ESO 208A: Computational Methods in Engineering Lecture 1

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Abhas Singh

Department of Civil Engineering IIT Kanpur

Acknowledgements: Profs. Saumyen Guha and Shivam Tripathi (CE)

ESO 208A: Computational Methods in Engineering

Numerical Integration

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Numerical Integration

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Numerical Integration:

Rectangular Rule

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f_i-1

f_i

f_i+1

f_i+2

a = x₀ x₁ x_i-1 x_i x_i+1 x_i+2 f₀

f_n

b = x_n f₁

f_i+1/2

x_i+1/2

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Rectangular Rule

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f_i+1/2

x_i+1/2 f_i-1

f_i

f_i+1

f_i+2

a = x₀ x₁ x_i-1 x_i x_i+1 x_i+2 f₀

f_n

b = x_n f₁

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Trapezoidal Rule

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f_i-1

f_i

f_i+1

f_i+2

a = x₀ x₁ x_i-1 x_i x_i+1 x_i+2 f₀

f_n

b = x_n f₁

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Trapezoidal Rule

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Simpson’s Rules

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f_i-1

f_i

f_i+1

f_i+2

a = x₀ x₁ x_i-1 x_i x_i+1 x_i+2 f₀

f_n

b = x_n f₁

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Simpson’s Rules

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Simpson’s Rules

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Simpson’s Rules

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Simpson’s Rules

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f_i-1

f_i

f_i+1

f_i+2

a = x₀ x₁ x_i-1 x_i x_i+1 x_i+2 f₀

f_n

b = x_n f₁

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Simpson’s Rules

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Simpson’s Rules

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Simpson’s Rules

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f_i-1

f_i

f_i+1

f_i+2

a = x₀ x₁ x_i-1 x_i x_i+1 x_i+2 f₀

f_n

b = x_n f₁

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Numerical Integration

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Rectangular Rule

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Rectangular Rule

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0 0

0

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Rectangular Rule

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0 0

0

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Rectangular Rule

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Local Truncation Error:

Trapezoidal Rule

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Local Truncation Error:

Trapezoidal Rule

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Numerical Integration: Simpson’s 1/3^rd Rule

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Global Truncation Error:

Trapezoidal Rule

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Global Truncation Error:

Trapezoidal Rule

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Romberg Integration

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Numerical Integration: Example

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