Numerical methods for ODEs

1
Numerical methods for ODEs

• Ordinary differential equations (ODEs)
• Dynamical system
• Widely applications in science engineering
• Mixture problem
• Population models
• Point-mass motion
• Mechanical vibration
• Electrical networks
• Pendulum motion, .

2
ODEs

• Different types
• First order ODE
• First order ODEs system
• High order ODE
• High order ODEs system
• Solutions
• Existence uniqueness
• Analytical solution
• Numerical solution

3
Different numerical methods

• For first order ODE
• Single-step method
• Euler, Trapezoidal method
• Runge-Kutta methods RK4 most popular
• Stability and convergence
• Time-splitting (split-step) method
• Integration factor method
• Multi-step methods

4
Different numerical methods

• For first order ODEs system
• Direct extension of all methods for first order
  ODE
• For high order ODE
• First order ODEs system
• Direct discretization
• For high order ODE system
• First order ODEs system
• direct discretization

5
For first order ODE

• Consider initial-value problem (IVP)
• Choose a time step and partition a,b into N
  pieces
• Denote

6
Basic numerical methods

• Ideas Derivative is the limit of difference
• Forward Euler method -- explicit method
• Backward Euler method implicit method

7
Basic numerical methods

• Trapezoidal method sum of forward backward
  Euler methods
• Taylor expansion
• Forward Euler method
• Backward Euler method

8
Numerical example

• The problem
• Time step
• Forward Euler method
• Backward Euler method
• Trapezoidal method

9
Numerical results

• Example

10
Order of accuracy

• Local truncation error
• Def The error at the given step if it is assumed
  that the previous results are all exact!!
• Ways to find
• Replace in the difference equation
• Define the local truncation error
• Do Taylor expansion use the ODE
• find the leading order term
• Order of accuracy p

11
Order of accuracy

• Forward Euler method
• Local truncation error
• Do Taylor expansion (see details in class)
• Order of accuracy first order explicit
• Backward Euler method (exercise)
• Order of accuracy first order implicit
• Trapezoidal method (exercise)
• Order or accuracy second order implicit

12
Runge-Kutta methods

• Ideas
• Integrate the ODE over the interval
• Choose r points in the interval
• Evaluate slopes at these points via (forward)
  Euler method

13
Runge-Kutta methods

• Construct the averaged slope
• Determine the parameters via Taylor expansion
  such that the difference method has a given
  accuracy!!

14
Runge-Kutta methods

• Forward Euler method -- RK1
• Backward Euler method
• Trapezoidal method

15
Runge-Kutta methods

• Second order Runge-Kutta method -- RK2
• Choose
• Construct slopes
• Averaged slope
• RK2 method
• Determine the three parameters via Taylor
  expansion (see details in class or exercise)

16
Runge-Kutta methods

• RK2 (modified Euler method) Order of accuracy
  2 explicit
• RK2 (modified midpoint method) Order of
  accuracy 2 explicit

17
Runge-Kutta methods

• Fourth order Runge-Kutta method -- RK4
• Choose
• Construct slopes
• Averaged slope
• RK4 method
• Determine the seven parameters via Taylor
  expansion (see details in class or exercise)

18
Runge-Kutta methods

• 4th Order Runge-Kutta method -- most popular!!
  --- exercise!!
• Local Truncation error
• Order of accuracy 4
• Explicit method

19
Numerical example

• The problem
• Exact Solution
• Different numerical methods
• Forward Euler method (FEM)
• Modified Euler method (MEM)
• 4th order Runge-Kutta method (RK4)

20
(No Transcript)
21
(No Transcript)
22
Numerical results

• Example

23
Numerical results

• Population of fruitflies

24
Numerical results

• Population of fruitflies

25
ODE demo

• Some very interesting ODE demo sites
• Interactive Differential Equations
• http//www.aw-bc.com/ide/
• More