Techniques overcome equipment limitations that restrict other computer techniques in solving trivial cases. The use of curve fitting by quadratic interpolation greatly reduces required digital storage space

Hammond, J. L., Jr.

Odowd, W. M.

English

NASA Technical Reports Server

November 1971	 Brief 7l-10424 
•	 .
NASA TECH BRIEF 
Marshall Space Flight Center 
NASA Tech Briefs announce new technology derived from the U.S. space program. They are issued to encourage 
commercial application. Tech Briefs are available on a subscription basis from the National Technical Information 
Service, Springfield; Virginia 22151. Requests for individual copies or questions relating to the. Tech Brief program may 
be directed to the Technology Utilization Office, NASA, Code KT, Washington, D.P. 20546. 
Hybrid Computer Techniques for Solving Partial Differential Equations 
is
Two hybrid (analog/digital) computer techniques 
can be used for generating approximate solutions to 
partial differential equations. A preliminary study 
indicates that these techniques may help overcome 
the equipment limitations that frequently restrict 
other computer techniques to solving trivial cases. 
In both techniques, the original equation is con-
verted into a set of coupled, ordinary differential 
equations by quantifying the space variable and 
using the method of finite difference approximation. 
Conceptually, the set of coupled equations can be 
easily solved. However, since the number of equa-
tions tends to be large, the required quantity of 
analog simulators or the required digital storage 
space can easily exceed the capacity of even the 
most modern equipment. 
The analog problem was surmounted by employ-
ing serial solution techniques in a hybrid computer. 
The equations were solved sequentially in the analog 
section. However, since each equation contained 
forcing functions that depended on the solutions to 
neighboring equations, it was necessary to store the 
succession of solutions in the digital section. This 
itself could require excessive storage space. 
Several techniques for reducing the required 
storage area were investigated. All of them stored 
the functions' values at only a few points and ap-
proximated the intermediate values of the functions 
as needed. Two principal approximation techniques 
were used: zero-order hold, and polynomial curve 
fitting. 
The test case used was the one-dimensional heat-
flow equation, with space quantized into seven dis-
crete intervals between zero and one. The following 
results were noted: (I) When a zero-order hold was 
used, the solution converged to the wrong answer. 
The error could be significantly reduced by shifting 
the data forward in time by precise amounts, but 
the error then depended strongly on the amount of 
shift. (2) Much better results were obtained using 
curve fitting by a process of linear interpolation. 
(3) Even better results were obtained using curve 
fitting by quadratic interpolation. 
Notes: 
1. For related work involving the application of 
finite difference approximations to solving ordi-
nary differential equations, refer to NASA Tech 
Brief B71-10423. 
2. Requests for further information may be directed 
to:
Technology Utilization Officer 
Code A&TS-TU 
Marshall Space Flight Center 
Huntsville, Alabama 35812 
Reference: TSP7I-10424 
Patent status: 
No patent action is contemplated by NASA. 
Source: W.M. O'Dowd and J.L. Hammond, Jr. of

Georgia Institute of Technology

under contract to

Marshall Space Flight Center

(MFS-21386) 
Category 09 
This document was prepared under the sponsorship of the National 
Aeronautics and Space Administration. Neither the United States 
Government nor any person acting on behalf of the United States
Government assumes any liability resulting from the use of the 
information contained in this document, or warrants that such use 
will be free from privately owned rights.
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Hybrid computer techniques for solving partial differential equations

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