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Project: Numerical Solutions to Ordinary Differential Equations in Hardware

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Project: Numerical Solutions to Ordinary Differential Equations in Hardware

Joseph Schneider

EE 800

March 30, 2010

- Function with one independent variable and derivatives of the dependent variable
- Eg. y’ = 1 + y/x
- Requires some initial condition in order to be solved
- Eg. y(1) = 2

- Found in many areas of engineering
- Radioactive decay, heat equation, motion…
- In electrical engineering, charge, flux, voltage, current, all intertwined by differential equations

- In some cases, original equation can be derived with relative ease; Exact solutions are then available
- In other cases, we will only have the ODE to work with
- Numerical solutions have been developed to deal with these cases

- Most basic case: Euler’s Method
- Selecting a step size h, iterate from initial value to desired value using the derivative function

- Euler’s Method most basic case – Simple, but inaccurate
- Variety of other methods that have been developed

- Error directly linked with step size
- As step size decreases, error decreases; However, takes longer for process to complete
- Implemented in software (eg. Matlab), more accurate methods take several seconds to complete for smaller scale cases; Several minutes for larger cases

- Original project goal: Implement the Runge-Kutta 4th order method in hardware for improved speed

- Euler’s method also implemented for comparisons on area, timing, and accuracy.

- Current implementation: Uses fixed-point number representation, state machine
- Further steps on this implementation
- Variable-point number representation to improve accuracy
- Modify for parallelism to examine impacts on area, timing

- Second implementation: Error-controlled design of Runge-Kutta method
- Error is specified at beginning of process, step size is varied to ensure final result meets error specifications

- For second implementation, desired to use a hardware design to improve on bottleneck of software design
- Comparisons to software on time vs. error threshold