There are numerous mathematical models in science and engineering which lead to a special class of time depending partial differential equations. This requires the development of effective and robust integration methods as well as to consider new problem classes. In the first part of this thesis a class of linearly implicit splitting methods for the numerical solution of higher space-dimensional parabolic initial-boundary value problems is developed. Following the method of lines the numerical solution of the system of ordinary differential equations arising from the semi-discretisation in space is considered. This system is large and stiff for small step-sizes in space and has a special structure. Using this structure splitting methods reduce the computational costs. In the thesis the stability of linear splitting methods is investigated and new stability concepts for linear splitting methods are given. The introduced new class of linearly implicit splitting methods have good stability properties, are consistent of classical order two and are easy to implement, where only linear equation systems with tridiagonal coefficient matrices have to be solved. Numerical examples illustrate the efficiency of the methods. In the second part of the thesis a special class of systems of partial differential equations is considered, which consist of a coupling of equations of different types. For example, timedepending partial differential equations are coupled with DAEs or ODEs or algebraic equations. These systems are also called partial differential algebraic equations (PDAEs). In this thesis linear PDAEs with constant coefficient matrices of the form Au(t,x)+Bu(t,x)+Cu(t,x)=g(t,x) mit A, B, C ∈ R^nxn are investigated. Hereby, at least one of the matrices A, B is singular. For a unique solvability these systems must be supplemented by suitable initial conditions and boundary conditions. Of importance for the treatment of PDAEs is the fact that, in contrast to systems with regular matrices A; B, for singular matrices A and/or B not for all components of the solution vector u initial and/or boundary conditions can be prescribed. Under some assumptions the linear PDAE can be reduced by a Laplace transformation or by a Fourier analysis to a sequence of differential algebraic systems (DAEs). On this basis a differential space index and an uniform differential time index for linear PDAEs are defined. These indexes are used to characterise the PDAE in several contexts. Furthermore, two numerical schemes for solving initial boundary value problems of linear PDAEs by means of the method of lines (MOL) are investigated. It is shown that there is a strong dependence of the order of convergence on these indexes. We present examples for the calculation of the order of convergence and give results of numerical calculations for several aspects encountered in the numerical solution of PDAEs.