Differential-algebraic equations index

Gear, C.W. Differential-algebraic equations index transformations. SIAM J. Sci. 

Brenan, K. E., and Petzold, L. R., The numerical solution of higher index differential/algebraic equations by implicit Runge-Kutta methods," UCRL-95905, preprint, Lawrence Livermore National Laboratories, Livermore, California (1987). 

Gritsis, D., Pantelides, C. C., and Sargent, R. W. H., The dynamic simulation of transient systems described by index two differential-algebraic equations, Proc. Third International PSE Symposium, Sydney, Australia, p. 132 (1988). 

Gritsis, D., The Dynamic Simulation and Optimal Control of Systems Described by Index Two Differential-algebraic Equations. PhD. Thesis, (Imperial College, University of London, 1990). 

Mattsson, S., Sdderlind, G. (1993). Index reduction in differential algebraic equations using dummy derivatives. SIAM Journal on Scientific Computing, 14(3), 677-692. 

Practically speaking, the index of a DAE system is an integer that represents the minimum number of differentiations of (at least part oO the DAE system (with respect to the independent variable) that reduces the DAE to a pure ODE system for the original algebraic and differential variables. Based on this definition, pure ODE systems are index 0. Eor index 1 systems, differentiating algebraic equations (14.3) with fixed values of u t) and p become... 

The operating point(s) [x z ] can be determined for prescribed input values u by solving (1-2) with x = 0 which means the solution of an algebraic system of equations. A necessary condition on the solvability of the system of equations above is that the number of differential (algebraic) equations equals to the number of differential (algebraic) variables (degree of freedom equals to zero), and tiie original DAE system has differential index 1. 

Differential-algebraic equations (DAEs) differ in main aspects from explicit or regularly implicit ordinary differential equations. This concerns theory, e.g. solvability and representation of the solution, as well as numerical aspects, e.g. convergence behavior under discretization. Both aspects depend essentially on the index of the DAE. Thus, we first define the index. 

Equation (6.4.3) together with the differential equation (6.4.2) can be interpreted as a system of differential-algebraic equations of index 2 for x,e) ... 

HIM97] Hanke M., Izquierdo Macana E., and Marz R. (1997) On asymptotics in case of linear index-2 differential-algebraic equations. Technical Report 97-3, Humboldt-Universitat Berlin, Institut fiir Mathematik, D-10088 Berlin. 

Note that the system (2.45) is a DAE system of nontrivial index, since z cannot be evaluated directly from the algebraic equations. A solution for the variables z must be obtained by differentiating the constraints k(x) = 0. For most chemical processes, such as reaction networks (Gerdtzen et al. 2004), reactive distillation processes (Vora 2000), and complex chemical plants (Kumar and Daoutidis 1999a), the z variables can be obtained after just one differentiation of the algebraic constraints ... 

Definition A.5. The index yj of the DAE system in Equation (A.10) with specified smooth inputs u(t) is the minimum number of times the algebraic equations or their subset have to be differentiated to obtain a set of differential equations for z, i.e., in order to be able to solve z = iF(x, z,t) for z. 

The initialisation of variables in a system of equations is important. While, in systems of ODEs all of the state variables must be initialised, in DAE systems only some of the variables need to be initialised, which is the same as the number of differential variables for index one system. The other variables can be determined using the algebraic equations. It is inconvenient for the user to be required to initialise all of the variables as this might require the solution of a set of nonlinear algebraic equations. Pantelides (1988) developed a procedure for consistent initialisation of DAE systems. Readers are directed to this reference for further details. 

The multi-mode model for a tubular reactor, even in its simplest form (steady state, Pet 1), is an index-infinity differential algebraic system. The local equation of the multi-mode model, which captures the reaction-diffusion phenomena at the local scale, is algebraic in nature, and produces multiple solutions in the presence of autocatalysis, which, in turn, generates multiplicity in the solution of the global evolution equation. We illustrate this feature of the multi-mode models by considering the example of an adiabatic (a = 0) tubular reactor under steady-state operation. We consider the simple case of a non-isothermal first order reaction... 

The index of a DAE is the number of times that the algebraic equations must be differentiated with respect to t to determine dyldt 31. An ODE, with no algebraic relations, has index zero. Differentiating Equation 4.125 with respect to time produces... 

Consider the algebraic equations appearing in Table 4.2 for the nonconstant-density cases 1. reactor volume constant, 2. reactor mass constant, and 3. Q specified. Show that, as expressed in Table 4.2, cases 1. and 2. are high-index DAEs and case 3. is index one. Without differentiating the equation of state, how can case 2. be modified to be index one Can you find a simple way to modify case 1. to be index one ... 

In [42], van Dijk has shown that the determinant of det(E ) = det(I — A22), is non-zero for bond graphs with causal paths between resistive ports. That means that the inverse of E exists and that differentiation of the algebraic equation (2.18) is sufficient to transform the DAE system (2.17) into a set ODEs. Accordingly, (2.17) is a DAE system of index 1. 

The form of Eq. (6) depends on the characterisation of the input and output variables as determined by the specifications of the selected control structure. A difficulty that arises in this point is related to the index of the differential-algebraic set of equations in Eq. (4) due to the selection of the input-output structure. 

It is possible and beneficial to reduce the system to index-one by replacing A with a new dependent variable , where A = 3/31 [13], The initial condition for is arbitrary, since itself never appears in the equations—a suitable choice is 4> = 0. Anywhere A appears, it is simply replaced with 3/31, which is conveniently done in the DAE software interface. The index reduction can be seen from the following procedure The continuity at the inlet boundary (an algebraic constraint) can be differentiated once with respect to t to yield an equation for dV/dt. Then dV/dt is replaced by substitution of the radial-momentum equation. This substitution introduces A = 34>/31, which makes the continuity equation (at the inlet boundary) an independent differential equation for 4>. Thus the modified system is index-one. This set of substitutions is not actually done in practice—it simply must be possible to do them to achieve the index reduction. 

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