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1 Electronic Transactions on Numerical Analysis. Volume 34, pp , Copyrigt 2008,. ISSN ETNA A NOTE ON NUMERICALLY CONSISTENT INITIAL VALUES FOR HIGH INDEX DIFFERENTIAL-ALGEBRAIC EQUATIONS CARMEN ARÉVALO Dedicated to Víctor Pereyra on te occasion of is 70t birtday Abstract. Wen differential-algebraic equations of index 3 or iger are solved wit backward differentiation formulas, te solution can ave gross errors in te first few steps, even if te initial values are equal to te exact solution and even if te stepsize is kept constant. Tis raises te question of wat are consistent initial values for te difference equations. Here we study ow to cange te exact initial values into wat we call numerically consistent initial values for te implicit Euler metod. Key words. ig index differential-algebraic equations, consistent initial values AMS subject classifications. 65L05 1. Introduction. A differential-algebraic equation (DAE) as te form F(t, x, ẋ) = 0, were te matrix F/ ẋ is singular. Here we sall consider te differential-algebraic equation of te form ṗ = U(t, v) v = F(t, p, v) + G(t, p, v)λ (1.1a) (1.1b) 0 = R(t, p), (1.1c) were t R, p R n, v R m, λ R s, and U : R R m R n, F : R R n R m R m, G : R R n R m R m s and R : R R n R s. Assume tat s min(m, n) in order to avoid an over-determined system. Te system (1.1) as index 3 if R/ p U/ v G is nonsingular for all t [t 0, T]. For simplicity we take t 0 = 0. Te algebraic variables appear linearly as in (1.1b) in important classes of pysical problems. Tis condition is fulfilled, for example, by te Euler-Lagrange equations of multibody mecanics, wic ave applications in biomecanics, te dynamics of macinery, robotics, and veicle design. To solve tis type of DAE, several tecniques ave been considered. One proposition as been to solve te system in its original formulation using a backward differentiation formula (BDF), as implemented in te DAE solver DASSL [4]; but suc a variable-step-size, variableorder code based on BDF metods presents some essential difficulties wen solving iger index DAEs, especially in te accuracy of te algebraic variables [2]. Te initial values (p 0, v 0, λ 0 ) are said to be consistent if te DAE as a differentiable solution (p(t), v(t), λ(t)) in te interval [0, T] suc tat (p(0), v(0), λ(0)) = (p 0, v 0, λ 0 ). Brenan and Engquist [3] defined numerically consistent starting values to order k + 1 for te k-step BDF applied to (1.1) as starting values [p k 1,..., p 0 ], [v k 1,..., v 0 ], [λ k 1,..., λ 0 ] Received Marc 31, Accepted August 6, Publised online on December 13, Recommended by Godela Scerer. Numerical Analysis, Centre for Matematical Sciences, Lund University, Box 118, SE Lund, Sweden. (Carmen.Arevalo@na.lu.se) 14

2 A NOTE ON NUMERICALLY CONSISTENT INITIAL VALUES 15 suc tat p j p(t j ) K 1 k+1 v j v(t j ) K 2 k+1 R(t j, p j ) K 3 k+2 (1.2a) (1.2b) (1.2c) for some constants K 1, K 2, K 3 and j = 0, 1,...,k 1. Tey proved tat te k-step BDF (k = 1,...,6) wit constant step size converges globally wit O( k ) accuracy to te solutions, but only after after k+1 steps, and provided te metod uses numerically consistent starting values to order k + 1. In particular, for te implicit Euler metod te state variables p and v ave O() accuracy after te first step, but te error in te algebraic variable λ is O(1), even wen te initial values are exact. By approximating te errors in te algebraic variable, a correction mecanism was devised in order to obtain O() accuracy even in te initial k steps [2]. Tese formulas correct te errors locally and produce O() accurate algebraic variables. Our view is tat tese O(1) errors are caused by initial values tat are inconsistent wit te difference equations. We attempt to redefine numerically consistent initial values as initial values tat are consistent wit te difference equations, as opposed to tose consistent wit te differential equation, as in (1.2). We explain te O(1) errors in te algebraic variables as a result of starting te BDF solver wit initial values tat are consistent wit te differential equation, but not wit te difference equations. Once tis is establised, we develop a sceme to construct numerically consistent initial values. We illustrate our results by solving te same problems tat appear in [1]. 2. Numerically consistent initial values. Initial values tat are numerically consistent wit te difference equation generated by an order p metod sould produce O( p ) accurate solutions. DEFINITION 2.1. Te set (x 0, x 1,..., x k 1 ) is a numerically consistent initial value at t = t 0 for a differential equation f(t, x, ẋ) = 0, solved wit a k-step multistep metod of order p, if te metod initiated wit (x 0, x 1,...,x k 1 ) generates te approximation x k suc tat x(t k ) x k = O( p ). (2.1) For differential-algebraic equations (DAEs) in semi-explicit form ẋ = f(t, x, λ) 0 = g(t, x, λ) of index less tan 2, consistent initial conditions are also numerically consistent; but for iger index DAEs, consistent initial conditions do not necessarily produce an approximation satisfying (2.1). For index 3 DAEs, in fact, BDF metods approximate te algebraic variables wit O(1) errors after te first step, even if te initial values are exact. 3. Euler-Lagrange equations and te implicit Euler formula. Te index 3 formulation of constrained multibody systems is ṗ = v M v = F(p, v) + G(p) T λ (3.1a) (3.1b) 0 = R(p), (3.1c)

3 16 C. ARÉVALO were M is te positive definite mass matrix, G = R/ p, and G(p)M 1 G(p) T is nonsingular. To simplify notation, we will denote G(p 1 ) by G 1 and F(p 1, v 1 ) by F 1. Wen te implicit Euler formula is applied to (3.1) using exact initial values, te algebraic variable λ 1 as an O(1) error ǫ λ = λ(t 1 ) λ 1 tat can be approximated wit O() accuracy by [2] δλ = (G 1 M 1 G T 1 ) 1 G 1 M 1 F 1 + λ 1. Te implicit Euler metod requires initial values only for state variables p and v. Taking te initial condition p 0 = p(0) + O( 3 ) v 0 = v(0) M 1 G T 1 δλ + O(2 ), (3.2a) (3.2b) te first step is p 1 = p 0 + v 1 v 1 = v 0 + (I B)M 1 F 1 + M 1 G T 1 δλ 0 = R(p 1 ), were te matrix B is dependent on p 1 and is defined by B = M 1 G T 1 (G 1 M 1 G T 1 ) 1 G 1. Te matrix B is a projector and B M 1 G T 1 = M 1 G T 1. We denote te global errors at t 1 by ǫ p = p(t 1 ) p 1, ǫ v = v(t 1 ) v 1 and ǫ λ = λ(t 1 ) λ 1. Te O() accurate correction term δλ may be expressed in terms of te exact value of te algebraic variable as δλ = 1 2 ((G 1M 1 G T 1 ) 1 G 1 M 1 F 1 + λ(t 1 )). By expanding p(t) and v(t) about t 1 and evaluating at t = 0 we obtain ǫ v = M 1 G T 1 (δλ + ǫλ ) + O( 2 ) ǫ p = ǫ v 2 2 M 1 (F(p(t 1 ), v(t 1 )) + G(p(t 1 )) T λ(t 1 )) + O( 3 ). Tese expressions lead to ǫ p = 2 2 M 1 (G T 1 (G 1 M 1 G T 1 ) 1 G 1 M 1 F 1 F(t 1 )) + 2 M 1 G T 1 ǫ λ + O( 3 ), and multiplying by G 1 we can conclude tat G 1 ǫ p = 2 G 1 M 1 G T 1 ǫλ + O( 3 ). If we expand R(p) about p 1 and use te fact tat ǫ p = O( 2 ) togeter wit R(p(t 1 )) = 0 and R(p 1 ) = 0, we finally obtain G 1 ǫ p = O( 4 ), tus implying tat ǫ λ = O(). In oter words, te initial condition (3.2) is numerically consistent for (3.1) solved wit te implicit Euler metod.

4 A NOTE ON NUMERICALLY CONSISTENT INITIAL VALUES 17 Te initial value (3.2b) may be written as Equation (3.3) in te form v 0 = v(0) + B(v(0) v 1 ) + O( 2 ). (3.3) v 0 v(0) + O( 2 ) = B (v 1 v(0)) (3.4) can be interpreted as an indication tat te initial values for v consistent wit te differential equation are also consistent wit te difference equation only in te case tat v(0) is in te null space of G 1. In practical terms, tis would require tat λ(t 0 ) = (G 0 M 1 G T 0 ) 1 G 0 M 1 F 0 + O(). As te cange in initial values for te state variable v is O(), it will not affect te accuracy of te solution in any of te state variables. 4. Te more general case. Te more general problem we consider ere is ṗ = U(t, q) q = F(t, p, q) + G(t, p, q)λ (4.1a) (4.1b) 0 = R(t, p). (4.1c) Tis is an index 3 DAE if R p U q G is non-singular. According to [2], te algebraic variable Λ is estimated by te first step of te implicit Euler metod wit an O(1) error ǫ Λ = Λ(t 1 ) Λ 1, wic is approximated wit O() accuracy by δλ = (R p U q G) 1 R p [U q (F + GΛ)U t ], were all functions are evaluated at t = t 1. Te first implicit Euler step is p 1 = p 0 + U(t 1, q 1 ) q 1 = q 0 + (F(t 1, p 1, q 1 ) + G(t 1, p 1, q 1 )Λ 1 ) 0 = R(t 1, p 1 ) wit numerically consistent initial condition p 0 = p(0) + O( 3 ) q 0 = q(0) GδΛ + O( 2 ). (4.2a) (4.2b) We may write (4.2b) as q 0 = q(0) AU q (q 1 q(0)) AU t + O( 2 ) wit A = G(R p U q G) 1 R p. Te matrix AU q is a projector wit AU q G = G. Te last equation can be written as q 0 q(0) = AU q q 1 q(0) AU t + O(), mirroring equation (3.4) wen U is independent of t. In tis case, te numerically consistent initial values for q are consistent wit te differential equation if AU q q = 0, tat is, only if q is in te null space of AU q. In practical terms, tis would require tat Λ(0) = (R p U q G) 1 R p U q F t=0.

5 18 C. ARÉVALO 5. Computational results. We illustrate our results wit te following two index 3 DAEs from [1]. PROBLEM 5.1. Te system of Euler-Lagrange equations as te solution ẍ = 2y + xλ ÿ = 2x + yλ (5.1a) (5.1b) 0 = x 2 + y 2 1 (5.1c) x = sin(1 + t) 2 y = cos(1 + t) 2 u = 2(1 + t)cos(1 + t) 2 v = 2(1 + t)sin(1 + t) 2 λ = 4(1 + t) 2. PROBLEM 5.2. Tis problem is defined in te form (4.1) wit p = [ x y z ] T q = [ u v w ] T Λ = [ λ β ] T U = [ 2u v w 1 ] T F = [ y 2x + y sin t 2 4yt 4zt sint 2] T [ ] T x 0 2z G = 0 2y 1 R = [ x 2 + y 2 + z 2 1 z 0.5 ] T. Its solution is 3 x = 2 cost2 y = 3 2 sin t2 z = u = 2 t sin t2 v = 3t cost 2 w = 1 λ = 2t 2 β = 0.5 sint 2. Problems 5.1 and 5.2 were solved wit implicit Euler using = and = After te first step te algebraic variable λ sows O(1) errors. Problem 5.1 was solved wit t 0 = 0, so G(P 0 ) T λ 0 = [sin(1) cos(1)] λ 0. Terefore, λ(0) would ave to be close to zero (but is instead equal to -4) for te exact initial values to be numerically consistent. Te numerically consistent initial values were calculated from (3.3) and te step was redone. Te exact initial values [ u v ] = [ ] were canged to [ ] for = and to [ ] for = Tis resulted in algebraic variables of O() accuracy (5.1). For Problem 5.2 wit t 0 = 1 and = 0.001, te initial values for q were canged from [ ] to [ ], producing algebraic variables of O() accuracy, as sown in Table 5.2. Te results presented in Figure 5.1 sow te O(1) errors wen Problem 5.1 was solved using exact initial values, and te O() errors in te algebraic variable obtained by using numerically consistent initial values.

6 A NOTE ON NUMERICALLY CONSISTENT INITIAL VALUES 19 TABLE 5.1 Absolute errors in te algebraic variable λ for Problem 5.1 using implicit Euler wit step sizes and O(1) errors and teir corrected values are displayed in bold face. = = t j numerically numerically exact IV consistent IV exact IV consistent IV TABLE 5.2 Absolute errors in te algebraic variable λ for Problem 5.2 using implicit Euler wit step sizes and O(1) errors and teir corrected values are displayed in bold face. = = t j numerically numerically exact IV consistent IV exact IV consistent IV Implicit Euler solution wit exact initial values log global error = = = time Implicit Euler solution wit numerically consistent initial values log global error = = = time FIG Implicit Euler used to solve an index 3 DAE test problem (see text) using numerically consistent initial values. Global errors wit exact initial values (top) and numerically consistent initial values (bottom). REFERENCES [1] C. ARÉVALO, Matcing te structure of DAEs and multistep metods, P. D. tesis, Department of Computer Science, Lund University, Lund, Sweden, [2] C. ARÉVALO AND P. LÖDSTEDT, Improving te accuracy of BDF metods for index 3 differential-algebraic equations, BIT, 35 (1995), pp [3] K. BRENAN AND B. ENGQUIST, Backward differentiation approximations of nonlinear differential/algebraic systems, Mat. Comp., 51 (1988), pp [4] L. PETZOLD, Differential/algebraic equations are not ODEs, SIAM J. Sci. Statist. Comput., 3 (1982), pp