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3.3 - High Order Accuracy - Cont’d N-th Order Accuracy: T.E. ∼ O(∆x N ). Requires N grid points to achieve the N-th order accuracy. The main difficulty with the higher order formulae occurs near boundaries of the domain. For example, the derivative of f at x 1 would require the value of f at x −1 which is not available. In practice, to alleviate this problem, we utilise lower order or non-central formulae near boundaries. 26
3.4 Second Order Derivative: ∂2 f ∂x 2 Similar formulae can be derived for second order or higher order derivatives. ( ) ∂f f i+1 = f i + ∆x ∂x ( ) ∂f f i−1 = f i − ∆x ∂x i i + 1 ( ∂ 2 f 2! ∆x2 ∂x )i 2 + 1 ( ∂ 2 f 2! ∆x2 ∂x )i 2 + 1 3! ∆x3 ( ∂ 3 f ∂x 3 )i − 1 3! ∆x3 ( ∂ 3 f ∂x 3 )i + H, + H. 27
Page 1 and 2: ES440/ES911: CFD Chapter 3. Finite
Page 3 and 4: 3.1 Introduction A generic transpor
Page 5 and 6: 3.1 Introduction A generic transpor
Page 7 and 8: 3.2 - 1D Discretisation Divide the
Page 9 and 10: 3.2 Basic Concept - Cont’d Unifor
Page 11 and 12: 3.2 Basic Concept - Cont’d Each n
Page 13 and 14: 3.2 Basic Concept - Cont’d Each n
Page 15 and 16: 3.2 Basic Concept - Cont’d A gene
Page 17 and 18: 3.2 - Finite Difference Approximati
Page 19 and 20: 3.2 - Finite Difference Approximati
Page 21 and 22: Taylor Series Expansion Taylor Seri
Page 23 and 24: 3.3 - Forward Difference: ∂f ∂x
Page 25 and 26: 3.3 - Forward Difference: ∂f ∂x
Page 27 and 28: 3.3 - Forward Difference - Cont’d
Page 29 and 30: 3.3 - Backward Difference: ∂f ∂
Page 31 and 32: 3.3 - Backward Difference - Cont’
Page 33 and 34: 3.3 - Central Difference: ∂f ∂x
Page 35 and 36: 3.3 - Truncation Errors Numerically
Page 37 and 38: 3.3 - Fourth Order Accuracy In gene
Page 39 and 40: 3.3 - Fourth Order Accuracy - Cont
Page 41 and 42: 3.3 - Higher Order Accuracy Second-
Page 43 and 44: 3.3 - Higher Order Accuracy Second-
Page 45: 3.3 - High Order Accuracy - Cont’
Page 49 and 50: 3.4 - Higher Order Accuracy: O(∆x
Page 51 and 52: 3.4 Third Order Derivative: ∂3 f
Page 53 and 54: 3.4 Third Order Derivative: ∂3 f
Page 55 and 56: First Order Accuracy: O(∆x) FORWA
Page 57 and 58: Second Order Accuracy: O(∆x 2 ) C
Page 59 and 60: Second Order Accuracy: O(∆x 2 ) C
Page 61 and 62: Fourth Order Accuracy: O(∆x 4 ) C
Page 63 and 64: Fourth Order Accuracy: O(∆x 4 ) C
Page 65 and 66: Finite Difference Method for Ellipt
Page 67 and 68: Finite Difference Method for Ellipt
Page 69 and 70: One-Dimensional FDM Elliptic PDE: L
Page 71 and 72: One-Dimensional FDM Elliptic PDE: L
Page 73 and 74: One-Dimensional FDM ∂ 2 u ∂x 2
Page 75 and 76: 3.8 Algebraic Equation System u 1
Page 77 and 78: 3.8 - Linear Algebraic Equation Fin
Page 79 and 80: 3.8 - Matrix Equations - Cont’d A
Page 83 and 84: 1-D Navier-Stokes Equation Elliptic
Page 85 and 86: 1-D Navier-Stokes Equation Elliptic
Page 87 and 88: One-Dimensional FDM 1 Re ∂ 2 u
Page 89 and 90: 3.8 Algebraic Equation System u 1
Page 91 and 92: 3.8 - Matrix Equations for 1D FDM T
Page 93 and 94: TDMA: Thomas Algorithm An upper tri
Page 97 and 98:
3.8 - Matrix Equations - Cont’d A
Page 99 and 100:
Solution to 1D Matrix Equations Dir
Page 101 and 102:
Example of First-Order ODE Heat Tra
Page 103 and 104:
3.11 Example The generic transport
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Slides - School of Engineering - University of Warwick

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