Partial differential algebraic equation

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

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

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

^ Wagner, Y. 2000. "A further index concept for linear PDAEs of hyperbolic type," Mathematics and Computers in Simulation, v. 53, pp. 287–291.
^ 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.
^ Lucht, W.; Strehmel, K.. 1998. "Discretization based indices for semilinear partial differential algebraic equations," Applied Numerical Mathematics, v. 28, pp. 371–386.
^ 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.
^ 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.
^ 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.

[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] Dieuvleveult, C.; Erhel, J.; Kern, M.. 2009. "A global strategy for solving reactive transport equations," Journal of Computational Physics, v. 228, pp. 6395–6410.

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