On Numerical Differential Algebraic Problems with Application to Semiconductor Device Simulation

This paper considers questions of conditioning of and numerical methods for certain differential algebraic equations subject to initial and boundary conditions. The approach taken is that of separating ``differential'' and ``algebraic'' solution components, at least theoretically. This yields conditioning results for differential algebraic boundary value problems in terms of ``pure'' differential problems, for which existing theory is well-developed. We carry the process out for problems with (global) index 1 or 2. For semi-explicit boundary value problems of index 1 (where solution components are separated) we give a convergence theorem for a special class of collocation methods. For general index 1 problems we discuss advantages and disadvantages of certain symmetric difference schemes. For initial value problems with index 2 we discuss the use of BDF schemes, summarizing conditions for their successful and stable utilization. Finally, the present considerations and analysis are applied to two problems involving differential algebraic equations which arise in semiconductor device simulation.