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:

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

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

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

• Test of stability

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

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:

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

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

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

• Test of stability

Absolutely stable

Growth factor is always less than one