Implicit Riquier bases for PDAE and their semi-discretizations
Systems of nonlinear partial differential equations with constraints (called PDAEs or partial differential algebraic equations) arise frequently in applications. Missing constraints appearing by prolongation of the corresponding PDAE have to be determined to initialize in a consistent way and to stabilize its numerical solution. In this paper, a symbolic-numeric approach to computing Riquier bases is discussed, which can be applied to the approximate solution of an initial boundary value problem defined by a certain class of PDAEs. The new procedure determines Riquier bases in implicit form, without the unstable eliminations which are common to other exact approaches, and can be applied to square systems of analytic PDAEs dominated by pure derivatives in one independent variable. When a discretization method is applied to the PDAE (i.e., finite differences or finite elements), one ends up with a differential algebraic equation (DAE) to which one of the many existing numerical solvers can be applied. In this sense, the methods developed in this paper provide a bridge between DAEs and PDAEs techniques. In fact, an interesting question posed (but not answered) in the paper is whether the structure analysis of the resulting DAE agrees with the structure analysis of the original PDAE. If this were the case, the discretization step and structure analysis may commute, and thus a very efficient approach would be to apply first the fast prolongation method developed here to the PDAE and then discretize the prolonged PDAE to yield a DAE system without hidden constraints.
partial differential algebraic equations
scientific article; zbMATH DE number 5566496