The Intrinsic Method

Returning to the possibility of solving the integral equation using the trapezoidal or trapezium rule, let us consider the case of a linear differential equation, such as our examples. For the decay equation we have

$$y_{n+1} = y_n - {\mathchoice{{\textstyle{\frac12}}}{{\texts... ...style{1/2}}}}\delta t\left[\alpha y_{n+1} + \alpha y_n\right]$$ (1.41)

which can be rearranged into an explicit equation for $y_{n+1}$ as a function of $y_n$

$$y_{n+1} = \left[ 1 - {\mathchoice{{\textstyle{\frac12}}}{{\... ...{{\scriptscriptstyle{1/2}}}}\delta t\cdot \alpha \right] y_n.$$ (1.42)

This intrinsic method is 2nd order accurate as that is the accuracy of the trapezoidal rule for integration. What about the stability? Applying the same methodology as before we find that the crucial quantity, $g$, is the expression in square brackets, $[]$, in (1.42) which is always $<1$ for the decay equation and has modulus unity in the oscillatory case (after substituting $\alpha\mapsto\pm\i\omega$). Hence it is stable in both cases. Why is it not used instead of the other methods? Unfortunately only a small group of equations, such as our examples, can be rearranged in this way. For non-linear equations it may be impossible, and even for linear equations when $\bi{y}$ is a vector, there may be a formal solution which is not useful in practice. It is always possible to solve the resulting non-linear equation iteratively, using (e.g. ) Newton-Raphson iteration, but this is usually not worthwhile in practice. In fact the intrinsic method is also stable for the growth equation when it is analysed as discussed earlier, so that the method is in fact stable for all 3 classes of equations.

Decay	Growth	Oscillation
stable	stable	stable