Linear partial differential-algebraic equations (PDAEs) of the form
where and are constant matrices, , and what is more interesting the matrix is singular, what leads to the differential-algebraic problem.
In this case it is impossible to prescribe initial and boundary conditions for all components of the solution vector, they must fulfill certain conditions. A practical example how to do it is shown for the so called superconducting coil.
The problem of linear PDAEs is investigated from the numerical point of view. The discretization in space via finite differences is used and then a Runge-Kutta approximation of the method of lines for differential algebraic equations (MOL-DAE) is derived. When is the number of space grid points, by a regular transformation the MOL-DAE of dimension is decoupled into systems of dimension and the Weierstrass-Kronecker transformation for each of these systems into a system of ordinary differential equations and an algebraic system. The so called differential time index of the PDAE is introduced which gives the Runge-Kutta approximation to these subsystems. The convergence of -stable Runge-Kutta discretizations with constant step sizes is proved. The obtained order of convergence in time depends on the differential time index of the PDAE and on the boundary condition if they are homogeneous or not.
Included numerical examples confirm the theoretical results for the backward Euler method and 3-stage Radau IIA method.