Efficient solution of large-scale, ill-conditioned and highly-indefinite algebraic equations often relies on high quality preconditioners together with iterative solvers. Because of their robustness, factorizationbased algorithms play a significant role in developing scalable solvers. We discuss the state-of-the-art, high performance sparse factorization techniques which are used to build spars...

Meta-S is a constraint solver cooperation system which allows the dynamic integration of arbitrary external (stand-alone) solvers and their combination with declarative languages. We sketch the main aspects of Meta-S including solver and language integration as well as its strategy definition framework for specifying solver cooperation and language evaluation strategies by means of an example. ...

A numerical method for variable coefficient elliptic problems on twodimensional domains is presented. The method is based on high-order spectral approximations and is designed for problems with smooth solutions. The resulting system of linear equations is solved using a direct solver with O(N1.5) complexity for the precomputation and O(N logN) complexity for the solve. The fact that the solver ...

We utilize the travelling-wave Ansatz to obtain novel analytical solutions linear diffusion–reaction equation. The reaction term is a function of time and space simultaneously, firstly in Lorentzian form secondly cosine form. new contain Heun functions first case Mathieu for second case, therefore are highly nontrivial. use these test some non-conventional explicit stable numerical methods agai...

Direct factorization methods for the solution of large, sparse linear systems that arise from PDE discretizations are robust, but typically show poor time and memory scalability for large systems. In this paper, we describe an efficient sparse, rank-structured Cholesky algorithm for solution of the positive definite linear system Ax = b when A comes from a discretized partial-differential equat...

Details of parallel-sparse Domain Decomposition (DD) with multi-point constraints (MPC) formulation are explained. Major computational components of the DD formulation are identified. Critical roles of parallel (direct) sparse and iterative solvers with MPC are discussed within the framework of DD formulation. Both symmetrical and unsymmetrical system of simultaneous linear equations (SLE) can ...