ONE-STEP 4-STAGE HERMITE-BIRKHOFF-TAYLOR DAE SOLVER OF ORDER 12

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1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 16, Number 4, Winter 2008 ONE-STEP 4-STAGE HERMITE-BIRKHOFF-TAYLOR DAE SOLVER OF ORDER 12 TRUONG NGUYEN-BA, HAN HAO, HEMZA YAGOUB AND RÉMI VAILLANCOURT ABSTRACT. The ODE solver HBT(12)4 [20] is expanded into the differential algebraic equation (DAE) solver, HBT(12)4DAE, for nonstiff and moderately stiff systems of fully implicit DAEs of arbitrarily high fixed index. Pryce s structural pre-analysis for DAEs is sketched. The stepsize is controlled by a local error estimator. Similarly, the Taylor series method of order 12, T12, and Dormand-Prince s DP(8,7)13M are expanded into T12DAE and DP(8,7)DAE, respectively, for DAEs. HBT(12)4DAE uses only the first nine derivatives of y as opposed to 12 for T12DAE. On the basis of the number of steps, CPU time, maximum global error and relative error at the end of the integration interval, HBT(12)4DAE is superior to DP(8,7)DAE, the super partitioned additive Runge- Kutta (SPARK), the projected implicit Runge-Kutta (PIRK), and methods based on Padé and Chebyshev series in solving several low- and high-index test DAEs. 1 Introduction The ODE solver HBT(12)4 [20] is expanded into a differential algebraic equation (DAE) solver, called HBT(12)4DAE, by the addition of a modification of Pryce s structural pre-analysis and automatic differentiation (AD) techniques. The independent variable is the real variable t. We denote real d- vectors as y = [ 1 y, 2 y,..., d y] T, f = [ 1 f, 2 f,..., d f] T. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada and the Centre de recherches mathématiques of the Université de Montréal. AMS subject classification: Primary: 65L06; Secondary: 65D05, 65D30.. Keywords: general linear method, non-stiff DAEs, Hermite-Birkhoff method, Taylor method, structural analysis.. Copyright c Applied Mathematics Institute, University of Alberta. 415

2 416 T. NGUYEN-BA ET AL. HBT(12)4DAE solves smooth nonstiff and moderately stiff systems of fully implicit DAEs of arbitrarily high fixed index of the form (1) f(t, y, y (1), y (2),...) = 0, with consistent initial conditions. A consistent point of (1) is defined as a scalar t together with a set of scalars j η l, where (j, l) are all the elements of a finite set S J, such that there exists a solution to (1) in a neighborhood of t = t with j y (l) (t ) = j η l for all (j, l) S J and the solution is unique. The set S J need not be minimal. The link between the Taylor and Runge Kutta methods, which make HBT(12)4DAE, is that values at off-step points are obtained by means of predictors which use y (1) to y (9) at the current point t n, and y (1) at the off-step points. The Taylor method for solving ODEs is briefly reviewed in [20] with references therein. More recently, improving on present DAE solvers, new Taylor series methods have been developed for solving nonlinear and fully implicit smooth high-index DAE initial value problems containing derivatives of any order [23, 2, 17, 18, 11]. Their cost is polynomial in the number of digits of accuracy [11]. The main cost in solving an ODE/DAE by a Taylor series method lies in the repeated evaluation of the Taylor coefficients of the functions involved. For smooth DAEs, Pryce [23] used automatic differentiation (AD) techniques with a structural analysis to compute the coefficients of Taylor series solutions about the current time-point. This procedure is implemented in [21] for a onestep 9-stage Hermite-Birkhoff-Taylor DAE solver of order 10. In this paper, increased efficiency is achieved by the addition of offstep points, where only y (1) is required, and the order of the necessary y (j) (t n ) is reduced to 9 as compared to 12 with the Taylor series method of order 12 for DAE, denoted by T12DAE. HBT(12)4DAE takes advantage of this fact and relies on a modification of Pryce s structural pre-analysis and automatic differentiation (AD) techniques at off-step points. Forcing an expansion of the numerical solution to agree with a Taylor expansion of the true solution to order 12 leads to a combination of Taylor- and Runge-Kutta-type order conditions which are reorganized into Vandermonde-type linear systems whose solutions are the coefficients of the method. These coefficients are obtained only once as solutions of these systems by means, say, of Gaussian elimination. Denote by DP(8,7)DAE an extension to DAEs of the Dormand Prince pair DP(8,7)13M [22]. The C++ performance of HBT(12)4DAE,

3 ONE-STEP 4-STAGE HBT DAE SOLVER OF ORDER DP(8,7)DAE and T12DAE, is compared. It is seen that HBT(12)4DAE is superior to T12DAE on some problems on the basis of CPU time and the number of steps. On the basis of the number of steps, CPU time, maximum global error, and relative error at the end of the interval of integration HBT(12)4DAE wins over DP(8,7)DAE, the super partitioned additive Runge Kutta (SPARK), the projected implicit Runge Kutta (PIRK), and methods based on Padé and Chebyshev series in solving several problems used to test DAE solvers, including high-index DAE problems. Section 2 sketches the ODE solver HBT(12)4 found in [20]. Section 3 summarizes Pryce s structural analysis for DAEs. Section 4 deals with the step control to advance integration. Section 5 presents numerical methods and compares DAE methods. 2 One-step HBT(12)4 The predictors P 2 to P 4, integration formula IF, order conditions, off-step points {c i }, i = 1, 2, 3, 4, Vandermondetype formulation and region of absolute stability of HBT(12)4 found in [20] are the same for the ODE and DAE methods. For DAEs, we need to add a step control predictor, P 5, which we construct from a Hermite Birkhoff polynomial of degree 12 to obtain ỹ n+1 to order 10, (2) ỹ n+1 = y n + h n+1 4 j=1 a 5,j y (1) n+c j + 9 j=2 h j n+1 γ 5,jy (j) n, where y n and y n+cj are numerical approximations to y(t n ) and y(t n + h n+1 c j ), respectively. The 12-vector of the reordered coefficients of predictor P 5 in (2) is ũ 5 = [a 5,4, a 5,3, a 5,2, a 5,1, γ 5,2, γ 5,3,..., γ 5,9 ] T. By setting a 5,j = b j +ω j, j = 3, 4, the vector ũ 5 reduces to the 10-vector u 5 which is the solution of the system of order conditions [ ( c i 1 ) [ ] ] 2 I9 (3) u (i 1)! 0 5 = r 5, i=1,2,...,10 1,9 where the 10 vector r 5 has components r 5 (i) = 1 4 i! c i 1 j (b j + ω j ), i = 1, 2,..., 10. (i 1)! j=3

4 418 T. NGUYEN-BA ET AL. For arbitrary nonzero ω 4 and ω 3, P 5 yields ỹ n+1 to order 10. A good experimental choice is ω 4 = and ω 3 = The numerical values of the coefficients in the formula of P 5 in (2) are listed in Appendix A of [20]. 3 Structural analysis for DAEs We assume that the system of DAEs (1) is sufficiently differentiable with respect to its independent variable t R and all its dependent variables j y (k) R in a neighborhood of a consistent point. The Taylor method for DAEs requires a pre-analysis which solves an assignment problem by Pryce s signature method, called Σ-method [24]. Let the nonnegative integer κ i, called equation-offset, be the number of times the ith equation i f = 0 has to be differentiated to reduce it to an ODE. Also, let the nonnegative integer δ j, called variable-offset, be the highest order derivative with respect to j y appearing in this ODE. We let (4) ρ ij = κ i δ j. The solvability of a DAE system by means of Taylor series is given by the following theorem [23]. Theorem 1. Let the DAE (1) be sufficiently smooth in a neighborhood of a consistent point. Then the Taylor series method applied at this point succeeds if and only if the d d system Jacobian matrix J is nonsingular at this point, where ( i f) (5) J ij = ( j y (ρij) ), if the variable j y (ρij) is present, 0, if j y (ρij) is absent or ρ ij < 0. We summarize Pryce s structural analysis and the corresponding algorithm [23, 24] as follows. (a) Form a d d signature matrix Σ = (σ ij ) with { highest order of the derivatives of j y present in i f, σ ij = if the variable j y is absent in i f. (b) Solve an assignment problem to determine a highest value transversal (HVT), which is a subset of indices (i, j) describing just one element in each row and each column, such that σ ij is maximized and finite. We assume that such an HVT exists.

5 ONE-STEP 4-STAGE HBT DAE SOLVER OF ORDER (c) Determine the smallest offsets κ i and δ j of the problem such that δ j κ i σ ij for all 1 i d, 1 j d, with equality on HVT. The structural index is then defined as ν = max κ i + i { 0, if all δ j > 0, 1, if some δ j = 0. (d) Form the system Jacobian J defined in (5). (e) Consider a consistent point at t. If J is non-singular at that point, then the solution can be computed by a Taylor series in a neighborhood of that point (Theorem 1). Substitute the expansion y(t) = l 0 1 l! y(l) (t )(t t ) l in equations (1) and expand in Taylor series to obtain f(t, y, y (1), y (2),...) = q 0 1 d q f q! dt q (t )(t t ) q = 0. The system d q f dt q (t ) = 0 has to be solved in a staggered way for y (l) (t ) by means of the following lockstep solution scheme: solve (6) d k+κi ( i f) dt k+κi (t ) = 0 for j y (k+δj ) (t ), i, j = 1,..., d, for stages k = 0, 1,..., where all other j y (l) (t ) occurring in (6) have already been found at earlier stages. In general, system (6) is linear in the corresponding unknowns and involves the same Jacobian J for k = 1, 2,..., where J is computed in step (d). It is to be noted that, at the initial stages k < 0, the initial guesses j y (s), s = 0, 1,..., δ j 1, are the result of summing the Taylor series from the previous step. The local error control will make these guesses close-to-consistent.

6 420 T. NGUYEN-BA ET AL. In a practical implementation of HBT(12)4DAE, starting with the pre-analysis stage we obtain the offsets κ i and δ j mentioned above. The offsets indicate which equations to solve for which unknowns, and give a systematic way of determining consistent initial conditions. The next step of the method consists in generating the system Jacobian J (defined in Theorem 1). Provided the Jacobians at off-step points and at step points are nonsingular at each integration step, the method succeeds and the Taylor coefficients can be computed to first order at off-step point and up to order 9 at step points. We remark that for each Taylor coefficient we solve a linear system involving the system Jacobian J but the matrix J is the same for every Taylor coefficient at one given off-step point or at one given step point. 4 Controlling stepsize and advancing integration For DAEs, the last accepted stepsize h n and the error estimate EEST = y n ỹ n are used to calculate the current stepsize, h n+1, by the following iterative procedure until the tolerance requirement EEST < TOL is satisfied. (a) The stepsize h n+1 is obtained by the following formula [13]: (7) h n+1 = min {h max, β h n [ TOL y n ỹ n ] 1/κ, 4 h n } where κ = 11, β = 0.81 is a safety factor, and ỹ n is obtained by predictor P 5 in (2) to order 10. The limitation h n+1 4h n is maintained as in [13] to limit the error growth. (b) At off-step points t n+ci, i = 2, 3, 4, the vectors y n+c2, y n+c3, y n+c4, y n+1, and ỹ n+1 are obtained successively by formulae in [20, Eqs. (2) (3)] and (2), each one being immediately used to obtain y (1) n+c 2, y (1) n+c 3, etc. as follows. (b.1) Given c i, the Taylor coefficients of y(t) about t n+ci are acquired by means of the components of y n+ci instead of using Pryce s method at stages preceding stage k = 0. (This step is a modification of Pryce s method.) (b.2) Then the Taylor coefficients at stage k = 0 or at stages k = 0 and k = 1 (depending on the problem) are computed by solving systems with the system Jacobian J defined in Theorem 1 by Pryce s method [23]. Then y (1) n+c i can be obtained (see Example 1).,

7 ONE-STEP 4-STAGE HBT DAE SOLVER OF ORDER (c) If EEST < TOL, then h n+1 is accepted. To satisfy the DAE constraints (system + hidden constraints), the close-to-consistent point y n+1 is mapped onto the DAE constraints by a root finding process which is iterated to convergence to get the final value of the vector y n+1 (in general, one iteration is enough). Pryce s linear systems with system Jacobian J are solved to obtain the Taylor coefficients of y(t) up to order 9 about t n+1. Then, the program returns to (a) with n replaced by n + 1. Otherwise the program returns to (a) with smaller h n+1 taken as 0.7h n+1. In summary, to solve a general DAE problem, HBT(12)4DAE uses two schemes to generate derivative vectors because of off-step points, compared to T12DAE which needs only one scheme. Scheme 1 generates a set of high-order derivatives y (k+1) n+1 for stage k 0 at step point t n+1. This scheme is built on the Taylor series method developed by Pryce [23, 24]. Scheme 2 generates y (1) n+c i at off-step points t n +c i h n+1. This scheme relies on modification (b.1) (in Section 4) of Pryce s structural preanalysis and automatic differentiation techniques. It is to be noted that for stages k 0 the systems to be solved for derivatives are, in general, not linear. But for stages k > 0 these systems are always linear. Schemes 1 and 2 are illustrated in Examples 1 and 2 below with their own notation. Example 1. (8) Solve the pendulum problem: x 1 + x 1 λ = 0, x 2 + x 2λ g = 0, x x2 2 L2 = 0. Solution. (a) Scheme 1 generates [ ] T y n (k+1) = x (k+1) 1,n, x (k+1) 2,n, x (k+2) 1,n, x (k+2) 2,n, λ (k) n, for k 0 at step points t n, where x (k) 1,n /k!, x(k) 2,n /k! and λ(k) n /k! are the kth Taylor coefficients of y 1 (t), y 2 (t) and λ(t) about t n, respectively.

8 422 T. NGUYEN-BA ET AL. Once the close-to-consistent vector (9) y n = [x 1,n, x 2,n, x 1,n, x 2,n, λ n ] T is obtained by integration formula IF, y n following two steps. is computed by means of the Step 1.1. Stages k = 2 and k = 1 of Pryce s method are completed as follows. At stage k = 2, modify the close-to-consistent values x 1,n or/and x 2,n to satisfy the algebraic constraint x 2 1,n + x 2 2,n L 2 = 0. At stage k = 1, modify the close-to-consistent values x 1,n or/and x 2,n to satisfy the hidden constraint holding x 1,n and x 2,n constant. 2x 1,n x 1,n + 2x 2,nx 2,n = 0, Step 1.2. Consider the enlarged DAE constraints system, with the modified values obtained in Step 1.1, (10) (11) (12) (13) (14) x 1,n + x 1,nλ n = 0, x 2,n + x 2,n λ n g = 0, 2 [ x 1,n x 1,n + x 2,n x 2,n + (x 1,n) 2 + (x 2,n) 2] = 0, x 2 1,n + x2 2,n L2 = 0, 2x 1,n x 1,n + 2x 2,n x 2,n = 0. This system is made of the original system (10), (11) and (13), taken from (8), and the hidden constraints (12) and (14) obtained by differentiating (13). At stage k = 0, solve Pryce s system (10) (12) written as (A, B, C ) = 0, where A = x 1,n + x 1,nλ n = 0, B = x 2,n + x 2,n λ n g = 0, C = 2 [ x 1,n x 1,n + x 2,nx 2,n + (x 1,n )2 + (x 2,n )2] = 0,

9 ONE-STEP 4-STAGE HBT DAE SOLVER OF ORDER for x 1,n, x 2,n and λ n. Then solve the full system (10) (14) for x 1,n, x 2,n, x 1,n, x 2,n and λ n by Newton s method taking the modified closeto-consistent vector y n as starting values. At stage k = 1, solve Pryce s linear system (A, B, C (3) ) = 0, obtained by differentiating (A, B, C ) = 0, for x 1,n, x 2,n and λ n to get (15) y n = [ x 1,n, x 2,n, x 1,n, x 2,n, λ n] T. For general k > 1, solve (A (k), B (k), C (k+2) ) = 0 for x (k+2) 1,n, x (k+2) λ (k) n as in [23]. The high-order derivatives y (k+1) n generated as [ ] T y n (k+1) = x (k+1) 1,n, x (k+1) 2,n, x (k+2) 1,n, x (k+2) 2,n, λ (k) n. 2,n and, k 0, of y n, can be (b) Scheme 2 generates y n+c i, i = 2, 3, 4. Once the vector y n+c2 = [x 1,n+c2, x 2,n+c2, x 1,n+c 2, x 2,n+c 2, λ n+c2 ] T is obtained by P 2, y n+c 2 is computed using known y n+c2 as follows. At stage k = 2 and k = 1, x 1,n+c2, x 2,n+c2, x 1,n+c 2 and x 2,n+c 2 should not be modified, At stage k = 0, solve Pryce s system for x 1,n+c 2, x 2,n+c 2 and λ n+c2, At stage k = 1, solve Pryce s linear system for x 1,n+c 2, x 2,n+c 2 and λ n+c 2. The output is (16) y n+c 2 = [x 1,n+c 2, x 2,n+c 2, x 1,n+c 2, x 2,n+c 2, λ n+c 2 ] T. HBT(12)4DAE uses vector (16) to compute y n+c3 by means of P 3, (17) y n+c3 = [x 1,n+c3, x 2,n+c3, x 1,n+c 3, x 2,n+c 3, λ n+c3 ] T. Then, y n+c 3 is computed using only stages k = 0 and k = 1 of Pryce s method as above with known y n+c3. Finally, y n+c i, i = 2, 3, 4, can be computed by the above procedure. In Scheme 2, the stages k = 2 and k = 1 of Pryce s method are not used in order to keep the components of y n+ci intact when used to compute y n+c i. The algorithm returns to Scheme 1 with n replaced by n + 1.

10 424 T. NGUYEN-BA ET AL. It is to be noted that, as in Runge Kutta methods, y n+ci are evaluated by numerical formulae which satisfy the order conditions listed in [20, Section 3]. Thus, the values x 1,n+ci, x 2,n+ci, x 1,n+c i, x 2,n+c i, x 1,n+c i, x 2,n+c i, λ n+ci at off-step points should not be projected onto the constraints manifold. It is also to be noted that, in this example, y n+1 is computed separately from y (k) n+1, k > 1. Thus a method which does not require high order derivatives, like DP(8,7)DAE, does not need general stages k > 1. 5 Numerical results The higher derivatives y (1) up to y (9) of Taylor series are calculated at each integration step by Pryce s AD techniques [23]. The error of a numerical solution at time t n is the norm y n y(t n ) of the difference between the numerical solution y n and a reference solution y(t n ) given by means of the analytic solution, or T12DAE or HBT(12)5DAE [19] at stringent tolerance, as appropriate. The maximum global error (MGE) is the maximum of the errors over all the integration steps. The relative error (RE) in the computed solution x at final time t f with respect to a reference solution r is RE = max{ x i r i / r i }. i Definition 1. The percentage efficiency gain of a quantity A obtained by methods 1 and 2 is defined by the function (cf. Sharp [25]), [ j (18) PEG(A, j) = 100 A ] 2,j j A 1. 1,j In this paper, j = log 10 (MGE) or j = log 10 (RE). For example, if PEG(A, j) is to be computed as a function of A = CPU and j = log 10 (MGE), the data (MGE, CPU) is used to plot the curve Γ = ( log 10 (MGE), log 10 (CPU)) in a least-squares sense, say, by Matlab s polyfit. For appropriate integers j in the range of log 10 (MGE), we have log 10 (MGE) = j and, hence, log 10 (CPU) on Γ. Computations were performed in C++ on a Sun Ultra 20 with Dual- Core AMD Opteron (tm) Processor 1218 and 4 Gb of RAM running on OS Fedora Core Release 6 (Zod). In this paper CPU time is measured in seconds.

11 ONE-STEP 4-STAGE HBT DAE SOLVER OF ORDER DAE test problems The following DAE test problems on the time interval 0 t t f will be used for comparing HBT(12)4DAE with other methods. (a) An example with t f = 4π found in [5, pp ] and [7] with initial conditions and exact solution: (19) v 1 tv 2 + v 1 (1 + t)v 2 = 0 v 1 (0) = 1, v 1 (t) = e t + t sin t, v 2 sin t = 0, v 2 (0) = 0, v 2 (t) = sin t. (b) An example with t f = 4π found in [8] with initial conditions and exact solution: y 1 + y 3y 2 (y 2 + 1)y 3 (20) +y 1 1 sin t = 0, y 1 (0) = 1, y 1 (t) = e t, (y 3 + 1)y 1 + y 1 y 2 + e t = 0, y 2 (0) = 0, y 2 (t) = sin t, y 1 y 2 y 3 e t sin t cos t = 0, y 3 (0) = 1, y 3 (t) = cos t. (c) A linear circuit with t f = 4π from [9] obtained by nodal analysis: (21) [ (cos t sin t)]e 1 + [ (sin t + cos t)] e 1 e 2 = 0, e 1 + e 2 j = 0, e 2 4 sin t 0.25 sin 2t = 0, with initial conditions and exact solution: e 1 (0) = 1, e 1 (t) = sin t + cos t, e 2 (0) = 0, e 2 (t) = 4 sin t sin 2t, j(0) = 3.5, j(t) = 3 cos t cos 2t + sin t. (d) The reduced van der Pol DAE with t f = 0.8 taken from [12] with initial conditions and exact solution: (22) y + z = 0, y(0) = 2 z2, log( z ) = t 2 + log 2, 3 2 ( ) z 3 y 3 z = 0, z(0) = 2, y = z3 3 z.

12 426 T. NGUYEN-BA ET AL. (e) An example with t f = 1 found in [1]: (23) [ x 1 + λ 1 ] t 1 + (2 t)λy + e t (3 t)/(2 t) = 0, 2 t x 2 + x 1 (1 λ)/(t 2) x 2 + (λ 1)y + 2e t = 0, (t + 2)x 1 + (t 2 4)x 2 (t 2 + t 2)e t = 0. The exact solution, with initial conditions x 10 = 1, x 20 = 1 at t 0 = 0, is x 1 (t) = e t, x 2 (t) = e t, y(t) = e t /(2 t). (f) An example with t f = 1 found in [14]: (24) y 1 = y 2 2y 2 1y 2 + 2y 1 y 2 2 2e 2t y 1 y 2 + y 1 y 2 2z 2 1 y 2 2z 1 + 2y 2 2z 2 1, y 2 = y2 1 + e t z 1 y 1 + y 2 1 y2 2 3y2 2 z 1 + z 1, 0 = y 2 1 y 2 1. The exact solution, with initial conditions y 10 = 1, y 20 = 1, z 10 = 1 at t 0 = 0, is y 1 (t) = e t, y 2 (t) = e 2t, z 1 (t) = e 2t. (g) A moderately stiff DAE of index 3 called the car-axis problem. It is a simple model of a car axis going over a bumpy road, taken from [15] and [16]. (h) An eight-node transistor amplifier problem which is a slightly stiff DAE system of index 1 found in [15] and [16]. (i) A nonlinear simple pendulum which is a nonstiff DAE system of index 3 with t f = 100 taken from [5]. (j) A five-node transistor amplifier problem which is a slightly stiff DAE system of index 1 with t f = 0.2 taken from [12]. 5.2 Numerical verification of the order of HBT(12)4DAE To show the relevance of the theoretical order of HBT(12)4DAE, we have applied the method with constant stepsize to test problem (19). This problem is often used to test several Runge Kutta methods which are likely candidates for solving stiff or differential-algebraic systems. In Figure 1, the error at t n = π is plotted for different stepsizes h in

13 ONE-STEP 4-STAGE HBT DAE SOLVER OF ORDER log 10 (global error) log 10 (h) FIGURE 1: Constant stepsize HBT(12)4DAE applied to the test problem (19) over t [0, π]. a log-log scale so that the curve, which fits the data (log 10 h, log 10 ( y n y(t n ) )) in a least-squares sense, appears as a straight line with slope k whenever the leading term of the error is of order k, that is, y n y(t n ) = O(h k ). For HBT(12)4DAE, we have a straight line with slope 12, thus confirming the order of the method. 5.3 CPU time versus maximum global error HBT(12)4DAE and DP(8,7)DAE, are applied to problems (i) and (j) listed in subsection 5.1. The CPU percentage efficiency gain (CPU PEG) defined by (18) is the function PEG(CPU, j) with j = log 10 (MGE). In Figure 2, CPU time is plotted versus log 10 (MGE) for the two DAE problems at hand. It is seen from the figure that HBT(12)4DAE compares favorably with T12DAE on the basis of CPU time versus MGE and also from the CPU PEG listed in Table The number of steps versus the number of significant correct digits The numerical performance of HBT(12)4DAE and other DAE solvers is compared over a set of tolerances on problems (g) and (h). The accuracy of the solutions by other DAE solvers is found in [15] and [16]. Let SCD denote the minimum number of significant correct digits in a numerical solution at t f [15, 16]. Then we have the approximation: (25) SCD log 10 (RE).

14 428 T. NGUYEN-BA ET AL log 10 (MGE) 4 6 Pend. log 10 (MGE) 4 6 Trans. amp CPU time CPU time HBT(12)4DAE, T12DAE and DP(8,7)DAE FIGURE 2: The pendulum and the five-node transistor amplifier DAE problems. DAE CPU PEG of HBT(12)4DAE over: problems T12DAE DP(8,7)DAE Pendulum 29% 104% Trans. amp. 52% 365% TABLE 1: CPU PEG of HBT(12)4DAE over T12DAE and DP(8,7)DAE for the listed DAE problems. Tables 5.4 and 5.4 list the relative tolerance (rtol), absolute tolerance (atol), SCD, total number of steps (Steps), and number of successful steps (Accept) of different solvers applied to the car-axis and the transistor amplifier problems, respectively. It is seen from these tables that, at stringent tolerance, HBT(12)4DAE compares favorably with other DAE solvers reported in [15] and [16] on the basis of the number of steps versus SCD. 5.5 The number of steps versus the relative error Car axis and eight-node transistor amplifier problems In Figure 3, the numerical performance of HBT(12)4DAE and T12DAE is compared at final time on the basis of the number of steps (NS) against the relative error (RE) for the car-axis and eight-node transistor amplifier problems [15, 16]. The number of step percentage efficiency gain (NS PEG) defined by

15 ONE-STEP 4-STAGE HBT DAE SOLVER OF ORDER Solver rtol atol SCD Steps Accept HBT(12)4DAE HBT(12)4DAE HBT(12)4DAE MEBDFDAE PSIDE RADAU RADAU TABLE 2: Results for different solvers for the car-axis problem (g). Solver rtol atol SCD Steps Accept HBT(12)4DAE HBT(12)4DAE HBT(12)4DAE HBT(12)4DAE DASSL MEBDFDAE PSIDE RADAU RADAU TABLE 3: Results for different solvers for the eight-node transistor amplifier problem (h). (18) is the function PEG(NS) with j = log 10 (RE). From Fig. 3, the NS PEG of HBT(12)4DAE over T12DAE are 44% and 63% for the car-axis and the eight-node transistor amplifier problems, respectively. 5.6 The number of constant steps versus the maximum global error Let NCS be the number of constant steps used in solving a DAE Comparing NCS for HBT(12)4DAE and DP(8,7)DAE Figure 4 shows the superiority of HBT(12)4DAE over T12DAE and DP(8,7)DAE on the basis of NCS versus log 10 (MGE) for problems (19), (21) and (22).

16 430 T. NGUYEN-BA ET AL. 0 4 log 10 (RE) Car axis log 10 (RE) Trans. amp NS HBT(12)4DAE and T12DAE NS FIGURE 3: The car-axis and eight-node transistor amplifier problems log 10 (MGE) log 10 (MGE) NCS NCS 2 4 log 10 (MGE) NCS HBT(12)4DAE, T12DAE and DP(8,7)DAE FIGURE 4: The DAE problems (19) (top left), (21) (top right) and (22) (bottom).

17 ONE-STEP 4-STAGE HBT DAE SOLVER OF ORDER The number of constant steps percentage efficiency gain (NCS PEG) defined by (18) is the function PEG(NCS, j) with j = log 10 (MGE). The NCS PEG are listed in Table 4 for the problems at hand. DAE NCS PEG of HBT(12)4DAE over: problems T12DAE DP(8,7)DAE Problem (19) 2% 113% Problem (21) 1% 145% Problem (22) 70% 13% TABLE 4: NCS PEG of HBT(12)4DAE over T12DAE and DP(8,7)DAE for the listed DAE problems. It is seen from Figure 4 and Table 4 that HBT(12)4DAE wins over T12DAE and DP(8,7)DEA Comparing HBT(12)4DAE with T12DAE and DP(8,7)DAE on a high-index problem Schemes 1 and 2 enable HBT(12)4DAE and DP(8,7)DAE to handle the following high-index problem. Example 2. The high-index DAE problem found in [23] and [6] can be rewritten in the following form with variables x 1, x 2, x 3, u 1, u 2, v, w, X, Y : 0 = D = x 1 [ v + X[a(x 3 ) + 2b(x 3 )] + a(x 3 )w ], 0 = E = x 2 [ v + X[1 3a(x 3 ) 2b(x 3 )] a(x 3 )w + u 2 ], 0 = F = x 3 [ v + X[a(x 3 ) 9b(x 3 )] 2x 2 1 c(x 3) d(x 3 )Y 2 [a(x 3 ) + b(x 3 )]w ], 0 = G = cos x 1 + cos Y p 1 (t), 0 = H = sin x 1 + sin Y p 2 (t), 0 = K = w (u 1 u 2 ), 0 = L = X (2x 3 x 2 ), 0 = M = Y (x 1 + x 3 ), 0 = N = v [ 2Y 2 c(x 3 ) + x 2 1 d(x 3) ],

18 432 T. NGUYEN-BA ET AL. where p 1 (t) = cos(e t 1) + cos(t 1), a(s) = c(s) = p 2 (t) = sin(1 e t ) + sin(1 t), 2 cos s 2 cos 2, b(s) = s 2 cos 2 s, sin s cos s sin s 2 cos 2, d(s) = s 2 cos 2 s. By construction x 1 (t) = 1 e t and x 3 (t) = e t t. Solution. Scheme 1 generates a set of high-order derivative vectors y n (k+1) k 0 at step point t n : for y (k+1) n Once the vector = [x (k+1) 1,n, x (k+1) 2,n, x (k+1) 3,n, w n (k+1), x (k+2) 1,n, x (k+2) 2,n, x (k+2) 3,n, w (k+2) n, x (k+3) 1,n, x (k+3) 3,n, x (k+4) 1,n, x (k+4) 3,n, u (k+1) 1,n, u (k+1) 2,n ] T. y n = [ x 1,n, x 2,n, x 3,n, w n, x 1,n, x 2,n, x 3,n, w n, x 1,n, x 3,n, x 1,n, x 3,n, u 1,n, u 2,n ] T is obtained by the integration formula IF, y n is computed as follows. At stage k = 4, solve (G, H) = 0 for x 1,n and x 3,n. At stage k = 3, solve (G, H ) = 0 for x 1,n and x 3,n. At stage k = 2, solve (G, H, D, F ) = 0 for x 1,n, x 3,n, w n, and x 2,n. At stage k = 1, solve (G, H, D, F ) = 0 for x 1,n, x 3,n, w n, and x 2,n. At stage k = 0, solve (G (4), H (4), D (2), F (2), E, K) = 0 for x (4) 1,n, x(4) 3,n, w n, x 2,n, u 1,n and u 2,n. At stage k = 1, solve (G (5), H (5), D (3), F (3), E (1), K (1) ) = 0 for x (5) x (5) 3,n, w(3) n, x (3) 2,n, u 1,n and u 2,n as in [23]. The output is (26) y n = [ x 1,n, x 2,n, x 3,n, w n, x 1,n, x 2,n, x 3,n, w n, x 1,n, x 3,n, x (4) 1,n, x(4) 3,n, u 1,n, u 2,n] T. 1,n,

19 ONE-STEP 4-STAGE HBT DAE SOLVER OF ORDER For general k > 1, solve (G (k+4), H (k+4), D (k+2), F (k+2), E (k), K (k) ) = 0 for x (k+4) 1,n, x (k+4) 3,n, w n (k+2), x (k+2) 2,n, u (k) 1,n and u(k) 2,n as in [23]. Then the high-order derivatives y n (k+1), k 0, can be generated as [ y n (k+1) = x (k+1) 1,n, x (k+1) 2,n, x (k+1) 3,n, w n (k+1), x (k+2) 1,n, x (k+2) 2,n, x (k+2) 3,n, w (k+2) n, ] T x (k+3) 1,n, x (k+3) 3,n, x (k+4) 1,n, x (k+4) 3,n, u (k+1) 1,n, u (k+1) 2,n. Scheme 2 generates first derivative vectors (using the notation of [23]), (27) y n+c i = [x 1,n+c i, x 2,n+c i, x 3,n+c i, w n+c i, x 1,n+c i, x 2,n+c i, x 3,n+c i, w n+c i, x 1,n+c i, x 3,n+c i, x 1,n+c i, x 3,n+c i, u 1,n+c i, u 2,n+c i ] T, i = 2, 3, 4, at off-step points t n + c i h n+1. Once the vector (28) y n+c2 = [x 1,n+c2, x 2,n+c2, x 3,n+c2, w n+c2, x 1,n+c 2, x 2,n+c 2, x 3,n+c 2, w n+c 2, x 1,n+c 2, x 3,n+c 2, x 1,n+c 2, x 3,n+c 2, u 1,n+c2, u 2,n+c2 ] T, is obtained by P 2, then y n+c 2 Using known y n+c2, is computed as follows. At stage k = 0, solve Pryce s system (G (4), H (4), D (2), F (2), E, K) = 0 for x (4) 1,n+c 2, x (4) 3,n+c 2, w n+c 2, x 2,n+c 2, u 1,n+c2 and u 2,n+c2. At stage k = 1, solve Pryce s linear system (G (5), H (5), D (3), F (3), E (1), K (1) ) = 0 for x (5) 1,n+c 2, x (5) 3,n+c 2, w (3) n+c 2, x (3) 2,n+c 2, u 1,n+c 2 and u 2,n+c 2.

20 434 T. NGUYEN-BA ET AL. The output vector y n+c 2 is then (29) y n+c 2 = [x 1,n+c 2, x 2,n+c 2, x 3,n+c 2, w n+c 2, x 1,n+c 2, x 2,n+c 2, x 3,n+c 2, w n+c 2, x 1,n+c 2, x 3,n+c 2, x (4) 1,n+c 2, x (4) 3,n+c 2, u 1,n+c 2, u 2,n+c 2 ] T. The vector y n+c 2 is used to compute y n+c3 by means of P 3, (30) y n+c3 = [x 1,n+c3, x 2,n+c3, x 3,n+c3, w n+c3, x 1,n+c 3, x 2,n+c 3, x 3,n+c 3, w n+c 3, x 1,n+c 3, x 3,n+c 3, x 1,n+c 3, x 3,n+c 3, u 1,n+c3, u 2,n+c3 ] T. Then, y n+c 3 is computed using only stages k = 0 and k = 1 of Pryce s method as above with known y n+c3. The derivatives y n+c i, i = 2, 3, 4, can be computed by the above procedure. In Scheme 2, stages k = 4, 3, 2, 1 of Pryce s method are not used in order to keep the components of y n+ci intact when used to compute y n+c i. The numerical performance of HBT(12)4DAE, T12DAE and DP(8,7)DAE with constant step over t [0, 1.49] is compared on the above typical high-index DAE problem. The errors in the solution at time t are listed in Table 5. It is seen that HBT(12)4DAE and T12DAE perform better than DP(8,7)DAE. These numerical results also show that HBT(12)4DAE and DP(8,7)DAE can handle high-index problems as well as T12DAE. By observing the error at different values of t in Table 5, we can appreciate the high order of a DAE solver and the number of high-order derivatives added to a DAE solver. In this case, HBT(12)4DAE and T12DAE, which use higher-order derivatives, produce good results Comparing HBT(12)4DAE with DP(8,7)DAE, SPARK and PIRK Table 6 lists the errors at t f = 1, of HBT(12)4DAE, T12DAE, DP(8,7)DAE and the Lobatto IIIA-B-C-C*-D super partitioned additive Runge Kutta (SPARK) s = 3 method [14] for problem (24). Table 7 lists the maximum errors in x 1 (t) at meshpoints in [0, 1] obtained by HBT(12)4DAE, DP(8,7)DAE and the projected implicit Runge Kutta (PIRK) [1]. Here x 1 (t) is one of three variables of problem (23) with λ = 50.

21 ONE-STEP 4-STAGE HBT DAE SOLVER OF ORDER Global error at t of t HBT(12)4DAE T12DAE DP(8,7)DAE e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-08 TABLE 5: Global errors, at time t, in the solution of HBT(12)4DAE, T12DAE and DP(8,7)DAE for the index-5 problem. No. of Global errors at t f = 1 of : steps HBT(12)4DAE DP(8,7)DAE SPARK e e e e e e e e-12 > e-06 > e-10 > e-12 TABLE 6: Global errors, at t f and SPARK for problem (24). = 1, of HBT(12)4DAE, DP(8,7)DAE

22 436 T. NGUYEN-BA ET AL. No. of Maximum error at meshpoints in x 1 (t), t [0, 1] of : steps HBT(12)4DAE DP(8,7)DAE PIRK Gauss k = e e e e e e e e-11 TABLE 7: Maximum errors in x 1(t) at meshpoints in [0, 1] obtained by HBT(12)4DAE, DP(8,7)DAE and PIRK for problem (23). In Tables 6 and 7, we observe that the lower order DAE solvers at hand require a larger number of steps than the higher order HBT(12)4DAE. Hence, the tables show the superiority of HBT(12)4DAE over the other methods Comparing HBT(12)4DAE with Padé and Chebyshev series methods The numerical performances of HBT(12)4DAE, DP(8,7)DAE, and Padé and Chebyshev series derived for a DAE in [8] and [10] in solving problem (20) are compared on various constant steps over t [0, 1]. The MGEs listed in Table 8 show that HBT(12)4DAE performs better than DP(8,7)DAE, and Padé and Chebyshev series. It is also observed that HBT(12)4DAE, with five times the stepsize of the Padé and Chebyshev series, obtains very favorable results. No. of Maximum global error of : steps HBT(12)4DAE DP(8,7)DAE Padé series Chebyshev series e e e e e e e e e-01 TABLE 8: Maximum global error over t [0, 1] of HBT(12)4DAE, DP(8,7)DAE, Padé and Chebyshev series for problem (20).

23 ONE-STEP 4-STAGE HBT DAE SOLVER OF ORDER Conclusion A one-step 4-stage Hermite-Birkhoff-Taylor (HBT) method of order 12, called HBT(12)4 constructed in [20] for solving ODEs is coupled with a modification of Pryce s pre-structural analysis and automatic differentiation to solve DAE s. The new method is named HBT(12)4DAE. The stepsize is controlled by a local error estimator. On the basis of CPU time versus maximum global error, and number of steps versus the error at final time t f, HBT(12)4DAE and T12DAE behave similarly on several problems. On the other hand, HBT(12)4DAE wins over DP(8,7)DAE and other known DAE solvers in solving the problems at hand. HBT(12)4DAE is a member of variable-order one-step 4-stage HBT(p)4 methods of order p which appear to be promising DAE solvers in the light of the numerical results obtained in this paper. Moreover, these methods can be derived and implemented efficiently. Bozic [3, 4] has programmed a variable-step variable-order 3-stage HBT ODE solver in C++ to arbitrary high order and extended precision. REFERENCES 1. U. M. Ascher and L. R. Petzold, Projected implicit Runge-Kutta methods for differential-algebraic equations, SIAM J. Numer. Anal. 28(4) (1991), R. Barrio, Sensitivity analysis of ODEs/DAEs using the Taylor series method, SIAM J. Sci. Comput. 27(6) (2006), V. Bozic, Three-Stage Hermite-Birkhoff-Taylor ODE Solver with a C++ Program, M.Sc. Thesis, University of Ottawa, Ottawa, ON, V. Bozic, Multiple-precision 1-step 3-stage VSVO Hermite-Birkhoff-Taylor ODE solver, Appl. Math. Comput. 216 (2010), doi: /j.amc K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initialvalue Problems in Differential-Algebraic Equations, SIAM, Philadelphia, second edition, S. L. Campbell and E. Griepentrog, Solvability of general differential algebraic equations, SIAM J. Sci. Comput. 16(2) (1995), E. Çelik and M. Bayram, Arbitrary order numerical method for solving differential-algebraic equation by Padé series, Appl. Math. Comput. 137 (2003), E. Çelik and M. Bayram, Numerical solution of differential-algebraic equation systems and applications, Appl. Math. Comput. 154 (2004), E. Çelik and M. Bayram, The numerical solution of physical problems modeled as a systems of differential-algebraic equations (DAEs), J. Franklin Inst. 342 (2005), E. Çelik and T. Yeloglu, Chebyshev series approximation for solving differentialalgebraic equations (DAEs), Int. J. Comput. Math. 83(8-9) (2006), R. M. Corless and S. Ilie, Polynomial cost for solving IVP for high-index DAE, BIT 48 (2008),29 49.

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