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3.3 - Fourth Order Accuracy - Cont’d Rearranging the above equations and dividing by the finite-difference ∆x gives ∂f ∂x = −f i+2 + 8f i+1 − 8f i−1 + f i−2 12∆x ∂f ∂x = −f i+2 + 8f i+1 − 8f i−1 + f i−2 12∆x + 4 5! ∂ 5 f ∂x 5 ∆x4 + H. + O(∆x 4 ). (3.14 F&P ) 24
3.3 - Fourth Order Accuracy - Cont’d Rearranging the above equations and dividing by the finite-difference ∆x gives ∂f ∂x = −f i+2 + 8f i+1 − 8f i−1 + f i−2 12∆x ∂f ∂x = −f i+2 + 8f i+1 − 8f i−1 + f i−2 12∆x + 4 5! ∂ 5 f ∂x 5 ∆x4 + H. + O(∆x 4 ). (3.14 F&P ) Fourth Order Accuracy: T.E. ∼ O(∆x 4 ). Requires four grid points to achieve the fourth order accuracy. 24
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: 3.3 - Fourth Order Accuracy In gene
Page 41 and 42: 3.3 - Higher Order Accuracy Second-
Page 43 and 44: 3.3 - Higher Order Accuracy Second-
Page 45 and 46: 3.3 - High Order Accuracy - Cont’
Page 47 and 48: 3.4 Second Order Derivative: ∂2 f
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 81 and 82: 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 95 and 96:
3.8 - Matrix Equations - Cont’d A
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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