A Special Case on the Stability and Accuracy for the 1 D Heat Equation Using 3-Level and θ-Schemes

We establish the conditions for the compute of the stability restriction and local accuracy on the time step and we prove the consistency and local truncation error by using θ -scheme and 3-level scheme for Heat Equation with smooth initial conditions and for some parameter [ ] 0,1 θ ∈ .


Introduction
In this paper we have considered the heat equation . Using θ -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 θ on the time step t ∆ .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 [ ] 0,1 θ ∈ .We compute the order of local accuracy in space and time as a function of θ and its the stability restriction.Until 1 T = , we compute the solution with some fixed L ∞ 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 δ and β we find the stability criterion of the scheme and its accuracy.

θ -Scheme
Let ( ) be the θ -scheme applied to the one-dimensional heat equation 0 Now for the order of local accuracy in space and time as a function of θ we write the local truncation error.
In time we have where 1 n U + represents the exact solution of the heat equation.Now we perform Taylor expansion of We can write the LHS of ( 1 2 6 here ( ) u represents the derivative with respect to time, of ( ) j u n .On the RHS, we have a centered differ- ence approximating second derivative of ( ) As we are solving the heat equation, the previous expression is Now, at time 1 n + we have Here RHS of (1) becomes After the elimination of some terms we have , u and moving all terms to the right side, we get Scheme (10) is first order in time, second order in space.If for example 1 2

θ =
, it becomes second order, this is due to cancellation of the τ ∆ .

Stability Restriction as a Function of θ
Here we will apply Von Neumann stability.Let ( ) ( ) Then Equation ( 1) can be written as Therefore by using ) ) ) By using the identity e e cos 2 We can say this scheme is stable only for 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.

Three-Level Scheme
We start by computing the stability restriction one has to impose on t ∆ .We apply Von Neumannstability analysis to the scheme.Let ( ) ( ) where or as ( ) ( ) The local truncation error for this scheme γ is as follow.U + represents the exact solution of the heat equation.Therefore we have ( ) ( ) Now expanding x M operator on the left side, we can isolate the forward difference in time at n j u , then ( ) ( ) ( ) Now applying Von Neumann stability again, the aim is to use ( , e , Using the cosine identity that 2cos e e We have a quadratic equation in G , where 2 1 G < , therefore