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Okay, let's start. Last time we started to discuss stability, which is a subject which is at least

of same importance as accuracy for at least for a time-dependent discretization scheme,

but in general as we have seen already that these things are very strongly connected.

And we are now on the way from the semi-discrete formulation to a fully discrete formulation have

already formulated the one-step theta method with this most important example explicit Euler,

theta equals to zero, implicit Euler, theta equals to one, and Cranck-Nicholson, theta equals to one

half. And we looked at the stability of these methods. Now we will do this a little bit more

precise today. Let's compare, let's repeat what we did. Maybe before I start, there will be the

old German transparencies, there will be a few new English transparencies, in particular this

chapter. As you might have seen, the English version is considerably extended compared to

the German version. And there is one discrepancy in the notation which I mentioned already last

time, but I hope it's not too bad. So the time step, the discretization parameter in time in

the German edition is denoted by K, and in the English edition it's denoted by tau. So you will

have a mixed, it will change from German to English transparencies and vice versa. I think

all the other notations should be the same. So one way to look at this, we started from another

direction, but one way to look at this is to say, actually we have to look at a system, a linear

system of equations. It is basically again written down here. It comes either from the finite element

or from the finite difference method. And what we consider are one-step methods as indicated,

that is we compute the approximate solution at the new time level only knowing the approximate

solution on the time level below. So we approximate, we computed n plus 1 by knowing an approximation

at the level n. And you see here the resolved difference quotient here in this formulation.

So behind, and we have seen already that behind the evaluation of this function phi, depending on

time discretization, time level, and old approximation, in general, in the implicit method,

there is a solution of a set of equations. You see this again. Here we have the exact formulas

for the theta methods, and you see a part of the case theta equals to zero. We have here to evaluate

an inverse matrix, that is we have to resolve a set of equations. So we have introduced the notion

of non-expensiveness, basically saying if we have a deviation in the data, for example, by measurement

perturbation, but of course also by rounding errors, that we want to have that this perturbation is not

enlarged, is not enhanced. We have seen in the equations, in looking at the stability of the

parabolic equations, often it's even better. We have something like parabolic smoothing, that is we

have that the initial data, and by this of course also differences of initial data, and by this of

course also perturbances in initial data, are damped exponentially, and the exponential term is

determined by the lowest positive eigenvalue, which we have, which of course assumes that all

eigenvalues are positive, which assumes that we are in the situation where we have at least some

Dirichlet boundary condition, or something similar, but here we require less, and we will see what we

have seen already what kind of assumptions this brings with this, so that means the error level

always stays below the, is not higher than the error level at the initial instance of time, and

the approach, oops, okay this we can skip, so we have seen at least in the situation where, a little

bit of an old notation here, the matrix is denoted by Q, I think in the English version it has another

name, but I forgot what the name of this matrix of the eigenvectors was, so now we assume that we are

in the diagonalizable case, we are always in the diagonalizable case of the matrix as symmetric,

of course, so if you have purely pure diffusive forces there's no problem with that, and one can

view now the situation we are looking at in this more abstract fashion as a situation which is not

so far away from the pure diffusive case, meaning that there's still, we still have a basis of

eigenvectors which forms this matrix Q, the eigenvalues forms the diagonal matrix lambda,

that is we can transform to the coordinates given by the eigenvectors, and that leads to the

decoupling of the system that is in this new coordinates we now have, m is our, is the number

of components which we have, we have m independent linear equations, and of course how they behave is

only now dependent on the eigenvalue which is in this, on this diagonal position, that is for positive