Numerical Solution of ODE Systems

1
Numerical Solution of ODE Systems

• ODE systems
• Extensions of single equation methods
• Stiff ODEs
• Implicit solution methods

2
ODE Systems

• Background
• Many chemical engineering models consist of
  coupled nonlinear ODEs
• Numerical solution usually is the only option
• Initial value problem
• Methods also applicable to high-order ODEs

3
Extensions of Single Equation Methods

• Methods developed for numerical solution of
  single ODEs can be extended directly to ODE
  systems
• Forward Euler
• Runge-Kutta

4
CSTR Example

• Van de Vusse reaction
• CSTR model
• Forward Euler

5
Stiff ODE Systems

• Stiff linear ODE systems
• Large difference in magnitudes of the smallest
  largest eigenvalues
• Necessitates small step size to maintain
  numerical stability
• Stiff nonlinear ODE systems
• Separation of time scales
• e ltlt 1
• Common in models of chemical engineering systems
• Slow fast physical phenomenon
• Need specialized solution methods

6
Implicit Solution Methods

• Motivation
• Implicit methods offer better stability
  properties than explicit methods for stiff ODE
  systems
• Often do not know if a particular ODE model is
  stiff
• Many large ODE systems are stiff
• May chose to use stiff ODE solver for all
  problems
• Backward Euler
• Simplest implicit method
• Requires repeated solution of nonlinear algebraic
  system
• Better stability properties but more
  computationally demanding than forward Euler

7
Stiff System Example

• CSTR model A ? B ? C
• Linear ODE system
• Eigenvalue analysis q/V 1, k1 1, k2 200

8
Explicit Solution

• Forward Euler
• First iterative equation
• Second iterative equation

9
Implicit Solution

• Backward Euler
• First iterative equation
• Second iterative equation