Convergence of One-Leg Hybrid Methods for Implicit Mixed Differential Algebraic Systems

Abstract

This paper focuses on a hybrid multistep and its twin one-leg methods and implementing them on implicit mixed differential algebraic equations. The orders of convergence for the above methods are discussed and numerical tests are solved.

1. Introduction

Consider the ordinary differential system: where the linear multistep method (LM) [1, 2] is given, and the generating polynomials have real coefficients and no common divisor. Also assume throughout the normalization that Then the associated one-leg (OL) method is defined by

The author presents hybrid multistep methods that take the form and the one-leg twin of (6a) and (6b) takes the form where , , , . , and , , are parameters to be determined as functions of and . Methods (6a) and (6b) with step have order and has order . To evaluate the value of at the off-step point, that is, , consider the nodes (double node), , and (simple nodes) [3, 4].

Applying Newton’s interpolation formula for this data gives the following scheme: where and (or ) is considered as a derivative of the solution .

The hybrid multistep method and its twin one-leg depend on two parameters, and , which control their convergence and stability; also the position of the stage point affects the stability regions of the methods. For optimal values of and , the methods have larger stability region compared to the hybrid backward differentiation formulae [3]; the corresponding one-leg twin is -stable for and ; see [5, 6].

Differential-algebraic equations (DAEs) often take place in highly scientific technology domains, such as automatic control engineering, simulation of electrical networks, and chemical reaction kinetics [7, 8]. Some systems can be reduced to ODE systems and can be solved by numerical ODE methods. Reduction to explicit differential system (1) in some other systems can be impossible or impractical 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 [9, 10].

Here LM (2) and OL (5) are defined for implicit mixed differential algebraic systems of the form where , , , and are vectors of the same dimension. Rewrite (2) and (5) in the form respectively, where , substituting for in (9a) and (9b): where acts only on backward data. Equations (12a) and (12b) can be solved for . In the one-leg form, the arguments are changed to , and . The implementation of OL (5) to (9a) and (9b) gives the equations

As a modification technique that applies the same arguments of on , this implementation can be written as The LMS and MOL formulations in (12a), (12b), (14a), and (14b) are easier to implement than OL method in (13a) and (13b) because both equations are evaluated on the same arguments.

In the following section, the hybrid multistep (HMS) method in (6a) and (6b) and its twin, hybrid one-leg (7) (HOL) method are defined for (9a) and (9b) and expressions for the local truncation errors of (HMS) and (HOL) are given.

2. The Hybrid Method

In the case of , the method in (6a) and (6b) takes the form where

Method (15a) has order and its truncation error is , and has order one and its truncation error is due to (8), where . Applying the method in (15a) and (15b) on implicit mixed differential algebraic equations (9a) and (9b) obtains the following: where

Let and be the exact solution of (9a) and (9b). The residues of (17a) and (17b) are the values of the left sides evaluated on and . Using Taytor expansion for the second argument of evaluated on , leads to where

Theorem 1. The order of convergence of the second-order hybrid method in (15a) and (15b) when applied to implicit mixed DAEs (9a) and (9b) is two.

Proof. Let the local truncation errors be defined by , , , and , where satisfies (17a) and (17b) with exact backward data: where ; expanding around implies that where the arguments of , , , and are The errors and satisfy the following equations: where and the arguments of are . If and exist, then Therefore, has the same order as . The substitution for in (25) implies that where Then and are third order; thus the method is of second order.
Therefore, is third order small and in accordance with classical theory we conclude that the global error in must be second order small. Furthermore, if denotes the global error in —note that the global errors satisfy the same algebraic constraints as the local errors , namely, (26)—consequently, is also second order small and thus method in (15a) and (15b) is second order small and accurate with respect to both and .
Modified Technique for Hybrid Method. It is noticed that the arguments of (17a) are and that of (17b) is . The arguments of (17a) and (17b) can be taken as which is called the modified technique for hybrid method.
In this case, Expanding around gives the following: Substitute for in (24): where and the global error affecting is .
Since thus the method is of second order.

3. The One-Leg Twin

In the case of , method (7) takes the form

Method (36a) has order and its truncation error is and has order one and its truncation error is due to (8).

Theorem 2. The order of convergence of the second-order one-leg twin in (36a) and (36b) when applied to implicit mixed DAE is two.

Proof. Applying the method in (36a) and (36b) on the implicit mixed differential algebraic equations (9a) and (9b) obtains the following: The residues of (37a) and (37b) are the values of the left sides evaluated on and : where and expanding around implies that where the arguments of are .
Expanding (37b) around , if exists, gives the following: substitute in (41): where thus the method is of second order.
Modified Technique for One-Leg Twin Method. Here the arguments of and are different if the arguments of are taken as , and (37a) and (37b) become the following: Substitute in (41): where Consequently, and the global error affecting is . However, since the global error is related to by the difference The solution of this difference equation is also since is a bounded operator; thus the method is of second order.

4. Numerical Tests

Here, some numerical results are presented to evaluate the performance of the proposed technique [11, 12].

Test 1. Consider the differential algebraic equations: with the initial conditions , , and the exact solution is , .

Test 2. Consider the nonlinear DAEs: with the initial conditions , , and .

The exact solutions are , , and .

Test 3. Consider the nonlinear DAEs: with subject to the initial conditions , , and .

The exact solution is ,  .

Test 4 (practical test). Consider rectifier diode circuit [13] in Figure 1 for transforming an AC voltage source into a DC voltage. It is designed in such a way that it damps the incoming sine-wave.

Figure 1 

Rectifier circuit.

In Figure 1, the circuit is a half wave rectifier circuit. The diode permits the flow of the current during the positive half cycle and stops the current flow during the negative half cycle. The capacitor is used to smooth the output voltage such that the output voltage at the load resistor is close to a DC voltage: Kirchhoff’s Current Law (KCL) at Assuming , KCL at : where is the supply current, is the diode current, is the current in the resistor , is the current in the capacitor , is the reverse current of the diode, and are node and node , is the reference node, and is the supply voltage. We solve this circuit with hybrid formula (6a) and (6b) method and draw the exact and the numerical solutions in Figures 2–4.

Figure 2 

The exact and approximate solutions for the current of the diode circuit.

Figure 3 

The exact and numerical solutions for the voltage of the diode circuit.

Figure 4 

The exact and approximate solutions of the voltage of the diode circuit.

The above tests are solved by the hybrid multistep method in (6a) and (6b) and its hybrid twin one-leg method (7) (with ,  , and ) and compared with HBDF method [3] (with and ) at different values of . In solving the tests, Newton-Raphson method is used and the initial guesses are obtained by an interpolating polynomial. The errors of solutions of tests 1, 2, and 3 are tabulated in Tables 1, 2, and 3.

Table 1 

The errors of solutions for the first test.

Table 2 

The errors of solutions for the second test.

Table 3 

The errors of solutions for the third test.

5. Conclusion

This paper focuses on the implementation of hybrid multistep classes and their twin one-leg classes on implicit mixed differential algebraic equations. The orders of convergence for these classes are discussed. Numerical tests are introduced, which show that the introduced methods give better results than HBDF.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.