Implicit Riquier Bases for PDAE and their semi-discretizations
Complicated nonlinear systems of PDE with constraints (called PDAE) arise frequently in applications. Missing constraints arising by prolongation (differentiation) of the PDAE need to be determined to consistently initialize and stabilize their numerical solution. In this article we review a fast prolongation method, a development of (explicit) symbolic Riquier Bases, suitable for such numerical applications. Our symbolic-numeric method to determine Riquier Bases in implicit form, Without the unstable eliminations of the exact approaches, applies to square systems which are dominated by pure derivatives in one of the independent variables. The method is successful provided the prolongations with respect to a single dominant independent variable have a block structure which is uncovered by Linear Programming and certain Jacobians are nonsingular when evaluated at points on the zero sets defined by the functions of the PDAE. For polynomially nonlinear PDAE, homotopy continuation methods from Numerical Algebraic Geometry can be used to compute approximations of the points. Our method generalizes Pryce's method for DAE to PDAE. Given a dominant independent time variable, for an initial value problem for a system of PDAE we show that its semi-discretization is also naturally amenable to our symbolic-numeric approach. In particular, if our method can be successfully applied to such a system of PDAE, yielding an implicit Riquier Basis, then under modest conditions, the semi-discretized system of DAE is also an implicit Riquier Basis. (C) 2008 Elsevier Ltd. All rights reserved.
riquier bases semi-discretization linear programming numerical algebraic geometry partial differential algebraic equation jet spaces ranking implicit function theorem method of lines algorithms design