Abstract
We use an unconditionally stable parallel difference scheme to solve telegraph equation. This method is based on domain decomposition concept and using asymmetric Saul'yev schemes for internal nodes of each sub-domain and alternating group implicit method for sub-domain's interfacial nodes. This new method has several advantages such as: good parallelism, unconditional stability and better accuracy than original Saul'yev schemes. The details of implementation and proving stability are briefly discussed. Numerical experiments on stability and accuracy are also presented.
1. Introduction
The large-scale scientific and engineering computations need the parallel computer with massively parallel processors of higher speed and large memory and also need effective parallel numerical methods and parallel algorithms. Some numerical methods directly have parallel character and have great efficiency on using in the parallel computer, but much numerical methods usually need to be reconstructed to be more appropriate for the parallel computing [1].
Consider the Telegraph equation of the form where and . Numerical solution of this equation based on finite difference has been successfully proposed in [2–4].
There are two types of schemes for numerical solution of time-dependent partial differential equations, implicit and explicit schemes. The former has no restriction on its time stepping. But in each time step one has to solve a global system of equations. The implementation on parallel computer is not straightforward due to its global nature. The latter is easy to program and implement on parallel computer. However, it suffers the severely restricted time step size from stability requirement [5]. Recently, a so-called alternating explicit-implicit scheme has been studied by many authors, which is based on the concept of domain decomposition and a combination of implicit schemes and explicit schemes [6, 7]. In these approaches, a physical domain is divided into several sub-domains. At each time step, the problem is solved in some sub-domain implicitly and others explicitly. For the next time step the solution scheme is changed (implicit is changed to explicit and vice versa). The main advantage of alternating explicit-implicit schemes is their parallelism. Although some families of these schemes are unconditionally stable, some families of these schemes still give rise to restriction involving the time step, however, this restriction is much less severe than for a fully explicit scheme. In spite of intrinsic parallel nature and good stability of alternating explicit-implicit schemes, these schemes still have implicit phase, that is, solution of problem implicitly over implicit subdomains. Implicit phase leads to solution of linear or nonlinear algebraic system of equations at each time level that is not desirable due to high computational cost, especially for large-scale problems. Therefore an unconditionally stable fully explicit scheme with good parallelism is desirable. In [7], Saul'yev has introduced two unconditionally stable fully explicit asymmetric schemes for solution of diffusion equation. Unfortunately the Saul'yev schemes are not intrinsically parallel.
In this paper we developed a new family of group explicit scheme for solution of Telegraph equations. The presented scheme is based on domain decomposition concept and using asymmetric Saul'yev schemes for internal nodes of each sub-domain and Evan's group explicit method [8] for sub-domain’s boundary nodes. The rest of this paper is organized as follows. In Section 2, we construct the scheme and describe details of implementation. In Section 3 the stability analysis is presented and in Section 4, some numerical examples are given to show the stability and accuracy of presented scheme.
2. Construction of New Group Explicit Scheme
In this section the numerical solution of the following problem is considered: where , , and are initial conditions, and and are prescribed boundary conditions. The domain will be divided into a mesh with spatial step size in direction and the time step size , respectively. Grid points are given by , and in which and are positive integers. We use to denote the finite difference approximations of . In [7], Saul’yev has introduced two asymmetric approximations for the second derivative as follows: By combining with the usual form for , one gets the Saul’yev formula, So for the Saul'yev scheme , we have for , where . Although the above approximation does not appear explicit, because and are on the left-hand side, a suitable use of the equation makes it explicit. Hence we write (2.4) in the following form: for . It is important to note that the Saul’yev scheme is explicit if the calculation begins at the left boundary, and moves to the right, so that only the single value is unknown. It can be easily seen that this explicit formula is unconditionally stable. Replacing Saul’yev in (2.3), then for . The computational molecule of formulae (2.4) and (2.6) is shown in Figure 1. The Saul’yev formula is in the following form: for . The Saul’yev scheme has a similar explicit nature if the calculations proceed to the left from the right-hand boundary. Application of Saul’yev formulae and alternatively give better accuracy and error distribution. Obviously the Saul’yev scheme or is not intrinsically parallel.
For parallel implementation of Saul’yev scheme, we use concept of domain decomposition. For convenience, we only decompose the domain into two sub-domains, which are and . If the formula is used for the sub-domain 1, , and the calculation begins at the left boundary, and moves to the right and in similar manner, the formula is used for the sub-domain 2, , and the calculations proceed to the left from the right-hand boundary, then the calculation of each sub-domain is independent from another. For the next time step the formula is used for the sub-domain 1 and formula is used for the sub-domain 2 and the calculation is started from points and for sub-domains 1 and 2, respectively. But we apply the initial condition only in the , therefore, we consider two cases and . In case 1, and , for the solution is approximated by (2.4) By initial condition , we have , then while for the solution is approximated by (2.6) Similarly, by initial condition , we have Also, in case 1, and , for sub-domain 1, we use formula (2.6) and for sub-domain 2, we use formula (2.4), then at node , the solution is approximated by (2.6) and at node , the solution is approximated by (2.4) Assum that , and , then the matrix form of (2.12) and (2.13) will be then, we have or, we have which can be diagrammatically represented by the computational molecule given in Figure 2.
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But, in case 2, (), for the solution at the nodes is approximated by (2.4), and the solution at the nodes is approximated by (2.6), where . Similary, at the node , the solution is approximated by (2.6), while at node the solution is approximated by (2.4), If (2.17) and (2.18) are rewritten in matrix form: whose matrix of coefficient can easily be inverted, the equation can be written in explicit form as This simplifies to which similarity can be diagrammatically represented by the computational molecule given in Figure 2. The flow chart of this method is displayed in Figure 3. We use to denote the scheme (2.4), to denote the scheme (2.6) and and to denote the scheme (2.16).
Implementation of this algorithm for more than two sub-domains is straightforward as follows (see Figure 4). For more than two sub-domains, we have :
3. Stability Analysis and Convergence
3.1. Stability Analysis
In this section we prove unconditional stability of presented scheme by the matrix method. Pay attention that in the best case, if the number of the sub-domains is reduced to one sub-domain, the presented scheme converted to asymmetric Saul'yev schemes that are unconditionally stable (see [9]) and in the worst case, if the number of sub-domains increased such that each sub-domain only has two nodes the presented scheme is converted to alternating group explicit method that is unconditionally stable (see [6]).
To prove the stability of presented scheme, suppose that the spatial computational domain is divided to number of sub-domains ( is even for simplicity) and each sub-domain contains number of nodes. Also the first sub-domain contains the left Neumann boundary node and the last sub-domain contains the right Neumann boundary node as additional nodes. Similarly, the presented scheme can be expressed in two cases. In the first case, considering , then we have : where , , and and in the second case, , where In both cases, and are
assume that then we have By eliminating from (3.1) we obtain where similarly, by eliminating from (3.3) we obtain where By Lax-Richtmyer condition of stability, the below inequality must be satisfy then we have Note that the matrixs and are singular, also must be , then for and , the eignvalues of and are It easy to see that [3, 4], the and are nonsingular and for all Then the following theorem satisfies for Telegraph equation.
Theorem 3.1. The parallel implementation of asymmetric Saul'yev schemes for the solution of Telegraph equation (2.1) is unconditionally stable for all .
3.2. Convergence
The real important of the concept of consistency lies in a theorem by Lax [10], which state that If a linear finite-difference equation is consistent with a properly posed linear initial value problem then stability guarantees convergence of to as the mesh lengths tend to zero, where and are numerical and exact solutions of problem. Consistency can be defined in either of two equivalent but slightly different ways.
The more general definition is as follows [11, pages 40–45]. Let represent the partial differential equation in the independent variables and , with exact solution , and represent the approximation finite difference equation with exact solution . Let be a continuous function of and with a sufficient number of continues derivatives to enable to be evaluated at the point .
Then the truncation error at the point is defined by If as then the difference equation is said to be consistent or compatible with the partial differential equation. With this definition gives an indication of the error resulting from the replacement of by .
Most authors put , because . It then follows that and the truncation error coincides with the local truncation error. The difference equation is then consistent if the limiting value of the local truncation error is zero as We now illustrate the convergence of covering difference equation (3.10). Substituting into (3.10), we get where and is the truncation error of (2.3), and for , we have . Then Using Theorem 3.1, we have , so we have where and is a constant. Therefore From this result we deduce unconditional convergence as and .
4. Numerical Experiments
Example 4.1. In this section the accuracy and stability of presented parallel scheme is numerically investigated by solution of a test problem. Consider the following problem: The exact solution is given by The average of absolute errors of this problem for , and , is given in Table 1. The figures of this data, that is, , are shown in Figure 5.
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5. Conclusion
In this paper, we have discussed a new three-level implicit parallel difference scheme of for the different solution of the Telegraph equation (2.1). The scheme is stable for , and applicable to singular equations. It is hoped that the new idea presented in this paper will be useful for the development of an unconditionally stable parallel difference formula of for the numerical solution of the Telegraph equation (2.1).