ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 32 Ordinary Differential Equations.

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Presentation transcript:

ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 32 Ordinary Differential Equations

Pendulum W=mg Ordinary Differential Equation

ODEs Non Linear Linearization Assume  is small

ODEs Second Order Systems of ODEs

ODE

ODE - OBJECTIVES Undetermined

ODE- Objectives Initial Conditions

ODE-Objectives Given Calculate

Runge-Kutta Methods New Value = Old Value + Slope X Step Size

Runge Kutta Methods Definition of  yields different Runge-Kutta Methods

Euler’s Method Let

Sources of Error Truncation: Caused by discretization Local Truncation Propagated Truncation Roundoff: Limited number of significant digits

Sources of Error Propagated Local

Euler’s Method

Heun’s Method PredictorCorrector 2-Steps

Heun’s Method Predict Predictor-Corrector Solution in 2 steps Let

Heun’s Method Correct Corrector Estimate Let

Error in Heun’s Method

The Mid-Point Method Remember: Definition of  yields different Runge-Kutta Methods

Mid-Point Method Predictor Corrector 2-Steps

Mid-Point Method Predictor Predict Let

Mid-Point Method Corrector Correct Estimate Let

Runge Kutta – 2 nd Order

Runge Kutta – 3rd Order

Runge Kutta – 4th Order

Boundary Value Problems

Fig 23.1 FORWARD FINITE DIFFERENCE

Fig 23.2 BACKWARD FINITE DIFFERENCE

Fig 23.3 CENTERED FINITE DIFFERENCE

xoxo Boundary Value Problems x1x1 x2x2 x3x3 x n-1 xnxn...

Boundary Value Problems xoxo x1x1 x2x2 x3x3 x n-1 xnxn...

Boundary Value Problems xoxo x1x1 x2x2 x3x3 x n-1 xnxn...

Boundary Value Problems xoxo x1x1 x2x2 x3x3 x n-1 xnxn...

Boundary Value Problems xoxo x1x1 x2x2 x3x3 x n-1 xnxn...

Boundary Value Problems Collect Equations: BOUNDARY CONDITIONS

Example x1x1 x2x2 x3x3 x4x4