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We establish the conditions for the compute of the Global Truncation Error (GTE), stability restriction on the time step and we prove the consistency using forward Euler in time and a fourth order discretization in space for Heat Equation with smooth initial conditions and Dirichlet boundary conditions.

In this paper we have considered the heat equation

space, we compute the Global Truncation Error (GTE), the stability restriction on the time step

Much attention has been paid to the development, analysis and implementation of accurate methods for the numerical solution of this problem in the literature. Many problems are modeled by smooth initial conditions and Dirichlet boundary conditions. A number of procedures have been suggested (see, for instance [

We’ll assume that the discretizations used near the boundaries have the same order [

There are three equivalent ways of computing the Global Truncation Error for this case.

Way 1. We can always go back to the definition of the GTE. Let

We consider de LTE

So that at stage

where

we get at stage

...

We now wish to estimate this quantity: first using the triangle inequality, we get

Now, taking stability into account, we can see that

Now, assuming that initial error is not too large, we have

Finally, we can conclude that the

Way 2. The GTE can be estimated by computing the LTE

Way 3. We can also compute the one-step-error for the scheme. This quantity is basically equal to

then substitute the true solution and compute the difference of the two sides

We can then estimate the GTE by summing up the one-step error at each stage

We start by computing the stability restriction one has to impose on

then

Now, let

So that

We know that a discretization scheme [

Thus,

Lastly, since we proved that the scheme is consistent and stable, by Lax equivalence theorem, we prove that

the scheme is convergent. (By the above, since the GTE is

that Lax Equivalence Theorem for PDEs holds provided the scheme is linear (which is the case here). It may not hold for non-linear schemes.

Another way to get the one-step error for the scheme is to combine the LTE for the temporal and spatial discretization, as follows.

LTE for forward Euler is

This is equivalent to the previous method for getting the one-step error.

I would like to thank the referee for his valuable suggestions that improved the presentation of this paper and my gratitude to Department of Mathematics of the Universidad Tecnológica de Pereira (Colombia) and the group GEDNOL.