Adams-Bashforth-Moulton Technique for systems.

[maistral]

Hello again, I have a new question (again, as usual). But let me give first an overview, haha. If not interested please skip the first two paragraphs. XD

I was trying to code a Bogacki-Shampine method for systems of 4 differential equations in my graphing calculator but apparently I lack the sufficient variables needed for it. I tried lowering the orders, ending up in an adaptive Euler technique for systems, but I guess we don't have to elaborate on the notorious inaccuracy of the predictor :|

So I finally gave up in using adaptive stepsizes in my programs (though I was able to successfully program a Runge-Kutta-Fehlberg for a single DE, yay. Quite useful, but not sufficient to my needs). I was reading certain textbooks about the ABM method and as far as I can understand, I am supposed to use the previous data from a normal Runge-Kutta which is somewhat easy to code, and the main advantage would be the number of evaluations is slashed by half (is what I understood correct?) Speed-wise this would be what I need, further this should allow me to use ridiculously small stepsizes without taking too long with the evaluations (at least, from what I understood), and lastly would be very advantageous with my calculator's limited variables.

Since I already have two separate running third and fourth order Runge-Kutta programs already for systems of 4 DE's the initial values isn't much of a problem. I was also able to run an Adams-Bashforth-Moulton iteration for a single DE in Excel (which would be a fair practice for me in preparation of coding it). The remaining thing is I don't know how to run an ABM iteration with systems of equations. Can someone refer me, or give me something that could teach me how to, or at least a working example that I can emulate? I have no clue on how to start.

Sorry for the gigantic wall of text, but I just had to tell it, lol. Thanks and more power guys :]

 

[gtoguy97]

The Adams-Bashforth-Moulton (ABM) method is a predictor-corrector approach for solving systems of differential equations. It uses the current and previous values of the system variables to predict the next value, and then adjusts this prediction using a correction loop. The ABM method is often used in combination with a Runge-Kutta method, as the Runge-Kutta method can provide the initial values needed for the ABM iteration. To use the ABM method with a system of 4 differential equations, you will need to calculate the derivatives of each equation at the initial point. These derivatives can then be used in the Adams-Bashforth formula, which will provide an approximate solution for the next point. This approximate solution can then be corrected using the Adams-Moulton formula, which uses the derivative of the equation at the new point to refine the solution. Once you have calculated the derivatives at both the initial and the new points, you can repeat this process until you have reached the desired level of accuracy. As the ABM method only requires a few evaluations of the derivatives to calculate the solution, it is often faster than other methods such as the Runge-Kutta. If you are having difficulty understanding the ABM method, there are many online tutorials and examples available. You may also find it helpful to look at the source code of existing implementations of the ABM method, as this can help you to understand how it works.