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The last times we discussed a little bit the qualitative behaviour of initial boundary value problems

as we are encountered with in conjunction now with the time-dependent version of the stationary elliptic problems we dealt with up till now.

And I indicated already that it is advisable to treat the time and that we also have seen in terms of the explicit solutions and the qualitative behaviour of the solutions.

It's advisable to treat the time different than the spatial coordinates and not to say, okay, we have a problem in D plus one and we apply finite elements in D plus one

or just finite difference methods which could also be possible.

So, doing so, it is also of advantage first to think of an intermediate step, which is not fully practical in the sense that what comes out is not fully computable,

but an intermediate step in terms of semi-discretization.

And there are two possible versions of semi-discretization, the horizontal one, the Waters method, and the vertical one.

In the horizontal one, which we are not dealing with here, one first discretizes in time but gets a sequence of connected elliptic boundary value problems,

which we know how to treat. Here we do it the other way around. We discretize in space first in the way we developed it.

So, at the moment we can do it either with a finite difference method or with a finite element method.

And you will see there is a strong parallelism between these two methods.

So, because in both cases, and in principle in more or less all the cases, one treats the time derivative by finite differences.

We have a very simple domain for the time in terms of an interval compared to the, in general, very complicated spatial domain.

So, if you later on will know more ways to discretize, for example, finite volume method, mixed finite elements, DG, or whatsoever,

the development is very much the same.

So, we start with the most simple situation we know, namely the finite difference method, and here to be specific to our model problem,

that is with the heat equation, so all coefficients in the equations are scaled to one, with Dirichlet boundary conditions,

and we take a rectangle such that it can apply finite differences without any problem.

We know what comes out, namely the five-point stencil in the stationary case, and exactly the same procedure we know too in the time dependent case.

So, we have, let's think of a smooth classical solution, we fix an instant of time, we keep the time derivative as it is,

and we perform the spatial derivative in the way we started with.

So, we get here the five-point stencil in the natural double index notation here, and correspondingly the right-hand side evaluated,

now also depending on time, and correspondingly the boundary condition with the right, with the heterogeneity, which now also might depend on time.

So, what we got here in this procedure, and this will be maybe a little bit clearer if we write this again in matrix notation,

in the stationary problem we got a linear set of equations for a tuple, for a grid function.

Now, we have a system of ordinary differential equations, system that is, the solution is a tuple or the solution is a grid function,

depends how we would like to see that, whether we fix an ordering on the grid points or not.

Okay, so without the time derivative you always written this problem as aH times vector, tuple vector u equals a right-hand side,

and now we do the same thing, now the solution becomes time-dependent, the matrix not, because we have assumed that our coefficients,

which are in this case, okay, constant equal to one, do not depend on time, but you see in some sense one could do basically the same thing

if we would have general coefficients and even if we would have time-dependent coefficients.

Then we would get here a time-dependent matrix, now we have just a constant matrix in time,

so here again the abbreviations which I use, so this is now as up till now in the finite difference context,

so it's just the evaluation of the right-hand side and of the heterogeneity of the boundary conditions at the grid points,

but now that is now the difference also t-dependent, so we get now not a vector but a function in t with images being a vector of being grid functions.

And correspondingly we need our initial data, correspondingly we have our initial data and again we approximate the initial data

by evaluating the function, of course this requires enough smoothness for the initial data

and we could think also of something different here, evaluate the initial data at the grid points.

Good. Is this end?

Okay, so now let's write this analogously in our matrix notation and the matrix A H is exactly the matrix A H we had before,

is exactly the matrix if we do for example the row wise ordering with the force on the diagonal,

the minus ones at the upper and lower diagonal and then as many points as we have in the vertical directions away another minus one.

So this is basically this block diagonal matrix here or a permutation of it and the difference now is our problem now does not read A H U H equals Q H

but it's now this system of ordinary differential equations.

And as we started with the linear problem and as we did discretizations which preserved linearity which seems quite reasonable,

actually more modern methods do not do that, there are also nonlinear approximations for linear problems.

If one wants to have more conditions to be fulfilled as one can do with linear approaches but any case here we have a linear approach

so we get a matrix here, we get a linear mapping, so this is a linear system of ordinary differential equations.