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6. Finite_differebces_formulas_for_unequall

# 6. Finite_differebces_formulas_for_unequall - i x x x i i i...

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2.2.4 Unequally spaced points All finite difference formulas (i.e., forward, backward, and central difference formulas) developed in this section and provided in the previous handout assume equal x Δ spacing (uniform grid). There are some applications which involve unequally spaced points (non-uniform grids). This is often the case with experimental measurements and points close to boundaries of a given system geometry. The formulas for derivative approximations have to be re-derived for a specific non-uniform grid. However, the derivation the derivation from Taylor series is very similar to that employed for uniform grids. Example: Find O( Δ x 2 ) accurate central difference formula for i x dx df (2) Equation ... ! 3 ! 2 ) ( (1) Equation ... ! 3 8 ! 2 4 2 ) 2 ( 3 3 3 2 2 2 1 3 3 3 2 2 2 1 + Δ Δ + Δ = Δ = + Δ + Δ + Δ + = Δ + = + i i i i i i x x x i i
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Unformatted text preview: i x x x i i i dx f d x dx f d x dx df x f x x f f dx f d x dx f d x dx df x f x x f f Eliminate Δ x 2 terms and solve for i x dx df , namely, Equation (1) - 4 x Equation (2) gives ... ! 3 2 6 4 3 ... ! 3 12 6 3 4 3 3 2 1 1 3 3 3 1 1 + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ − Δ − + = + Δ + Δ + − = − − + − + i i i i x i i i x x x i i i dx f d x x f f f dx df dx f d x dx df x f f f The dominant error term is O( Δ x 2 ), thus the approximate formula has 2 nd order accuracy. Conclusion: the procedure is the same. Only the algebraic details of the elimination change when we have unequal spacing between the points. f i f i-1 2 Δ x Δ x x y x i-1 x i x i+1 f i+1 exact derivative approximate formula error terms...
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