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:

${\displaystyle 0=\mathbf {F} \left(\mathbf {x} ,\mathbf {y} ,{\frac {\partial y_{i}}{\partial x_{j}}},{\frac {\partial ^{2}y_{i}}{\partial x_{j}\partial x_{k}}},\ldots ,\mathbf {z} \right),}$

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.