Understand Adam Moulton & Bashforth Methods for Numerical Analysis

[steven187]

hello all

I have been researching into numerical analysis, differential equations in particular, I underdstand how the Runge kutta methods work geometrically but I don't quit understand what is the idea behind Adam moultons method And Adam Bashforth method, Is there a graphical way of understanding it? I find it very confusing?

steven

 

[steven187]

hello all

so far after a bit more research I have come to understand that the adams bashforth requires information about the solution at more than one point, If I assume that we already know these points then what this method does is find an interpolating polynomial that passes through these points and then uses it to obtain the solution of the next point- this occurs when the resulting polynomial is extrapolated over the next interval h, this is pretty understandable graphically but I still don't understand what adams moultons method does graphically does anybody have any ideas

steven

 

[saltydog]

Hello Steven,

My understanding of these multi-step methods is that they're similar in that they use previous points to calculate a polynomial curve as a trajectory to the next point as opposed to a "weighted" straight line for Runge-Kutta. However, the Milne's method and Adam-Moulton methods correct the predicted value of the next y before going on to the next step.

I think a nice way of gleaming a geometrical notion of these methods is to work with ODEs that I know what the analytical solution is and then go through the numerical methods and actually plot (in Mathematica of course) the quadratic and cubic curves that are these methods, as well as the "predictors" and "correctors", and see how they follow the actual path.

Why not just work through a few points for y'=Ay. Maybe post some plots too.

 

[steven187]

Hello Saltydog

well that aint a bad idea, there is always one way of finding out how something works and that is if we demonstrated it graphically, I have been writting a program over the last few days, writting the predictor corrector code wasnt difficult but trying to plot each step all on the same graph is a bit complicated, I have given it some thought but i think the way i have structured my program does not accommodate for my expected outcome which is to see the convergence for each point after running the predictor corector many times consecutively for the same point until it satisfies the stopping criteria, anyway this is my current output which is based upon increasing the subintervals

steven

 

[steven187]

hello all

to answer my original question, a predictor-corrector algorithm, consists of basically two parts, the predictor extrapolates the solution over some finite range h based on the information at prior points and is inherently unstable and the corrector which allows for this local instability and makes a correction to the solution at the end of the interval which is also based on prior information as well as the extrapolated solution.

steven

 

[saltydog]

Hello Saltydog

well that aint a bad idea, there is always one way of finding out how something works and that is if we demonstrated it graphically, I have been writting a program over the last few days, writting the predictor corrector code wasnt difficult but trying to plot each step all on the same graph is a bit complicated, I have given it some thought but i think the way i have structured my program does not accommodate for my expected outcome which is to see the convergence for each point after running the predictor corector many times consecutively for the same point until it satisfies the stopping criteria, anyway this is my current output which is based upon increasing the subintervals

steven

Hello Steven,

My first though is why didn't you just save the plots in Mathematica as bitmaps and/or convert them to JPEG as I do and then just post the plots directly here without going through Microsoft Office?

Also, the plots seems to have straight line approximations. I was expecting quadratic or cubic approximations for these numerical methods.

 

[steven187]

hello saltydog

well the reason why Iv got straight line approximations is because I have structured my program with the 2nd order runge kutta and the second order adam bashforth and second order adam moulton, upgrading the program shouldn't be too difficult anyway

steven