Why does finite differences work

Another way to solve the ODE boundary value problems is the finite difference method , where we can use finite difference formulas at evenly spaced grid points to approximate the differential equations.

This way, we can transform a differential equation into a system of algebraic equations to solve. In the finite difference method, the derivatives in the differential equation are approximated using the finite difference formulas.

Commonly, we usually use the central difference formulas in the finite difference methods due to the fact that they yield better accuracy. The differential equation is enforced only at the grid points, and the first and second derivatives are:. If the differential equation is nonlinear, the algebraic equations will also be nonlinear.

EXAMPLE: Solve the rocket problem in the previous section using the finite difference method, plot the altitude of the rocket after launching. The ODE is. Therefore, we have 11 equations in the system, we can solve it using the method we learned in chapter See the calculation below. Take the neighboring pairwise differences. So, for instance, Then take the neighboring pairwise differences of those values, and so on.

It turns out that the n th level will consist entirely of the same nonzero value if, and only if, the polynomial has degree n. Notice that after three levels, we yield the constant value So, I had to work it out myself.

Notice that in my example, the final value is. We will prove that it is true for a polynomial of degree n. Notice that because the leading coefficient acn is nonzero, q x has degree n By our inductive hypothesis, after n -1 pairwise differences, the polynomial q x will yield a constant value Thus, for p, the process terminates after n steps with the constant value This proves the theorem.

After playing around with this, I googled it, and—no surprise—the mathematics of finite differences has a long history. Also, it is not difficult to see the resemblance of these calculations to the calculation of the derivative using the definition of the derivative. This gives a convenient way to extrapolate a sequence. If your terms actually do come from a polynomial then this method can continue the sequence despite never actually calculating the coefficients of the polynomial.

Sadly, the course Numerical Analysis has fallen by the wayside. I regret most the chapter on propagation of errors falling by the wayside. Finite differences can help to detect the propagation of one kind of error.

A ghost of that chapter shows up in machine architecture when discussing single- versus double-precision floating point numbers. Otherwise, nothing, sadly.

Skip to content It is interesting watching my kids go through the school math curriculum.