[Math] Iterative trapezoidal method for differential equations

Question

I am studying numerical methods for differential equations. I came accros the trapezoidal method in two forms, an explicit and an iterative one. I would like to know the advantages and disadvantages of each of those methods. Furthermore, how can I study the stability for the iterative method? Which stability definition is the better one and why?. I explain both methods below.

Consider an initial value problem given by $y' = f(t, y)$ and $y(a) = t_0$, where $f$ is defined in $[a, b]\times[\alpha, \beta]$ and satisfies the Lipschitz property with Lipschitz's constant $L$.

Given a natural number $n$ and $h = \frac{b-a}{n}$, we are trying to approximate the unique solution of the problem at $t_i = a + ih \ \forall i = 0, 1 \ldots n$. If $y$ is the solution, we call $y_i = y(t_i)$ and we denote $w_i$ to the approximations obtained by the applied method.

One of the methods studied is the explicit trapezoidal method. It follows the following rule:

$w_{i+1} = w_i + \frac{h}{2} \left[f(t_i,w_i) + f(t_{i}+h, w_i + h f(t_i,w_i))\right]$

We have proved that it has a local error of order 3 and, hence, a global error of order 2.

Then, reading some books I came accros the iterative trapezoidal method, which solves the following implicit equation:

$ w_{i}= w_{i-1} + \frac{h}{2} \left[f(t_{i-1}, w_{i-1}) + f(t_i, w_{i})\right] $

The idea is taking an initial approximation $w_i^{(0)}$ and defining the following sequence:

$w_{i} ^{(j+1)} = w_{i-1} + \frac{h}{2} \left[f(t_{i-1}, w_{i-1}) + f(t_i, w_{i}^{(j)})\right]$

The limit of that sequence is taken as $w_i$. I have proved that the sequence converges if $hL/2 < 1$ and that if we use $w_i$, then the local error is $O(h^3)$. However, why is this method useful and how can I study its stability?

Answer 1

Your questions contains multiple subquestions.

1. The advantage of the explicit method is that it requires only two function evaluations pr. time step. The advantage of the implicit method is that it is A-stable.
2. The disadvantage of the explicit method is that the stability region is small. The disadvantage of the implicit method is that typically requires more than two function evaluations per time step.
3. The stability of these method are studied by examining their behavior when applied to the simple test equation $y' = \lambda y$. When the real part of $\lambda$ is strictly less than zero, then the exact solution decays to zero as $t$ tends to infinity. The implicit method will reproduce this behavior regardless of the time step $h$. The explicit method will only produce this behavior for sufficient small values of $h$.

Using the implicit method requires less knowledge about the problem, but it will frequently take more time to obtain a solution. You can often reduce the time to solve the nonlinear equation by using the explicit method to generate a good initial guess. If you are dealing with one-dimensional problem, then the secant method can further reduce the solve time. If you have a multi-dimensional problem, then you should try to apply Newton's method.

When judging the quality of a scheme we should be concerned with the extent to which it reproduces the key features of the physical reality. There are several kinds of stability and which one is relevant to you depends on what kind of problem you are trying to solve.

Most textbooks which discuss stiff equations will have much more information about the terms "stability", "stability region", "A-stable methods". The simple test equation may appear trivial, but appearances are deceptive and it actually covers an amazing large class of physical problems.