Browsing by Author "Keenan, Philip T."

Now showing 1 - 7 of 7
  • Thumbnail Image
    Item
    A Remark on Collocation and Upwinding in First Order Hyperbolic Systems
    (1992-11) Keenan, Philip T.
    Keenan[2] defines and analyzes a new numerical method for coupled systems of nonlinear first order hyperbolic partial differential equations with one degenerate eigenvalue. That work extends in a certain direction the collocation method described by Luskin[3], which applies to systems with all the eigenvalues uniformly bounded away from zero. Luskin's method and Keenan's method both have direct application to the study of one dimensional fluid flow through pipelines. The pressure and velocity of an isothermal fluid in a pipeline can be described by a coupled pair of nonlinear first order hyperbolic partial differential equations. When thermal effects are important a third equation for temperature is added. While Luskin's method works well for the isothermal situation he discussed, it does not apply in certain common cases when thermal effects are modeled. The analysis of the new method shows how the difficulties that come from the application of standard collocation can be overcome by using upwinded piecewise constant functions for the degenerate component of the solution. Experiments indicate that this method is a substantial improvement over standard collocation. A number of technical details obscure the analysis presented in Keenan [2], because that work treats the general nonlinear case. The present paper describes and analyzes the method in the context of a linear, constant coefficient system of equations. In this special case the proof simplifies considerably.
  • Thumbnail Image
    Item
    An Efficient Postprocessor for Velocities from Mixed Methods on Triangular Elements
    (1994-05) Keenan, Philip T.
    Certain finite difference methods on rectangular grids for second order elliptic equations are known to yield superconvergent flux approximations. A class of related finite difference methods have recently been defined for triangular meshes by applying special quadrature rules to an extended version of a mixed finite element method [1]; the flux vector fields from these methods are not superconvergent. This report presents empirical evidence indicating that a simple local postprocessing technique recovers higher order accurate vector velocities at element centers on many meshes of triangular elements, with approximately second order accuracy on three lines meshes.
  • Thumbnail Image
    Item
    C++ and Fortran 77 Timing Comparisons
    (1993-01) Keenan, Philip T.
    Recently there has been considerable debate within the scientific computation community over the suitability of C++ for large scale numerical computation. This note reports on timing studies of Fortran 77 and C++ conducted on the Intel IPSC/3 Hypercube, the IBM RS-6000 and the Sun Sparc Station 2. Timings are presented for two fundamental algorithms including a dense vector inner product and multiplication of a dense vector by a sparse matrix. Comparison to hand coded assembler routines is also provided in selected cases. The results demonstrate that C and C++ can be just as efficient as FORTRAN and therefore deserve serious consideration.
  • Thumbnail Image
    Item
    Conventions for Using PIERS
    (1991-12) Keenan, Philip T.
  • Thumbnail Image
    Item
    Mixed Methods on Quadrilaterals and Hexahedra
    (1992-04) Keenan, Philip T.
    We describe a new family of discrete spaces suitable for use with mixed methods on general quadrilateral and hexahedral elements. The new spaces are natural in the sense of differential geometry, so all the usual mixed method theory, including the hybrid formulation, carries over to these new elements with proofs unchanged. Because transforming general quadrilaterals into squares introduces nonlinearity and because mixed methods involve the divergence operator, the new spaces are more complicated than either the corresponding Raviart-Thomas spaces for rectangles or corresponding finite element spaces for quadrilaterals. These new elements may be useful in topologically regular grids, where initially rectangular grids are deformed to match features of the physical region.
  • Thumbnail Image
    Item
    RUF 1.0 User Manual
    (1994-08) Keenan, Philip T.
  • Thumbnail Image
    Item
    Thermal Simulation of Pipeline Flow
    (1991-09) Keenan, Philip T.
    A new numerical method for studying one dimensional fluid flow through pipelines is presented and analyzed. This work extends in a certain direction the collocation method described by Luskin ["An Approximation Procedure for Nonsymmetric, Nonlinear Hyperbolic Systems with Integral Boundary Conditions," SIAM J. Numer. Anal. 1976]. The pressure and velocity of an isothermal fluid in a pipeline can be described by a coupled pair of nonlinear first order hyperbolic partial differential equations. When thermal effects are important a third equation for temperature is added. While Luskin's method works well for the isothermal situation he discussed, it does not apply in certain common cases when thermal effects are modeled. The analysis of this new method shows how the difficulties that come from the application of standard collocation can be overcome. Experiments indicate that this method is a substantial improvement over standard collocation. It also describes an approach to analyzing nonlinear evolution equations with smooth solutions which produces convergence theorems about the nonlinear system from the corresponding linear theorems with relatively little extra work. This technique also yields an H¹ estimate in the isothermal case.