Index reduction for differential–algebraic equations by substitution method

Abstract Differential–algebraic equations (DAEs) naturally arise in many applications, but present numerical and analytical difficulties. The index of a DAE is a measure of the degree of numerical difficulty. In general, the higher the index is, the more difficult it is to solve the DAE. Therefore, it is desirable to transform the original DAE into an equivalent DAE with lower index. In this paper, we propose an index reduction method for linear DAEs with constant coefficients. The method is applicable to any DAE having at most one derivative per equality. In contrast to the other existing methods, it does not introduce any additional variables. Exploiting a combinatorial property of degrees of minors in polynomial matrices, we show that the method always reduces the index exactly by one. Thus the paper exhibits an application of combinatorial matrix theory to numerical analysis of DAEs.

[1]  V. Mehrmann,et al.  Index reduction for differential‐algebraic equations by minimal extension , 2004 .

[2]  Satoru Iwata,et al.  Computing the Maximum Degree of Minors in Matrix Pencils via Combinatorial Relaxation , 1999, SODA '99.

[3]  M. Günther,et al.  The DAE-index in electric circuit simulation , 1995 .

[4]  Pawel Bujakiewicz,et al.  Maximum weighted matching for high index differential algebraic equations , 1994 .

[5]  C. W. Gear,et al.  Differential-Algebraic Equations , 1984 .

[6]  R. Sargent,et al.  The mathematical modelling of transient systems using differential-algebraic equations , 1988 .

[7]  Sven Erik Mattsson,et al.  Index Reduction in Differential-Algebraic Equations Using Dummy Derivatives , 1993, SIAM J. Sci. Comput..

[8]  Satoru Iwata,et al.  Primal-Dual Combinatorial Relaxation Algorithms for the Maximum Degree of Subdeterminants , 1996, SIAM J. Sci. Comput..

[9]  C. W. Gear,et al.  ODE METHODS FOR THE SOLUTION OF DIFFERENTIAL/ALGEBRAIC SYSTEMS , 1984 .

[10]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

[11]  Peter J. Gawthrop,et al.  Systematic construction of dynamic models for phase equilibrium processes , 1991 .

[12]  L. Mirsky,et al.  The Theory of Matrices , 1961, The Mathematical Gazette.

[13]  Linda R. Petzold,et al.  Numerical solution of initial-value problems in differential-algebraic equations , 1996, Classics in applied mathematics.

[14]  Werner C. Rheinboldt,et al.  Nonholonomic motion of rigid mechanical systems from a DAE viewpoint , 1987 .

[15]  W. Marquardt,et al.  Structural analysis of differential-algebraic equation systems—theory and applications , 1995 .

[16]  Rafiqul Gani,et al.  Modelling for dynamic simulation of chemical processes: the index problem , 1992 .

[17]  Kazuo Murota,et al.  Matrices and Matroids for Systems Analysis , 2000 .

[18]  C. Tischendorf,et al.  Structural analysis of electric circuits and consequences for MNA , 2000 .

[19]  C. W. Gear,et al.  Differential-algebraic equations index transformations , 1988 .

[20]  Steffen Schulz,et al.  Four Lectures on Differential-Algebraic Equations , 2003 .

[21]  P. Rentrop,et al.  Differential-algebraic Equations in Vehicle System Dynamics , 1991 .

[22]  Albert E. Ruehli,et al.  The modified nodal approach to network analysis , 1975 .

[23]  Kazuo Murota,et al.  Combinatorial relaxation algorithm for the maximum degree of subdeterminants: Computing Smith-Mcmillan form at infinity and structural indices in Kronecker form , 1995, Applicable Algebra in Engineering, Communication and Computing.

[24]  C. W. Gear,et al.  Simultaneous Numerical Solution of Differential-Algebraic Equations , 1971 .

[25]  C. W. Gear,et al.  The index of general nonlinear DAEs , 1995 .

[26]  V. Mehrmann,et al.  Canonical forms for linear differential-algebraic equations with variable coefficients , 1994 .

[27]  E. Hairer,et al.  Solving Ordinary Differential Equations I , 1987 .

[28]  Roswitha März,et al.  Numerical methods for differential algebraic equations , 1992, Acta Numerica.