Mathematical Methods for Engineers II/Lecture 1

Ordinary Differential Equations
Given initial values
 * $$u(t=0)$$

What is the equation associated with evolution?
 * There can be a large number of equations

Linear equation:
 * $$u' = au\;$$


 * $$a\;$$ - scalar
 * $$a=\frac{\partial f}{\partial u}$$

N x N matrix
 * $$u' = Au$$
 * Know all the constants
 * Symmetric matrices associated with real eigenvalues
 * Negative eigenvalues mean the solution decays

The constant $$a$$ can be complex
 * Convection term may not be symmetric

Non-stiff Ordinary Differential Equations

 * $$u' = 4u$$
 * Equations relatively easy to solve
 * Use explicit methods
 * Compute $$u_{n+1}$$ directly from $$u_n$$ and $$f(u_n, t_n)$$
 * Calculate from formula
 * Fast methods of calculation
 * Types of methods
 * Euler
 * Minimum accuracy
 * First order
 * Families of methods
 * Adams-Bashforth
 * Multi-step method
 * Runga-Kutta
 * Half-steps to calculate $$u_{n+1}$$
 * ODE45 in Matlab
 * Fourth order Runga-Kutta
 * Varies $$\delta t$$ based on behavior
 * The code uses internal checks to estimate the error
 * Relative accuracy of $$10^{-3}$$
 * Absolute accuracy of $$10^{-6}$$

Stiff

 * $$u(t) = e^{-t} + e^{-99t}$$
 * $$e^{-t}$$ control $$u$$ but $$e^{-99t}$$ control $$\delta t$$
 * Stiff problems arise in process where there is a dynamic range in rates
 * Ill-conditioned
 * Implicit methods
 * Formula involves the previous value and slope
 * The equation can be non-linear
 * Methods are not as fast
 * Types of methods
 * Backward Euler
 * Families of Methods
 * Adams Moulton
 * Backward differences
 * ODE15s in Matlab
 * Varies $$\delta t$$

Trade-off of speed versus stability

Euler's method

 * Construction of method:

$$\frac{u_{n+1} - u_n}{\Delta t} = f(u_n, t_n) = a u_n$$

$$u_{n+1} = (1 + a \Delta t) u_n\;$$

$$u_n = (1 + a \Delta t)^n u_o\;$$


 * Test of stability

$$|1 + a \Delta t| \le 1$$

Limit when $$a \Delta t = 2\;$$

Too big a time step results in an estimated value with too great a difference

Backward Euler method

 * Construction of method:

$$\frac{u_{n+1} - u_n}{\Delta t} = f(u_n, t_{n+1}) = a u_{n+1}$$

$$(1 - a \Delta t) u_{n+1} = u_n\;$$

$$u_n = \left ( \frac{ 1}{1-a \Delta t} \right )^n u_o$$


 * Test of stability

Absolutely stable

Growth factor is always less than one