A Special Case on the Stability and Accuracy for the 1D Heat Equation Using 3-Level and θ-Schemes ()
1. Introduction
In this paper we have considered the heat equation
with
. Using
-scheme and 3-level scheme in space we compute the order of local accuracy in space and time and stability restriction as a function of
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 [1] - [3] ). 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 (see [4] and [5] ). The last one has the advantage of high accuracy attained by the resulting discretization for a given number of nodes [6] - [8] .
We consider Scheme (1) for the 1D heat equation for some parameter
. We compute the order of local accuracy in space and time as a function of
and its the stability restriction. Until
, we compute the solution with some fixed
error with the smallest amount of CPU time, and finally we can see this findings producing the relevant convergence and efficiency plot. For the 3-level scheme we consider (11) for the 1D heat equation and we compute the local truncation error. For different values of
and
we find the stability criterion of the scheme and its accuracy.
2.
-Scheme
Let
(1)
be the
-scheme applied to the one-dimensional heat equation
(2)
Now for the order of local accuracy in space and time as a function of
we write the local truncation error. In time we have
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where
represents the exact solution of the heat equation. Now we perform Taylor expansion of
at
.
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We can write the LHS of (1) as
(3)
(4)
here
represents the derivative with respect to time, of
. On the RHS, we have a centered difference approximating second derivative of ![]()
(5)
As we are solving the heat equation, the previous expression is
(6)
Now, at time
we have
(7)
(8)
therefore, applying Taylor expansion with respect to
we can write
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So (8) becomes
(9)
Here RHS of (1) becomes
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After the elimination of some terms we have
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Now simplifying we obtain
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Cancelling
and moving all terms to the right side, we get
(10)
Scheme (10) is first order in time, second order in space. If for example
, it becomes second order, this is due to cancellation of the
.
Stability Restriction as a Function of ![]()
Here we will apply Von Neumann stability. Let
and
. Then Equation (1) can be written as
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Now dividing by
we have
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Therefore by using
and
we can rewrite the expression as
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By using the identity
we have
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We can say this scheme is stable only for
. Now, let
. The inequality is
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thus
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Now multiplying by the denominator we have
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The expression in the absolute value becomes
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Therefore by the Von Neumann stability condition, the scheme is stable if
.
In this case we can say the following about the best combination for
and
. In order to have both local accuracy and stability, the optimal value of
is
and therefore this scheme represents the Crank-Ni- cholson scheme. Here
and
appear in the form
.
In Figure 1 the convergence plot equation (varying the radio r) is
![]()
with matrix A described in the heat equation. We can say the scheme is unconditionally stable. We can see in Figure 1 that we have a linear convergence with respect to r.
3. Three-Level Scheme
We start by computing the stability restriction one has to impose on
. We apply Von Neumannstability analysis to the scheme.
Let
(11)
where
(12)
and
(13)
By using (12) and (13) we can rewrite (11) as
(14)
or as
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The local truncation error for this scheme
is as follow.
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where
represents the exact solution of the heat equation. Therefore we have
(15)
Now expanding
operator on the left side, we can isolate the forward difference in time at
, then
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however,
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Expanded this expression becomes
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Finally we have
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4. Stability Criterion for the Three-Level Scheme and Its Accuracy When
and ![]()
By using Equation (14) we have
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Now applying Von Neumann stability again, the aim is to use
and
, therefore
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Multiplying both sides by
and write
we obtained
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Using the cosine identity that
we have
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We have a quadratic equation in
, where
, therefore
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After some cancellations, we can write
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Here, if all
we need
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Acknowledgements
We would like to thank the referee for his valuable suggestions that improved the presentation of this paper.