =0 ((1 ± ( − 1) n ) ( m Δ x ) n n ! u ( n ) (3.57) u i + m + u i- m 2 = ∞ summationdisplay n =0 , 2 , 4 ( m Δ x ) n n ! u ( n ) (3.58) u i + m − u i- m 2 m Δ x = ∞ summationdisplay n =0 , 2 , 4 ( m Δ x ) n ( n + 1)!...
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compact - 46 CHAPTER 3 FINITE DIFFERENCE APPROXIMATION OF...

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