Mathematical Methods for Engineers II/Lecture 1

Ordinary Differential Equations

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Given initial values

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What is the equation associated with evolution?

  • There can be a large number of equations

Linear equation:

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  - scalar
 

N x N matrix

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  • Know all the constants
  • Symmetric matrices associated with real eigenvalues
  • Negative eigenvalues mean the solution decays

The constant   can be complex

  • Convection term may not be symmetric

Non-stiff Ordinary Differential Equations

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  • Equations relatively easy to solve
  • Use explicit methods
    • Compute   directly from   and  
      • 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  
          • ODE45 in Matlab
            • Fourth order Runga-Kutta
            • Varies   based on behavior
            • The code uses internal checks to estimate the error
            • Relative accuracy of  
            • Absolute accuracy of  

Stiff

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  •  
    •   control   but   control  
  • 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  

Trade-off of speed versus stability

Euler's method

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  • Construction of method:

 

 

 

  • Test of stability

 

Limit when  

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

Backward Euler method

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  • Construction of method:

 

 

 

  • Test of stability

Absolutely stable

Growth factor is always less than one