JAMPJournal of Applied Mathematics and Physics2327-4352Scientific Research Publishing10.4236/jamp.2014.213135JAMP-52435ArticlesPhysics&Mathematics A Survey of the Implementation of Numerical Schemes for the Heat Equation Using Forward Euler in Time edroPablo Cárdenas Alzate1*Department of Mathematics, Universidad Tecnológica de Pereira, Pereira, Colombia* E-mail:ppablo@utp.edu.co2212201402131153115816 September 201417 October 2014 22 October 2014© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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.

Global Truncation Forward Euler Heat Equation
1. Introduction

In this paper we have considered the heat equation on with smooth initial conditions and Dirichlet boundary conditions. Using forward Euler in time and fourth order discretization in

space, we compute the Global Truncation Error (GTE), the stability restriction on the time step, also we prove consistency and finally we prove the convergence for this scheme.

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 can say that three classes of solution techniques have emerged for solution of PDE: the finite difference techniques, the finite element methods and the spectral techniques. The last one has the advantage of high accuracy attained by the resulting discretization for a given number of nodes  -  . Let denote the grid-size in the spatial direction and the gridsize in the time direction. By using forward Euler in time, and the fourth order discretization from the previous problem in space, the heat equation reads:

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

2. Global Truncation Error (GTE)

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 be the true solution at stage, and be the solution returned by the scheme at stage. Therefore

We consider de LTE

So that at stage, we have

where

is a vector taking care of the boundary conditions and is a matrix. Since

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. Letting we get

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 and imposing stability to it

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

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

3. Stability Restriction

We start by computing the stability restriction one has to impose on. We apply Von Neumann stability analysis to the scheme: Letting denote the wave number, we get

then

Now, let and

So that when. This guarantees that. Now, in order to make sure that, we must have

4. Consistency and Convergence

We know that a discretization scheme  for a PDE is consistent provided that as, where is the LTE. We compute it by substituting the true solution in the scheme and by using Taylor expansions

Thus, obviously goes to 0 as and go to 0. Therefore, we can say that the scheme is consistent.

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, it goes to 0 as). We can see

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 and the LTE for the spatial discretization is

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

Acknowledgements

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.

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