A Differentiation Index for Partial Differential-Algebraic Equations

A differentiation index for nonlinear partial differential-algebraic equations is presented. Determination of the differentiation index with respect to a direction in the space of independent variables uncovers all equations that must be satisfied by Cauchy data on the hyperplane orthogonal to that direction. This index is thus a generalization of the differentiation index of differential-algebraic equations. The motivation for this work is consistent initialization of partial differential-algebraic systems. The analysis is demonstrated through several examples, including some taken from fluid dynamics and chemical engineering.

[1]  John B. Fraleigh.  A first course in abstract algebra , 1967 .

[2]  C. Doering,et al.  Applied analysis of the Navier-Stokes equations: Index , 1995 .

[3]  S. Campbell.  Singular Systems of Differential Equations , 1980 .

[4]  C. Pantelides.  The consistent intialization of differential-algebraic systems , 1988 .

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

[6]  溝畑 茂,et al.  The theory of partial differential equations , 1973 .

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

[8]  H. Elmqvist,et al.  Automated formula manipulation supports object-oriented continuous-system modeling , 1993, IEEE Control Systems.

[9]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

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

[11]  Wade S. Martinson,et al.  Index and Characteristic Analysis of Partial Differential Equations , 2000 .

[12]  Edward J. Longwell.  Dynamic modeling for process control and operability , 1994 .

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

[14]  Eustathios S. Kikkinides,et al.  Simultaneous sulfur dioxide/nitrogen oxide (NOx) removal and sulfur dioxide recovery from flue gas by pressure swing adsorption , 1991 .

[15]  Ernst Hairer,et al.  The numerical solution of differential-algebraic systems by Runge-Kutta methods , 1989 .

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

[17]  Vincent G. Grassi.  Dynamic simulation as a tool to integrate process design and process control , 1993 .

[18]  Alan Jeffrey,et al.  Quasilinear hyperbolic systems and waves , 1976 .