However, due to the famous. Dahlquist barrier in , an explicit linear multistep method cannot be A-stable. Therefore, we try to find the new explicit linear multistep methods with the longest interval of stability region in this paper. And some numerical experiments are given to compare the proposed methods with existing methods such as Adams-Bashforth method, Adams-Moulton methods, and BDF methods.
Practical calculations have shown that these proposed methods are adaptive. Applying the linear multistep fc-step methods to the initial value problem 1. The corresponding numerical method is said to be absolutely stable.
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Definition 2. The intersection of q with the real axis in the complex h-plane is called the interval of absolute stability. The linear multistep methods 2. Since an explicit linear multistep method cannot be A-stable, we focus our attention on its absolute stability. The most convenient method for finding regions of absolute stability is the boundary locus technique BLT.
If this map is one to one, then r is a Jordan curve, which can constitute the boundary of q, see .
Linear multistep method
Theorem 3. Furthermore, if the real interval. Although this criterion is not easy to prove, we always use condition 3. Fortunately, the true stability interval is just the same as that estimated by 3. Therefore, the longest interval of absolute stability can be evaluated by condition. For example, consider the two-order explicit linear two-step methods, whose first and second generating polynomials can be written as. Corollary 3. The estimated interval of absolute stability of two-order explicit linear two-step methods 3.
The maximal length of stability interval of two-order explicit linear two-step methods. In fact, we can also study the interval of absolute stability of linear multistep methods from definition directly. Therefore, the interval of absolute stability of methods 3. It means that the maximal length of stability interval can not exceed 2, which is in agreement with Corollary 3.
In this section, we mainly consider the three-order explicit linear four-step methods with the first and second generating polynomials. Due to 2. Here, the error constant is.
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To guarantee the convergence, according to [2, Theorem 2. So, these two conditions are always assumed to be held in the following.
Appeal to the well-known Routh-Hurwitz criterion see Section 1. Lemma 4. For convergent methods 4. For convergent three-order four-step methods 4. From condition 4. Corollary 4. The interval of absolute stability of convergent LMMs 4. As we known, explicit linear multistep methods cannot be A-stable. In fact, LMMs 4. At the end of this section, we use BLT to study the stability region of 4. In Figure 1, we plot the region S, the interval length a and the maximal height along the positive imaginary axis Im h of absolute stability.
It demonstrates that the map h d is no longer one to one on this interval, which means that the interior of this loop does not belong to the stability region. Therefore, the curve constructs one single connected region of corresponding numerical method only when The corresponding maximums a and Ih are also given out. Especially, several stability regions of LMMs 4. It is shown that the maximal length of stability interval for methods 4. These results are in agreement with the conclusions of Corollaries 4.
As we all know, this equation is stiff because its solutions contain a component which decays much more rapidly than the other. III Linear four-step methods 4. Then, we attempt to solve 5. This implies that the proposed methods 4. Then, system 5. To compare the proposed methods 4. It can be noted that the accuracy of AM3 is indeed better than that of LMM1 in the beginning interval. However, LMM1 can repair the initial error and obtain the same accuracy as AM3 eventually, which can be seen from the second component the bottom figure.
This phenomenon reveals that the proposed methods LMM1 have good stability. Example 5. To illustrate the efficiency of the proposed methods 4. It is well known that the BDF methods are central to the construction of efficient algorithms for handling stiff systems.
In fact, they play the same role in stiff problems as the Adams methods do in nonstiff ones. In Table 4, the numerical solutions for 5. Meanwhile, the opposite happened for the explicit linear multistep methods. Therefore, people are willing to use the explicit Runge-Kutta methods in the physical problems.
Linear Multistep methods.
However, the maximal length of stability interval for methods 4. Furthermore, Corollary 4.
Their formulations include both multistage nature of Runge-Kutta methods as well as the multi-value nature of linear multistep methods. We also write y [n-1] for the vector of approximations imported into step n and y [n] for the quantities computed in this step and exported for use by the following step.
The detailed computation is based on the formula. Also is the Kronecker product of two matrices. With a slight notation, equations 1 and 2 are often written in the form. This formulation of general linear methods was first introduced by . The structure of the leading coefficient matrix A, determines the implementation cost of these methods, similar to that of the matrix A in the Runge-Kutta methods.
The V matrix determines the stability of these methods. The B matrix gives the weights. The U matrix is simply e. In order for general linear methods to unify the study of the traditional methods, Runge-Kutta and linear multistep methods it must be possible to represent these traditional methods using the partitioned matrix M.
As shown in  Runge-Kutta methods are very simple to rewrite as general linear methods. The matrix A of the general linear method is the same as the matrix A of the Runge-Kutta method.
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The matrix B is b T where b is the vector of weights of the Runge-Kutta method. Assuming the input vector is an approximation to , the U matrix is simply e, a vector of ones in Runge-Kutta case. The V vector consists only of the number 1.
It is thus written as:. Similarly we can also easily represent the linear multistep methods in form of general linear methods. Definition 1. If a general linear method A,U,B,V takes the special form. For methods with this property the step size is never restricted by stability on linear constant coefficient problems, regardless of the stiffness. Material and Method.
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Consider the general first order system of initial value problem of ordinary differential equations in the vector form. A particularly useful class of discrete method for 5 is the class of linear multistep method of the form. We consider an approximant of the form in 6 formulated by  which was a generalization of .
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From the interpolation and collocation conditions 8a - 8b and the expression for in 7 , the following conditions are imposed on x , and x :. Writing 10a - 10b in a matrix form, we have. The matrix C is the solution vector output and D is the non-singular matrix termed the Data input , which is assumed to be non-singular for the existence of the inverse matrix C.
An efficient algorithm for obtaining the elements of the inverse matrix C is found in . We now derive the continuous formulation of the general linear methods following the derivation techniques discussed in section 3. Construction of the Continuous General Linear Methods. Here we propose a more computationally and attractive procedure, which leads to a class of A -stable continuous general linear methods for systems of stiff initial value problems. If we evaluate the proposed continuous scheme in 14 , we first obtain the block finite difference method  associated with the continuous scheme and converted the block method into uniformly accurate order A-stable continuous general linear method and we display in the formalism of  as follows:.
We plotted the region of absolute stability of the general linear method 15 using the method used in  and display in figure 1. Figure 1.