Differential Equations Lecture Work Solutions 291

Differential Equations Lecture Work Solutions 291 - 8...

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Unformatted text preview: 8 Finite Differences 8.1 Taylor Series 8.2 Finite Differences Problems 1. Verify that ∂3u ∆3 ui,j |i,j = x 3 + O (∆x). ∂x3 (∆x) 2. Consider the function f (x) = ex . Using a mesh increment ∆x = 0.1, determine f (x) at x = 2 with forward-difference formula, the central-difference formula, and the second order three-point formula. Compare the results with the exact value. Repeat the comparison for ∆x = 0.2. Have the order estimates for truncation errors been a reliable guide? Discuss this point. 3. Develop a finite difference approximation with T.E. of O (∆y ) for ∂ 2 u/∂y 2 at point (i, j ) using ui,j , ui,j +1, ui,j −1 when the grid spacing is not uniform. Use the Taylor series method. Can you devise a three point scheme with second-order accuracy with unequal spacing? Before you draw your final conclusions, consider the use of compact implicit representations. 4. Establish the T.E. for the following finite difference approximation to ∂u/∂y at the point (i, j ) for a uniform mesh: −3ui,j + 4ui,j +1 − ui,j +2 ∂u ≈ . ∂y 2∆y What is the order? 291 ...
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