Numerical Methods for High-Index Differential-Algebraic Equations: Sparse, Stiff, and Hybrid Systems

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Modeling and simulation of physical systems is becoming highly automated. In environments such as Dymola, Simulink, Maplesim, and OpenModelica, a user can build a model by interconnecting prebuilt components into a network structure in a hierarchical way. When such a model is compiled, typically it results in a large, sparse system containing both differential and algebraic equations, or DAEs. The index of a DAE measures how difficult it is to solve it compared to solving an ordinary differential equation, which has index 0. The higher the index, the harder is a numerical solution by standard methods. Established DAE software can solve index-1 problems without difficulty, and some special forms of index 2 and 3. Frequently, DAEs of index 2 and higher arise, and the common approach is to perform index reduction or remodel the problem to arrive to an index-1 problem.The applicant has been working on solving numerically initial-value problems of any index, arbitrary order DAEs. This has resulted in the C++ solver DAETS for solving such problems and in the DAESA Matlab tool for structural analysis (SA) of DAEs. An advantage of treating DAE systems directly, rather then by index reduction and then integrating an index-1 system, is that one can generate and simulate models in their natural form, without worrying about the numerical difficulties arising when solving high-index problems. Moreover, due to its generality, our approach could simplify the automatic generation of equations by modeling software.Although the technology implemented in DAETS has been shown to be very accurate and capable of solving high-index problems, with index 47 being the highest attempted, to solve industrial-size problems, it must be extended in several key directions, which are the main objectives of the proposed research program. They include developing methods for parallel solution of large sparse systems of DAEs, stiff high-index DAEs, and hybrid systems, and improving the underlying structural analysis.Stiff systems contain both very slow and very fast components and arise e.g. in modeling chemical reactions, multibody dynamics with contact, joints and friction, and electronic circuits. Their efficient solution requires a method that permits large step sizes, and our goal is to construct such within our framework for solving high-index DAEs.Many complex dynamical systems are naturally modeled by systems of DAEs, in which the continuous model may change at discrete points in time, when discrete events occur. Such systems are called hybrid or switched systems. They arise when modeling e.g. electric circuits and mechanical systems such as robots and gear boxes. Our goal is to solve numerically hybrid systems whose continuous behavior is described by high-index DAEs.Before a numerical simulation, typically some form of SA is applied. Although the widely used Pantelides's algorithm and the more general Pryce's method determine correctly structural data on many problems of interest, there are problems arising in practice in which the SA fails. We shall investigate the reasons for failure and search for heuristics to transform a DAE problem on which it fails into a form on which it succeeds.With the success of the proposed research program, we will be able to solve large systems of DAEs without restrictions on their index. Success in improving the SA can have a direct impact on simulation environments by widening the class of problems they can solve. Furthermore, we aim to extend DAETS to an industrial-strength solver. Provided that we can solve large systems efficiently, we expect to influence the way in which systems of equations are produced by modeling software, since we can solve them without any index or order reduction.