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## Numerical Differentiation

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**Numerical Differentiation**Let us compute dy/dx or df/dx at node i Denote the difference operators:**Numerical Differentiation: Finite Difference**Approximate the function between as: Forward Difference: Approximate the function between as: Backward Difference:**Numerical Differentiation: Finite Difference**Approximate the function between three points: Now, evaluate df/dx at x = xi:**Numerical Differentiation: Finite Difference**Central Difference: For regular or uniform grid: Let us assume regular grid with a mesh size of h**Numerical Differentiation: Finite Difference**• Accuracy: How accurate is the numerical differentiation scheme with respect to the TRUE differentiation? • Truncation Error analysis • Modified Wave Number, Amplitude Error and Phase Error analysis for periodic functions • Recall: True Value (a) = Approximate Value + Error (ε) • Consistency: A numerical expression for differentiation or a numerical differentiation scheme is consistent if it converges to the TRUE differentiation as h → 0.**Truncation Error Analysis: First Derivative**Truncation error for forward difference scheme for the 1st Derivative is: O(h) Truncation error for backward difference scheme for the 1st Derivative is: O(h)**Truncation Error Analysis: First Derivative**Truncation error for the central difference scheme for the 1st Derivative is: O(h2)**Truncation Error Analysis: Second Derivative**Truncation error for this central difference scheme for the 2nd Derivative is: O(h2)**Truncation Error Analysis: Non-Uniform Grid**For regular or uniform grid: Truncation error for this central difference scheme for the 1st Derivative is O(h) for non-uniform grid and O(h2) uniform grid**Numerical Differentiation: Finite Difference**Consistency: A numerical expression for the derivative is consistent if the leading order term in the Truncation Error (TE) satisfies the following: If the leading order term in the truncation error is: TE = Khp or O(hp) where, the numerical differentiation scheme is consistent if , p ≥ 1**General Technique for Construction of Finite Difference**Scheme of Arbitrary Order Method of Undetermined Coefficients General finite difference scheme for uniform grid size h: or or Let us take an example with q = 1, m = 2 and n = 0**General Technique for Construction of Finite Difference**Scheme of Arbitrary Order: Example General finite difference scheme for uniform grid size h: Expand all the function values evaluated at nodes other than i using Taylor’s series:**General Technique for Construction of Finite Difference**Scheme of Arbitrary Order: Example**General Technique for Construction of Finite Difference**Scheme of Arbitrary Order: Example Consider Take q = 2, m = 2 and n = 0 (for example)**General Technique for Construction of Finite Difference**Scheme of Arbitrary Order: Example General finite difference scheme for uniform grid size h: LHS RHS RHS RHS**Richardson’s Extrapolation**Combine terms to result in derivative estimates of TE O(h2)**Richardson’s Extrapolation**Combine terms to result in derivative estimates of TE O(h3)**Richardson’s Extrapolation**In order to cancel the term of order hp from the truncation errors of two successive interval halving or doubling, the general formula is given by: Order of the resulting approximation may be (p + 1) or (p + 2) depending on the sequence of terms in the truncation error of the original approximation!**Partial Derivatives**• Same expressions can be used for partial derivatives as well. • Example: a function of two variables f(x, y), use indices i and j, grid sizes hx and hy for x and y: • 1st order accurate forward difference at (xi, yj): • 2nd order accurate forward difference at (xi, yj):**Partial Derivatives**• 2nd order accurate central difference at (xi, yj): • 2nd order accurate central difference at (xi, yj): • 2nd order accurate backward difference at (xi, yj):