In this paper, we consider systems of algebraic and non-linear partial differential equations and inequations. We decompose these systems into so-called simple subsystems and thereby partition the set of solutions. For algebraic systems, simplicity means triangularity, square-freeness and nonvanishing initials. Differential simplicity extends algebraic simplicity with involutivity. We build upon the constructive ideas of J. M. Thomas and develop them into a new algorithm for disjoint decomposition. The given paper is a revised version of Bächler et al. (2010) and includes the proofs of correctness and termination of our decomposition algorithm. In addition, we illustrate the algorithm with further instructive examples and describe its Maple implementation together with an experimental comparison to some other triangular decomposition algorithms.Keywords: disjoint triangular decomposition, simple systems, polynomial systems, differential systems, involutivity input by means of regular chains (if the input only consists of equations) or regular systems. However, the Thomas decomposition differs noticeably from this decomposition, since the Thomas decomposition is finer and demands disjointness of the solution set. For a detailed description of algorithms related to regular chains, we refer the reader to Moreno Maza (1999).The disjointness of the Thomas decomposition combined with the structural properties of simple systems provide a useful platform for counting solutions of polynomial systems. In fact, the Thomas decomposition is the only known method to compute the counting polynomial introduced by Plesken (2009a). We refer to §2.3 for details on this structure, counting and their applications.During his research on triangular decomposition, Thomas was motivated by the Riquier-Janet theory (cf. Riquier (1910); Janet (1929)), extending it to non-linear systems of partial differential equations. For this purpose he developed a theory of (Thomas) monomials, which generate an involutive monomial division nowadays called Thomas division (cf. Gerdt and Blinkov (1998a)). He gave a recipe for decomposing a non-linear differential system into algebraically simple and passive subsystems (cf. Thomas (1937)). A modified version of the differential Thomas decomposition was considered by Gerdt (2008) with its link to the theory of involutive bases (cf. Gerdt and Blinkov (1998a);Gerdt (2005Gerdt ( , 1999; Seiler (2010)). In this decomposition, the output systems are Janet-involutive in accordance to the involutivity criterion from Gerdt (2008) and hence they are coherent. For a linear differential system it is a Janet basis of the corresponding differential ideal, as computed by the Maple package Janet (cf. Blinkov et al. (2003)).The differential Thomas decomposition differs from that computed by the Rosenfeld-Gröbner algorithm (cf. Boulier et al. (2009, 1995)). The latter decomposition forms a basis of the diffalg, DifferentialAlgebra and BLAD packages (cf. Hubert (1996-2004); Boulier (2004Boulier ( -2009). Experimentally, we f...

Abstract. In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring R, i.e., a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over R. For a finitely generated maximal ideal m in a commutative ring R we show how solving (in)homogeneous linear systems over R m can be reduced to solving associated systems over R. Hence, the computability of R implies that of R m . As a corollary we obtain the computability of the category of finitely presented R m -modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a by-product, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Mora's algorithm.

In this paper we consider disjoint decomposition of algebraic and non-linear partial differential systems of equations and inequations into so-called simple subsystems. We exploit Thomas decomposition ideas and develop them into a new algorithm. For algebraic systems simplicity means triangularity, squarefreeness and non-vanishing initials. For differential systems the algorithm provides not only algebraic simplicity but also involutivity. The algorithm has been implemented in Maple

We combine algebraic and geometric approaches to general systems of algebraic ordinary or partial differential equations to provide a unified framework for the definition and detection of singularities of a given system at a fixed order. Our three main results are firstly a proof that even in the case of partial differential equations regular points are generic. Secondly, we present an algorithm for the effective detection of all singularities at a given order or, more precisely, for the determination of a regularity decomposition. Finally, we give a rigorous definition of a regular differential equation, a notion that is ubiquitous in the geometric theory of differential equations, and show that our algorithm extracts from each prime component a regular differential equation. Our main algorithmic tools are on the one hand the algebraic resp. differential Thomas decomposition and on the other hand the Vessiot theory of differential equations.

SUMMARY We present an eigen‐decomposition of the quasi‐linear convective flux formulation of the completely coupled Reynolds‐averaged Navier–Stokes and kω turbulence model equations. Based on these results, we formulate different approximate Riemann solvers that can be used as numerical flux functions in a DG discretization. The effect of the different strategies on the solution accuracy is investigated with numerical examples. The actual computations are performed using a p‐multigrid algorithm. To this end, we formulate a framework with a backward‐Euler smoother in which the linear systems are solved with a general preconditioned Krylov method. We present matrix‐free implementations and memory‐lean line‐Jacobi preconditioners and compare the effects of some parameter choices. In particular, p‐multigrid is found to be less efficient than might be expected from recent findings by other authors. This might be due to the consideration of turbulent flow. Copyright © 2012 John Wiley & Sons, Ltd.