ESO 208A: Computational Methods in Engineering Lecture 1

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Ordinary Differential Equation

 The methods for Initial Value Problems (IVPs):

 Multi-step Methods

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Explicit: Euler Forward, Adams-Bashforth

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Implicit: Euler Backward, Trapezoidal and Adams-Moulton

 Backward Difference Formulae (BDF)

 Runge-Kutta Methods

 Applications, Startup, Combination Methods (Predictor- Corrector)

 Consistency, Stability, Convergence

 Application to System of ODEs

 Boundary Value Problems (BVPs)

 Shooting Method

 Direct Methods

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ESO 208A: Computational Methods in Engineering

Ordinary Differential Equation: System of IVPs and Higher Order IVPs

Abhas Singh

Department of Civil Engineering IIT Kanpur

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

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System of IVPs

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Higher Order IVP

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Higher Order IVP: Example

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Numerical Methods for System of

• IVPs

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• IVPs

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• IVPs

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• IVPs

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• IVPs

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Example application: Higher Order

• IVP

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Example application: Higher Order

• IVP

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• IVP

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IVP •

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• IVP

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• IVP

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• IVP

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Example application: Higher Order

• IVP

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• IVP

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Example application: Higher Order IVP

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Stability of the System of IVPs

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 In higher order IVP or in a system of IVP, the solutions are characterized by the eigenvalues.

 One or two equation in the system typically have high eigenvalues close to λ

_max

and the rest of the equations may have eigenvalues of much lower magnitudes!

 The time step is restricted by λ

_max

.

 Stiff system: large value of (λ

_max

/λ

_min

); typically > 100

 As the time progresses, larger time step can be used for the problem but is limited by the stability criteria!

 This is the utility of the BDFs:

 They are stable for all real and negative λ

_R

up to 6

^th

order

 Region of stability for imaginary λ

_I

increases as one increases

the time step (increasing λ

_R

h)!

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Example: Stiff System

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Stability: BDF Methods Example

 For all the BDFs: Stability Region is outside the enclosed region!

 For real λ, all the BDFs are unconditionally stable!

 One can use any h without having to worry about the stability!

 Useful for stiff equations!

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Method h t u v

0 1.00E+00 0.0000

BDF1 0.01 0.01 6.67E-01 -0.3329

0.01 0.02 4.44E-01 -0.5544

0.01 0.03 2.96E-01 -0.7015

0.01 0.04 1.98E-01 -0.7991

0.01 0.05 1.32E-01 -0.8636

BDF2 0.05 0.1 -5.92E-02 -1.0460

0.05 0.15 -4.60E-02 -1.0217

0.05 0.2 -1.56E-02 -0.9776

BDF3 0.1 0.3 5.49E-02 -0.8725

0.1 0.4 2.46E-02 -0.8588

0.1 0.5 -1.99E-03 -0.8319

0.1 0.6 -3.61E-03 -0.7706

BDF4 0.2 0.8 -2.97E-02 -0.6428

0.2 1 -5.90E-03 -0.4285

0.2 1.2 4.52E-03 -0.1917

0.2 1.4 -2.44E-04 0.0648

0.2 1.6 -1.24E-03 0.3596

BDF5 0.4 2 1.85E-03 1.0548

0.4 2.4 3.26E-03 1.8780

0.4 2.8 -3.90E-04 2.8208

0.4 3.2 -4.31E-04 3.8870

0.4 3.6 3.63E-04 5.0691

0.4 4 -4.63E-05 6.3605

BDF6 0.8 4.8 -4.08E-03 9.2541

0.8 5.6 -8.42E-04 12.5474

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Method h t u v

0 1.00E+00 0.0000

BDF1 0.02 0.02 5.00E-01 -0.4986

0.02 0.04 2.50E-01 -0.7463

BDF2 0.04 0.08 0.00E+00 -0.9903

0.04 0.12 -3.57E-02 -1.0181

0.04 0.16 -2.04E-02 -0.9932

BDF3 0.08 0.24 4.66E-02 -0.9024

0.08 0.32 2.92E-02 -0.8900

0.08 0.4 1.88E-03 -0.8815

0.08 0.48 -3.89E-03 -0.8451

BDF4 0.16 0.64 -3.77E-02 -0.7767

0.16 0.8 -9.44E-03 -0.6222

0.16 0.96 6.24E-03 -0.4570

0.16 1.12 3.92E-04 -0.2904

0.16 1.28 -2.01E-03 -0.0977

BDF5 0.32 1.6 -1.98E-04 0.3607

0.32 1.92 4.53E-03 0.9086

0.32 2.24 8.73E-05 1.5310

0.32 2.56 -1.05E-03 2.2378

0.32 2.88 5.08E-04 3.0257

0.32 3.2 1.29E-04 3.8877

BDF6 0.64 3.84 -4.81E-03 5.8257

0.64 4.48 -1.45E-03 8.0484

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ESO 208A: Computational Methods in Engineering

Partial Differential Equations:

Introduction, Parabolic Equation

Department of Civil Engineering

IIT Kanpur

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Introduction

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Parabolic PDE: semi- discretization

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