Take the 2-minute tour ×
Mathematica Stack Exchange is a question and answer site for users of Mathematica. It's 100% free, no registration required.

What is NDSolve's mode of operation? I use it to solve partial differential equations and never gave it too much thought. Recently, I came across this question. Accordingly, I used Trace[...] and realized that my 4th order, non linear partial differential equation was solved using an LSODA method.

Further digging revealed that LSODA is a method for ORDINARY DIFF EQS which chooses between Adams and BDF methods to solve the problem.

So... how in the first place did NDSolve convert my PDE to an ODE?

Any references would be useful.

share|improve this question
    
Welcome to Mathematica.SE –  Mr.Wizard Mar 18 '12 at 20:20
    
I think "mode of operation" is not very standard terminology. I didn't understand what you were talking about before reading the question. I edited the title, please review it. –  Szabolcs Mar 18 '12 at 20:20
    
@Szabolcs the new title works! –  drN Mar 19 '12 at 18:42
    
Just as a quick comment, MATHEMATICA uses the LSODA method by default if the methodof solution option in NDSolve[..] is set to Automatic, and I don't just mean for stiff equations. Please do leave a comment with your thoughts. –  drN Mar 22 '12 at 20:08

1 Answer 1

up vote 10 down vote accepted

The methods NDSolve uses are documented in detail here:

This section says that PDEs are solved using the "method of lines", and explains which kinds of problems this method can deal with. There's also a detailed example of how the method works.

The numerical method of lines is a technique for solving partial differential equations by discretizing in all but one dimension, and then integrating the semi-discrete problem as a system of ODEs or DAEs.

...

It is necessary that the PDE problem be well posed as an initial value (Cauchy) problem in at least one dimension, since the ODE and DAE integrators used are initial value problem solvers. This rules out purely elliptic equations such as Laplace's equation, but leaves a large class of evolution equations that can be solved quite efficiently.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.