Project numerical solutions to ordinary differential equations in hardware
This presentation is the property of its rightful owner.
Sponsored Links
1 / 14

Project: Numerical Solutions to Ordinary Differential Equations in Hardware PowerPoint PPT Presentation


  • 69 Views
  • Uploaded on
  • Presentation posted in: General

Project: Numerical Solutions to Ordinary Differential Equations in Hardware. Joseph Schneider EE 800 March 30, 2010. Ordinary Differential Equations. Function with one independent variable and derivatives of the dependent variable Eg . y’ = 1 + y/x

Download Presentation

Project: Numerical Solutions to Ordinary Differential Equations in Hardware

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Project numerical solutions to ordinary differential equations in hardware

Project: Numerical Solutions to Ordinary Differential Equations in Hardware

Joseph Schneider

EE 800

March 30, 2010


Ordinary differential equations

Ordinary Differential Equations

  • 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


Ordinary differential equations1

Ordinary Differential Equations

  • Found in many areas of engineering

  • Radioactive decay, heat equation, motion…

  • In electrical engineering, charge, flux, voltage, current, all intertwined by differential equations


Ordinary differential equations2

Ordinary 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


Ordinary differential equations3

Ordinary Differential Equations

  • Most basic case: Euler’s Method

  • Selecting a step size h, iterate from initial value to desired value using the derivative function


Ordinary differential equation

Ordinary Differential Equation

  • Euler’s Method most basic case – Simple, but inaccurate

  • Variety of other methods that have been developed


Ordinary differential equations4

Ordinary Differential Equations


Ordinary differential equations5

Ordinary Differential Equations


Ordinary differential equations6

Ordinary Differential Equations

  • 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


Ordinary differential equations7

Ordinary Differential Equations

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


Project

Project

  • 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


Project1

Project

  • 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


Matlab implementations

Matlab Implementations


Project2

Project

  • For second implementation, desired to use a hardware design to improve on bottleneck of software design

  • Comparisons to software on time vs. error threshold


  • Login