Slides - School of Engineering - University of Warwick

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Taylor Series Expansion Taylor Series Expansion Any continuous differentiable function f(x) can, in the vicinity of x i , be expressed as a Taylor series: f(x) = f i +∆x ( ) ∂f + 1 ( ∂ 2 f ∂x i 2! ∆x2 ∂x )i 2 + 1 ( ∂ 3 f 3! ∆x3 ∂x )i 3 · · · + 1 ( ∂ n f n! ∆xn ∂x )i n + H. (3.3 F&P ) where H means all remaining HIGHER ORDER TERMS. + 13
Taylor Series Expansion Taylor Series Expansion Any continuous differentiable function f(x) can, in the vicinity of x i , be expressed as a Taylor series: f(x) = f i +∆x ( ) ∂f + 1 ( ∂ 2 f ∂x i 2! ∆x2 ∂x )i 2 + 1 ( ∂ 3 f 3! ∆x3 ∂x )i 3 · · · + 1 ( ∂ n f n! ∆xn ∂x )i n + H. (3.3 F&P ) where H means all remaining HIGHER ORDER TERMS. + Expressions for the variable values in terms of the variable and its derivatives at x i . ( ) ( ∂f ∂ 2 f f i , , ∂x i ∂x 2 )i , ( ∂ 3 f ∂x 3 )i , . . . , ( ∂ n f ∂x n )i . 13
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: 3.2 - Finite Difference Approximati
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 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:
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:
Page 97 and 98:
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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