Solving nonlinear systems of PDEs without linearising the equations
Faculty of Sciences. Mathematics and Computer Science
Edinburgh , 2000
5th International Conference on Computational Structures Technology/2nd International Conference on Engineering Computational Technology, September 6-8, 2000, Leuven, Belgium
The numerical study and exploration of new classes of systems of partial differential equations (PDE) or new constitutive equations is often hampered by the development cost of building a new solver or extending existing ones. In this paper is it is demonstrated how automatic differentiation techniques can be used to relieve the user of the burden of manually linearising PDE systems without severely deteriorating computational efficiency. A multiple point AD-tool  based on operator overloading is built into a class hierarchy for the solution of quasilinear PDE systems by means of the least-squares finite element method (LSFEM) and classical Galerkin FEM. Whereas classical solution techniques for nonlinear PDE systems require the programming of linearised terms, these classes require only the nonlinear terms which are generally smaller in number and complexity than the linearised ones. Hence the amount of programming and debugging is drastically reduced. The linearisation is carried out automatically "under the hood" by the AD-tool. In addition no extra programming is needed to implement successive substitution and the user can switch easily at runtime between Newton Raphson linearisation and successive substitution. The described class hierarchies are ideal candidates for the exploration of new PDE based mathematical models and for the prototyping of new solvers. The LSFEM class hierarchy is especially interesting since it also performs the discretisation automatically and can be used irrespective of the mathematical type of the PDE system. It is also demonstrated that solvers for manually linearised PDE systems, when implemented using a multiple point AD-tool, will exhibit an improved computational efficiency and that new constitutive equations can be explored with a reduced programming effort. Also, two linearisation schemes (Newton-Raphson and successive substition) are obtained at once.