Theta Method

This method is also known as the weighted method. Both Euler and Trapezoidal rules fit an equation of the form

$$y_{n+1}=y_n+h\big[\theta f(t_{n+1}y_{n+1})+(1-\theta) f(t_{n}, y_{n})\big]\quad n=0, 1\cdots$$

where $\theta\in[0,1]$. When $\theta=0$(explicit), where $\theta\ne 0$ (implicit). Note that $\theta=0$ is Euler, $$\theta=\frac12$$ is Trapezoidal.

Order of Method: (Exercise) Show that the difference between the exact solution and the above approximation at $t=t_n$ is

$$(\theta -\frac12)h^2 y''(t_n) + \left(\frac12\theta -\frac13\right) h^3 y'''(t_n)+\theta(h^4)$$

hence method is order 2 for $\theta=\displaystyle \frac12$ (corresponding to Trapezoidal) and otherwise is of order 1. $\hfill\Box$

If we go through the usual argument (exercise), for $h>0$ and sufficiently small, then

$$e_{n+1}=e_n+\theta h[f(t_n, y(t_n)+e_n)-f(t_n, y(t)n))]$$

$$+(1-\theta)h\bigg[f(t_{n+1}, y(t_{n+1})+e_{n+1})$$

$$-f(t_{n+1}, y(t_{n+1}))\bigg]$$

$$\left\{ \begin{array}{ll} -\frac{1}{12}h^3 y'''(t_n)+{\cal O}... ...y''(t_n)+{\cal O} (h^3) & \theta \ne \frac12 \end{array}\right.$$

Now, take $e_{n+1}$ as an unknown and apply implicit function theorem.

Ok, since $f$ is analytic and for $h>0$ sufficiently small, the matrix

$$I-(1-\theta)h\frac{\partial f(t_{n+1}, y(t_{n+1}))}{\partial y} \mbox { is nonsingular}.$$

Then, using the implicit function theorem one can can show (try it!)

$$e_{n+1}=e_n\left\{ \begin{array}{ll} \displaystyle -\frac{1}{... ...h^2 y''(t_n)+\theta (h^3) & \theta\ne\frac12 \end{array}\right.$$

Why bother with the Theta Method i.e. with $\theta$ taking any value in $[0,1]$, not just $1/2$ and $1$?

1)

The concept of order is based on assumption that error is concentrated on the leading order of Taylor series expansion (on real computers, $h$ is small, but finite). e.g. $\theta=\displaystyle \frac23$ gets rid of ${\cal O}(h^3)$ while retaining ${\cal O}(h^2)$. Hence, for different types of $f(t,y)$ one can tune $\theta$ to control whether ${\cal O}(h^3)$ and higher order terms or ${\cal O}(h^2)$ and higher order terms contribute to the overall error when $h$ is finite. It may be possible to choose a $\theta$ that generates a more optimal or smaller error $\ldots$.

2)

Theta Method is an example of a general approach to designing algorithms in which geometric intuition is replaced by Taylor series expansion. Invariably the implicit function theorem is also used in the design and analysis of scheme.

3)

The $\theta=1$ Case is very practical:

$$y_{n+1}=y_{n}+h f(t_{n+1}, y_{n+1})\quad n=0,1, \cdots$$

This is the ``Backward Euler'' or ``Implicit Euler'' scheme, a simple yet robust method for solving STIFF ODES (Stiffness will be discussed later in detail).

4)

Comparison of the Trapezoidal and Euler methods (see reftreu) will be done later, but the Euler method is more dissipative than the trapezoidal and in some problems a little more or a little less dissipation is appropriate or wanted.