HMD Mathematics / HMD Software Blog: Recent posts

The solution of a second-order ODE y'' + y = 0 was done in my last post with the predictor/corrector trapezoid rule. Here I show how to use the HMD stepper for solving the same problem as a Taylor series.

For the HMD stepper the Taylor series is just a one stage method (one step) method. But there needs to be more than just one callback function, y' = f(x,y). We need to differentiate f(x,y) to as many orders as the order of the Taylor series to be used. I will keep terms only up to second order in Taylor's formula,... read more

Consider the differential equation

y'' + y = 0

where y' means the first derivative of y, y'' is the second derivative of y.

We decompose this equation into two first-order equations with the following method:

y1 = y
y2 = y'

Now there will be two coupled differential equations, let's see,

y1' = y2
y2' = -y1

This is the simplest ODE system of equations because we already know the answer, so it is an easy matter to check the numerical solution. The solution is a sinusoid of type that depends on the initial conditions.... read more

HMD began as a finite element tool. With the release of version 4.1.0 HMD now has solvers for ODE (ordinary differential equations). There are ODE front-ends for the Euler, Heun, Runga-Kutta, and Adams methods. These are just front-ends because no modifications to the method can be made with them. They have limited use, except for generating data for engineering purposes. But as R. W. Hamming wrote,

The purpose of computing is insight, not numbers

The HMD program provides elementary functions for piecing together custom ODE models. At the moment, the elementary functions are aimed at custom predictor-corrector models, and the generic Taylor series method. This allows you to specify the number of stages (separate equations) and the number of past solution values, and their weights, to be used in the formula.

See the man page hmd_ode.