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Gstability oneleg hybrid methods for solving DAEs
Advances in Difference Equations volume 2019, Article number: 103 (2019)
Abstract
This paper introduces the solution of differential algebraic equations using two hybrid classes and their twin oneleg with improved stability properties. Physical systems of interest in control theory are sometimes described by systems of differential algebraic equations (DAEs) and ordinary differential equations (ODEs) which are zero index DAEs. The study of the first hybrid class includes the order of convergence, A(α)stability, stability regions, and Gstability for its oneleg twin in two cases: for step (\(k=1\)) and steps (\(k=2\)). For the second class, Gstability of its oneleg twin is studied in two cases: for steps (\(k=2\)) and steps (\(k=3\)). Test problems are introduced with different step size at different end points.
Introduction
Consider the initial value problems of the form
where \(a\in R^{m}\) is a consistent initial value for (1) and the function f: \(R^{{m}} \times R^{{m}} \times [t _{0} ; T] \rightarrow R^{{m}}\) is assumed to be sufficiently smooth. If \((\partial f/\partial x)\) is nonsingular, then it is possible to formally solve (1) for x in order to obtain an ordinary differential equation. However, if \((\partial f/\partial x)\) is singular, it is no longer possible and the solution x has to satisfy certain algebraic constraints; therefore, equations (1) are referred to as differential algebraic equations.
Systems of differential algebraic equations arise from many applications such as physics, engineering, and circuit analysis. Some systems can be reduced to an ODE system, which are zero index DAEs, and can be solved by numerical ODE methods after reduction. Other systems, in which reductions to an explicit differential systems are in the form \(x' = f(x; t)\), are either impossible or impractical, that is because the problem is more naturally posed in the form
and a reduction might reduce the sparseness of Jacobian matrices. These systems are then solved directly [16, 17].
A fundamentally important concept in the algorithms of the numerical solutions of DAEs is the index of a DAE. In a sense, this tells us how far away the DAE is from being an ODE. The index of a DAE is the minimum number of times all or part of the DAE system must be differentiated with respect to time in order to convert the DAE into an explicit ODE. The higher the index is, the further it is from an ODE and the more difficult it is in general to solve the DAE [12].
The first general method applied to the numerical solution of DAEs is backward differentiation formula (BDF). Ebadi andGokhale presented in [9,10,11] class \(2+1\) hybrid BDFlike methods, hybrid BDF methods (HBDF), and new hybrid methods for the numerical solution of IVPs. These methods have wide stability regions and good performance in solving CPU time compared to the extended BDF (EBDF) and modified extended BDF (MEBDF) methods [3].
In Sect. 2 the first hybrid class is derived, its orders of convergence are investigated, and its stability analysis is studied. In Sect. 3 some basic notions of oneleg schemes and Gstability are mentioned. The oneleg twin of the first class is derived and its Gstability is discussed in Sect. 4. The oneleg twin of the second class [14] is derived and its Gstability is discussed in Sect. 5. Numerical tests are investigated in Sect. 6. Finally, a conclusion is introduced.
The first hybrid class
The first hybrid class takes the form
where \(f_{n+s} = f(t _{n+s}; y _{n+s})\); \(t_{n+s} = t _{n} + s h\); \(1 < s\) and \(\beta_{s}\), \(\beta_{1}\), \(\alpha_{ n j}\), \(j = 1, 2, \dots , k\), are parameters to be determined as functions of s and \(\beta _{0}\). The methods for step k and order \(p=k+1\) (from \(k = 1\) up to 6) will be derived and \(y_{n+s}\) has order \(k  1\). To evaluate the value of \(y_{n+s}\) at an offstep point, i.e. \(t _{n+s}\), we will consider the nodes \(t _{n}\) (double node), \(t_{ n1 }, \dots , t _{ n k}\) (simple nodes).
Applying Newton’s interpolation formula for this data gives the following scheme:
Differentiate (5) with respect to t:
Using (5) and (6) to evaluate \(y_{n+s}\) and \(f_{n+s}\) gives
where f (or \(f(t, y)\)) is considered as a derivative of the solution \(y(t)\), \(\nabla y _{n} = y _{n}  y_{ n1 }\).
Method (4) is of order p if and only if
The coefficients of the methods for steps \(k= 1\) up to 6 are tabulated in Tables 1, 2, 3, and 4.
Since formula (3) is of order k and formula (4) is of order \(k+1\), then it is easy to see that method (3)–(4) has order \(k+1\).
Stability analysis
Consider the scalar test problem \(y' = \lambda y\), \(y(0) = y_{0}\). From equations (3) and (4) the corresponding characteristic equation is as follows:
where \(\overline{h} = \lambda y\), i.e.
where
The absolute stability regions for this class for \(k=1\) up to 7 are given in Fig. 1 for the optimal s and \(\beta_{0}\) using the boundary locus method.
The angle α of A(α)stability for different methods, BDF, EBDF, AEBDF, MEBDF, Enright methods, HEBDF, and The class (3–4) for various orders are tabulated in Table 5.
We recall some basic notions of oneleg schemes and Gstability.
Oneleg schemes
Suppose that a linear kstep method
is given. Oneleg methods can be formulated in a compact form by introducing the polynomials
with real coefficients \(\alpha_{i}, \beta_{i} \in R\) and no common divisor. There is also the assumption throughout the normalization that
The associated oneleg methods are defined by
In the oneleg methods, the derivative f is evaluated at one point only, which makes it easier to analyze. The oneleg method (15) may have stronger nonlinear stability properties such as Gstability [12, 15]. On the other hand, it is known that to obtain a oneleg method of high order, the parameters \(\alpha_{i}\), \(\beta_{i}\) have to satisfy more constraints than those for linear multistep methods, see [7, 8, 14]. The conditions \(\rho (1)= 0\), \(\rho '(1) = \sigma (1)= 1\) imply the consistency of the scheme \(( \rho , \sigma )\).
Gstability analysis
The Gstability analysis, announced at the 1975 Dundee conference and published in [6], uses the test problem \(\mathrm{d}y/\mathrm{d}x = f(x, y)\), where \(\langle y  z, f(x, y)  f(x, z)\rangle \leq 0\). In the same publication, oneleg methods were introduced and related to corresponding linear multistep methods. Stable behavior for this problem was defined as Gstability. A more detailed account of this work will be given.
If the differential equation satisfies the onesided Lipschitz condition
with \(\nu = 0\), then the exact solutions are contractive. Consider the multistep method as a mapping \(R^{n,k}\rightarrow R^{n,k}\). Let \(Y _{m} =(y_{m+k1}, \dots , y _{m})^{{T}}\) and consider inner product norms on \(R^{n,k}\)
where \(\langle \cdot , \cdot \rangle \) is the inner product on \(R^{{n}}\) used in (16) and kdimensional matrix \(G = (g _{ij})\) \(i, j=1,\dots, k\) is assumed to be real, symmetric, and positive definite. The inner product \(\langle \cdot , \cdot \rangle \), on which \(\langle \cdot , \cdot \rangle_{ {G}}\) is built, is supposed to have a corresponding norm defined by \(\u\_{2} = \langle u, u\rangle \). Similarly we will write \(\\cdot , \cdot \_{{G}}\) as the norm corresponding to \(\langle \cdot , \cdot \rangle_{{G}}\).
Definition 1
([6])
The oneleg method (15) is called Gstable if there exists a real, symmetric, and positive definite matrix G such that, for two numerical solutions \(\{Y_{m}\}\) and \(\{\hat{Y} _{m}\}\), we have
for all step sizes \(h > 0\) and for all differential equations satisfying (16) with \(\nu = 0\).
Theorem 1
([2])
Gstability implies Astability.
Theorem 2
([12])
Consider a method \(( \rho , \sigma )\). If there exists a real, symmetric, and positive definite matrix G, and real numbers \(a _{0},\dots ,a _{{k}}\) such that
then the corresponding oneleg method is Gsable.
Theorem 3
([5])
If ρ and σ have no common divisor, then the method \(( \rho , \sigma )\) is Astable if and only if the corresponding oneleg method is Gstable.
Oneleg method for the first hybrid class
Here, the oneleg twin of the first class is studied when \(k=1\) and \(k=2\).
In the case of \(k=1\), method (4) takes the form
where
Method (20) has order 2, its truncation error takes the form
The oneleg twin of (20) takes the form
and has order 2 and its truncation error takes the form
To discuss Gstability of (22),
Substitute \(f_{n+s}\) in equation (20), it becomes
The corresponding characteristic equations are
Applying Theorem 2, the variables \(a _{i}\), \(i=0,1\) and \(g_{ij}\), \(i,j=1\) satisfy the relations
Choose \(\beta_{0}<1/2\), \(g_{11}=1/2 > 0\). So, method (22) is Gstable.
In the case of \(k=2\), method (4) takes the form
After normalization
Method (23) has order 3, its truncation error takes the form
The oneleg twin of (23) takes the form
and has order 2 if \(\beta_{0}=(2+3s)/(6(1+s))\) and its truncation error takes the form
To discuss Gstability of (25),
Substitute \(f_{n+s}\) in equation (23), it becomes
The corresponding characteristic equations are
Applying Theorem 2, the variables \(a _{i}\), \(i=0,1,2\) and \(g_{ij}\), \(i,j=1,2\) satisfy the relations
Choosing \(\beta^{*}=0.4\) and \(s=0.9\) makes that \(a_{0}=0.079714\), \(a_{1}=0.0561286\), \(a_{2}=0.0235855\), \(g_{11}> 0\), and $Det\left(\left(\begin{array}{cc}{g}_{11}& {g}_{12}\\ {g}_{21}& {g}_{22}\end{array}\right)\right)>0$. Therefore, the matrix G is positive definite and method (25) is Gstable.
The second hybrid class
The second hybrid class takes the form
where \(f_{n+s} = f(t _{n+s} ; y _{n+s})\); \(t_{n+s} = t _{n} + s h\); \(1 < s < 1\) and \(\beta_{s}\), \(\alpha_{ n j}\), \(j = 1, 2, \dots , k\), are parameters to be determined as functions of s and \(\beta ^{*}\). The method with step k has order \(p = k\) and \(y_{n+s}\) has order \(k  1\). To evaluate the value of \(y_{n+s}\) at offstep point, i.e. \(t _{n+s}\), consider the nodes \(t _{n}\) (double node), \(t_{ n1 }, \dots , t _{ n k}\) (simple nodes) [14].
Here, the oneleg twin of the second class is studied when \(k = 2\) and \(k = 3\).
In the case of \(k=2\), the method takes the form
where
Method (28) has order 2, its truncation error takes the form
The oneleg twin of (28) takes the form
and has order 2 and its truncation error takes the form
If
then method (30) has order 3 and its truncation error becomes
To discuss Gstability of (30), using (8), we have
Substitute \(f_{n+s}\) in equation (28), it becomes
The corresponding characteristic equations are
Applying Theorem 2, the variables \(a _{i}\), \(i = 0, 1, 2\) and \(g _{ij}\); \(i, j = 1, 2\) satisfy the relations
Choosing \(\beta ^{*}= 0.3\) and \(s =  0.1\) makes \(a _{0} =  0.583636\),
Therefore, the matrix G is positive definite and method (30) is Gstable.
In the case of \(k = 3\), the method takes the form
where
Method (31) has order 3 and its truncation error takes the form
The oneleg twin of (31) takes the form
and has order 2, its truncation error takes the form
To discuss Gstability of (33), using (8), we have
Substitute \(f_{n+s}\) in equation (31), it becomes
The corresponding characteristic equations are
Applying Theorem 2, the variables \(a _{i}\), \(i = 0, 1, 2, 3\) and \(g _{ij} \); \(i, j = 1, 2, 3\) satisfy the relations
Choosing \(s =  0.3\) and \(\beta ^{*}= 0.2\) makes \(a _{0} =  0.231455\);
Therefore, the matrix G is positive definite and method (33) is Gstable.
Numerical tests
Here, some numerical results are presented to evaluate the performance of the proposed technique [1, 4, 13]. The numerical tests are solved after reduction.
Test 1
Consider the differential algebraic equations:
with the initial condition \(y_{1}(0) = 1\); \(y_{2}(0) = 1\); \(y_{3}(0) = 0\); and the exact solution is
Test 2
Consider the differential algebraic equations:
with the initial condition \(x(1)=1\); \(y(1)=0\); and the exact solution is \(x(t)= t \cos (1t^{2})\), \(y(t) =1t^{2}\).
Test 3
Consider the differential algebraic equations:
where \(x'(t) = (x_{1},x_{2})^{\mathrm{T}}\); \(f(x; t) = (1+ (t 1/2) \exp (t), 2t + (t^{2}1/4 ) \exp (t))^{\mathrm{T}}\); \(B(x; t) = (x' _{1},x'_{2})^{\mathrm{T}}\); \(g(x; t) = 1/2 (x^{2} _{1} + x ^{2} _{2}  (t  1/2)^{2} (t^{2}  1/4 )^{2})\); with the initial condition \(x_{1}(0) =  1/2\); \(x_{2}(0) = 1/4\); and the exact solution is \(x_{1}(t) = (t  1/2)\); \(x_{2}(t) = t^{2} 1/4\); \(y(t) = \exp (t)\).
The above tests are solved by the two hybrid classes and their oneleg twins of the two classes with \(k=2\) at different values of t. In the first method, \(\beta_{0} =0.8\), \(s = 2\), and in the second method, \(\beta^{*} = 0.4\), \(s =  0.3\). The errors of numerical solutions of tests 1, 2, and 3 are tabulated in Tables 6, 7, and 8, with different stepsizes h, respectively.
Conclusion
In this paper, the first hybrid class is studied for \(k=1\) up to 6. It has large stability regions. Its oneleg twin for \(k=1\) (\(p=2\)) and \(k=2\) (\(p=3\)) is Gstable. In the second class, for \(k = p = 2\), the oneleg twin has order 2 except when \(s =\frac{1}{3} (3+\sqrt{3} \sqrt{(1 2 \beta ^{*} + \beta ^{*2})})\) it has order 3. For \(k = p = 3\), the oneleg twin has order 2 and if \(\beta^{*} = 0\), it leads to one leg hybrid BDF, and they are Gstable. The numerical tests show that the hybrid method (4) gives good result with small steps.
Abbreviations
 DAEs:

Differential algebraic equations
 ODEs:

Ordinary differential equations
 BDF:

Backward differentiation formula
 HBDF:

Hybrid BDF methods
 EBDF:

Extended BDF
 MEBDF:

Modified extended BDF
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We are really thankful to the reviewers for their useful suggestions and corrections.
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MSC
 65L05
 65L06
 65L20
Keywords
 Hybrid methods
 Oneleg methods
 DAEs
 Gstability
 A(α)stability