Ordinary Differential Equations

For a first order differentiation $\frac{\partial f}{\partial t}$, we might have many finite differencing methods.

Euler Method

For linear first ODE,

$$ \frac{dy}{dx} = f(x, y), $$

we can discretize the equation using a step size $\delta x \cdot$ so that the differential equation becomes

$$ \frac{y_{n+1} - y_n }{ \delta x } = f(x_n, y_n), $$

which is also written as

$$ y_{n+1} = y_n + \delta x \cdot f(x_n, y_n). \label{euler-method-discretized-form-y-n-plus-1} $$

This is also called forward Euler differencing. It is first order accurate in $\Delta t$.

Generally speaking, a simple iteraction will do the work.

Adams’ Method

For linear first ODE,

$$ \frac{dy}{dx} = f(x, y), $$

This equation can always be written as a integral form

$$ y(x_{n+1}) - y(x_n) = \int_{x_n}^{x_{n+1}} f(x,y) dx, $$

which is basically a very general idea of how to numerically solve such an equation, as long as we can solve the integral efficiently and accurately. In other words, we are dealing with

$$ y(x_{n+1}) = y(x_n) + \int_{x_n}^{x_{n+1}} f(x,y) dx. $$

The problem is how exactly do we calculate the integral or the iteraction. Two methods are proposed as explicit method Adams-Bashforth Method and implicit method Adams-Moulton Method.

What can be done is to Taylor expand the integrand. At first order of $f(x,y)$, we would have

$$ y(x_{n+1}) = y(x_n) + \int_{x_n}^{x_{n+1}} f(x_{n},y(x_n)) dx = y(x_n) +(x_{n+1}- x_n) f(x_{n},y(x_n)) , $$

which is the Euler method. For simplicity step size is defined as

$$ \begin{equation} \delta x = x_{n+1}- x_n. \label{adams-method-step-size-def} \end{equation} $$

Also to simplify the notation, we introduce the notation

$$ y_n = y(x_n). $$

For second order, we have at least two different methods to approximate the integral.

• Adams-Bashforth method is to approximate the integral using

  $$ \int_{x_n}^{x_{n+1}} f(x,y) dx \sim \frac{1}{2} ( 3 f( x_n - f( x_{n-1}, y_{n-1} ) , y_n) ) \delta x $$

  where we used the definition of step size equation ($\ref{adams-method-step-size-def}$).

• Adams-Moulton method uses trapezoidal rule, which approximates the integral as

  $$ \int_{x_n}^{x_{n+1}} f(x,y) dx \sim \frac{1}{2} f( x_{n+1} + f(x_n, y_n) , y_{n+1} ), $$

  which is similar to backward Euler method but of second order.

In fact the AB and AM methods to the first order are

• Adams-Bashforth Method First Order = Forward Euler Method;
• Adams-Moulton Method First Order = Backward Euler Method.

Runge-Kutta

Adaptive Stepsize for R-K

Modified Midpoint Method

$$ \begin{align} z_0 &= y(x) \\ z_1 &= z_0 + h f(x,z_0) \\ z_{m+1} &= z_{m-1} + 2h f(x+mh,z_m) \\ y(x+H) &\approx y_n = \frac{1}{2} \left( z_n + z_{n-1} + h f(x+H,z_n) \right) . \end{align} $$

This method contains only the even powers of $h$ thus we can gain two orders of precision at a time by calculating one more correction.