Partial differential algebraic equation

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In mathematics a partial differential algebraic equation (PDAE) set is an incomplete system of partial differential equations that is closed with a set of algebraic equations.

Definition[edit]

A general PDAE is defined as:

0 = \mathbf F \left( \mathbf x, \mathbf y, \frac{\partial y_i}{\partial x_j}, \frac{\partial^2 y_i}{\partial x_j \partial x_k}, \ldots, \mathbf z \right),

where:

  • F is a set of arbitrary functions;
  • x is a set of independent variables;
  • y is a set of dependent variables for which partial derivatives are defined; and
  • z is a set of dependent variables for which no partial derivatives are defined.

The relationship between a PDAE and a partial differential equation (PDE) is analogous to the relationship between an ordinary differential equation (ODE) and a differential algebraic equation (DAE).

PDAEs of this general form are challenging to solve. Simplified forms are studied in more detail in the literature.[1][2][3] Even as recently as 2000, the term "PDAE" has been handled as unfamiliar by those in related fields.[4]

Solution methods[edit]

Semi-discretization is a common method for solving PDAEs whose independent variables are those of time and space, and has been used for decades.[5][6] This method involves removing the spatial variables using a discretization method, such as the finite volume method, and incorporating the resulting linear equations as part of the algebraic relations. This reduces the system to a DAE, for which conventional solution methods can be employed.

References[edit]

  1. ^ Wagner, Y. 2000. "A further index concept for linear PDAEs of hyperbolic type," Mathematics and Computers in Simulation, v. 53, pp. 287–291.
  2. ^ W. S. Martinson, P. I. Barton. (2002) "Index and characteristic analysis of linear PDAE systems," Siam Journal on Scientific Computing, v. 24, n. 3, pp. 905–923.
  3. ^ Lucht, W.; Strehmel, K.. 1998. "Discretization based indices for semilinear partial differential algebraic equations," Applied Numerical Mathematics, v. 28, pp. 371–386.
  4. ^ Simeon, B.; Arnold, M.. 2000. "Coupling DAEs and PDEs for simulating the interaction of pantograph and catenary," Mathematical and Computer Modelling of Dynamical Systems, v. 6, pp. 129–144.
  5. ^ Jacob, J.; Le Lann, J; Pinguad, H.; Capdeville, B.. 1996. "A generalized approach for dynamic modelling and simulation of biofilters: application to waste-water denitrification," Chemical Engineering Journal, v. 65, pp. 133–143.
  6. ^ de Dieuvleveult, C.; Erhel, J.; Kern, M.. 2009. "A global strategy for solving reactive transport equations," Journal of Computational Physics, v. 228, pp. 6395–6410.