TY - JOUR
T1 - Differential elimination-completion algorithms for DAE and PDAE
AU - Reid, Gregory J.
AU - Lin, Ping
AU - Wittkopf, Allan D.
N1 - dc.publisher: Wiley-Blackwell
PY - 2001
Y1 - 2001
N2 - Differential–algebraic equations (DAE) and partial differential–algebraic equations (PDAE) are systems of ordinary equations and PDAEs with constraints. They occur frequently in such applications as constrained multibody mechanics, spacecraft control, and incompressible fluid dynamics. A DAE has differential index r if a minimum of r+1 differentiations of it are required before no new constraints are obtained. Although DAE of low differential index (0 or 1) are generally easier to solve numerically, higher index DAE present severe difficulties. Reich et al. have presented a geometric theory and an algorithm for reducing DAE of high differential index to DAE of low differential index. Rabier and Rheinboldt also provided an existence and uniqueness theorem for DAE of low differential index. We show that for analytic autonomous first-order DAE, this algorithm is equivalent to the Cartan–Kuranishi algorithm for completing a system of differential equations to involutive form. The Cartan–Kuranishi algorithm has the advantage that it also applies to PDAE and delivers an existence and uniqueness theorem for systems in involutive form. We present an effective algorithm for computing the differential index of polynomially nonlinear DAE. A framework for the algorithmic analysis of perturbed systems of PDAE is introduced and related to the perturbation index of DAE. Examples including singular solutions, the Pendulum, and the Navier–Stokes equations are given. Discussion of computer algebra implementations is also provided.
AB - Differential–algebraic equations (DAE) and partial differential–algebraic equations (PDAE) are systems of ordinary equations and PDAEs with constraints. They occur frequently in such applications as constrained multibody mechanics, spacecraft control, and incompressible fluid dynamics. A DAE has differential index r if a minimum of r+1 differentiations of it are required before no new constraints are obtained. Although DAE of low differential index (0 or 1) are generally easier to solve numerically, higher index DAE present severe difficulties. Reich et al. have presented a geometric theory and an algorithm for reducing DAE of high differential index to DAE of low differential index. Rabier and Rheinboldt also provided an existence and uniqueness theorem for DAE of low differential index. We show that for analytic autonomous first-order DAE, this algorithm is equivalent to the Cartan–Kuranishi algorithm for completing a system of differential equations to involutive form. The Cartan–Kuranishi algorithm has the advantage that it also applies to PDAE and delivers an existence and uniqueness theorem for systems in involutive form. We present an effective algorithm for computing the differential index of polynomially nonlinear DAE. A framework for the algorithmic analysis of perturbed systems of PDAE is introduced and related to the perturbation index of DAE. Examples including singular solutions, the Pendulum, and the Navier–Stokes equations are given. Discussion of computer algebra implementations is also provided.
U2 - 10.1111/1467-9590.00159
DO - 10.1111/1467-9590.00159
M3 - Article
SN - 0022-2526
VL - 106
SP - 1
EP - 45
JO - Studies in Applied Mathematics
JF - Studies in Applied Mathematics
IS - 1
ER -