Transcript MATH 685/CSI 700 Lecture Notes

MATH 685/ CSI 700/ OR 682
Lecture Notes
Lecture 10.
Ordinary differential equations.
Initial value problems.
Differential equations

Differential equations involve derivatives of unknown solution
function

Ordinary differential equation (ODE): all derivatives are with
respect to single independent variable, often representing time

Solution of differential equation is function in infinite-dimensional
space of functions

Numerical solution of differential equations is based on finitedimensional approximation

Differential equation is replaced by algebraic equation whose
solution approximates that of given differential equation
Order of ODE

Order of ODE is determined by highest-order
derivative of solution function appearing in ODE

ODE with higher-order derivatives can be
transformed into equivalent first-order system

We will discuss numerical solution methods only for
first-order ODEs

Most ODE software is designed to solve only firstorder equations
Higher-order ODEs
Example:
Newton’s second law
u1 = solution y of the original
equation of 2nd order
u2 = velocity y’
Can solve this by methods for 1st order equations
ODEs
Initial value problems
Initial value problems
Example
Example (cont.)
Stability of solutions
Solution of ODE is
 Stable if solutions resulting from perturbations of
initial value remain close to original solution

Asymptotically stable if solutions resulting from
perturbations converge back to original solution

Unstable if solutions resulting from perturbations
diverge away from original solution without bound
Example: stable solutions
Example: asymptotically
stable solutions
Example: stability of solutions
Example:
linear systems of ODEs
Stability of solutions
Stability of solutions
Numerical solution of ODEs
Numerical solution to ODEs
Euler’s method
Example
Example (cont.)
Example (cont.)
Example (cont.)
Numerical errors in ODE
solution
Global and local error
Global vs. local error
Global vs. local error
Global vs. local error
Order of accuracy. Stability
Determining stability/accuracy
Example: Euler’s method
Example (cont.)
Example (cont.)
Example (cont.)
Example (cont.)
Stability in ODE, in general
Step size selection
Step size selection
Implicit methods
Implicit methods, cont.
Backward Euler method
Implicit methods
Backward Euler method
Backward Euler method
Unconditionally stable
methods
Trapezoid method
Trapezoid method
Implicit methods
Stiff differential equations
Stiff ODEs
Stiff ODEs
Example
Example, cont.
Example (cont.)
Example (cont.)
Example (cont.)
Numerical methods for ODEs
Taylor series methods
Taylor series methods
Runge-Kutta methods
Runge-Kutta methods
Runge-Kutta methods
Runge-Kutta methods
Extrapolation methods
Multistep methods
Multistep methods
Multistep methods
Example of multistep methods
Example (cont.)
Multistep Adams methods
Properties of multistep
methods
Properties of multistep
methods
Multivalue methods
Multivalue methods
Example
Example (cont.)
Example (cont.)
Example (cont.)
Multivalue methods, cont.
Variable-order/Variable-step
methods