- Professor Emeritus, Colorado School of MinesExecutive Editor Emeritus, Mathematical Reviews, American Mathematical So... moreProfessor Emeritus, Colorado School of MinesExecutive Editor Emeritus, Mathematical Reviews, American Mathematical SocietyResearch interests: numerical analysis and scientific computingedit
Research Interests:
Publisher Summary This chapter focuses on the Richardson extrapolation for parabolic Galerkin methods. The chapter discusses that approximations generated by certain discrete Galerkin procedures can be modified using Richardson... more
Publisher Summary This chapter focuses on the Richardson extrapolation for parabolic Galerkin methods. The chapter discusses that approximations generated by certain discrete Galerkin procedures can be modified using Richardson extrapolation to obtain approximations whose time discretization error is 0((Δt)p(s+1)), where s is a positive integer. As an example, the heat equation in the cylinder D × (0,T] subject to homogeneous Dirichlet boundary conditions is considered. D is a bounded domain in Rn. The techniques described in the chapter can be used to analyze a family of three-level methods proposed by Dupont, Fairweather, and Johnson. These methods are algebraically linear even for nonlinear problems. It can be shown that for a wide class of nonlinear problems one extrapolation may be performed to obtain an approximation which is fourth-order correct in time.
Research Interests:
Research Interests:
The software package COLNEW [G. Bader and U. Ascher, SIAM J. Sci. Statist. Comput., 8 (1987), pp. 483–500], which implements the method of spline collocation at Gauss points, has been widely used for the solution of mixed-order systems of... more
The software package COLNEW [G. Bader and U. Ascher, SIAM J. Sci. Statist. Comput., 8 (1987), pp. 483–500], which implements the method of spline collocation at Gauss points, has been widely used for the solution of mixed-order systems of boundary value problems in ordinary differential equations. Considerable attention has been given in recent years to the development of efficient algorithms for the solution of the almost block diagonal (ABD) linear systems of the type which arise in this method, and some of these algorithms have been designed to exploit parallelism. However, in COLNEW, the solution of the ABD systems contributes only a modest amount to the overall execution time, thus limiting the potential of these new solvers. In this paper, we examine the linear system setup, which is the primary bottleneck in COLNEW and has not been addressed previously. We describe PCOLNEW, a modified version of COLNEW which incorporates coarse-grained parallelism to minimize the cost of this bottleneck. Numerical results of tests conducted on a variety of shared-memory machines are presented which demonstrate the efficacy of this parallel implementation.
Research Interests:
Research Interests:
Research Interests: Engineering, Mathematics, Applied Mathematics, Partial Differential Equations, Computational Physics, and 15 moreMathematical Sciences, Newton Method, Physical sciences, Discretization, PARTIAL DIFFERENTIAL EQUATION, Second Order, Linearization, Finite Difference, Boundary Value Problems, Boundary Value Problem, Crank Nicolson, Shallow Water, Perturbation Theory, Nonlinear Equations, and Shallow water equations
... imajna.oxfordjournals.org Downloaded from Page 8. 532 G. FAIRWEATHER AND JC LOPEZ-MARCOS Proof. Let (V°, V1,..., VN), (W°, Wl,..., WN) be in the ball B(uh, Mh) of the space Xh. We set E" = V-WeUJ+l, O^n^N, <Ph(V°,... more
... imajna.oxfordjournals.org Downloaded from Page 8. 532 G. FAIRWEATHER AND JC LOPEZ-MARCOS Proof. Let (V°, V1,..., VN), (W°, Wl,..., WN) be in the ball B(uh, Mh) of the space Xh. We set E" = V-WeUJ+l, O^n^N, <Ph(V°, V1,..., VN) = {P°, Po, P\...,PN), ...
Research Interests:
... Page 10. 252 G. Faitweather, JC L6pez-Marcos / Galerkin methods ... Let R be a fixed positive constant. Under the hypotheses of theorem 3.1, there exists a constant S such that, for h and k sufficiently small, if (V°,... VN) and... more
... Page 10. 252 G. Faitweather, JC L6pez-Marcos / Galerkin methods ... Let R be a fixed positive constant. Under the hypotheses of theorem 3.1, there exists a constant S such that, for h and k sufficiently small, if (V°,... VN) and (W°,... W iv) belong to the ball B((~,. N),R) of Xhk then ...
Research Interests: Mathematics, Applied Mathematics, Computer Science, Finite element method, Finite Element, and 9 moreComputational Mathematics, Mathematical Analysis, Computational Science and Engineering, Computational, Numerical Analysis and Computational Mathematics, Galerkin Method, Numerical Solution, Boundary Value Problem, and Crank Nicolson
Research Interests:
Research Interests: Mechanical Engineering, Civil Engineering, Mathematics, Computational Mechanics, Potential Theory, and 13 moreMultidisciplinary, Method of fundamental solutions (MFS), Mathematical Analysis, Fundamental Solution, Three Dimensional, Acoustic Scattering, Interdisciplinary Engineering, Rotational Symmetry, Boundary Condition, Boundary Value Problem, Isotropy, Approximate solution, and Elliptic integral
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Fast direct methods are proposed for the solution of linear systems arising when orthogonal spline collocation with piecewise Hermite bicubics is employed for the approximate solution of Poisson’s equation in a rectangle. The new methods,... more
Fast direct methods are proposed for the solution of linear systems arising when orthogonal spline collocation with piecewise Hermite bicubics is employed for the approximate solution of Poisson’s equation in a rectangle. The new methods, which are matrix decomposition algorithms involving fast Fourier transforms, require $O(N^2 \log N)$ arithmetic operations on an $N \times N$ uniform partition and are highly parallel in nature. The treatment of more general elliptic problems is also discussed and the results of numerical experiments using parallel implementations of the new algorithms on several shared memory machines are presented.
Research Interests:
Research Interests:
Research Interests:
New numerical techniques are presented for the solution of a class of linear partial integro-differential equations (PIDEs) with a positive-type memory term in the unit square. In these methods, orthogonal spline collocation (OSC) is used... more
New numerical techniques are presented for the solution of a class of linear partial integro-differential equations (PIDEs) with a positive-type memory term in the unit square. In these methods, orthogonal spline collocation (OSC) is used for the spatial discretization, and, for the time stepping, new alternating direction implicit (ADI) methods based on the backward Euler, the Crank-Nicolson, and the second order BDF methods combined with judiciously chosen quadrature rules are considered. The ADI OSC methods are proved to be of optimal accuracy in time and in the L2 norm in space. Numerical results confirm the predicted convergence rates and also exhibit optimal accuracy in the L°° and H1 norms and superconvergence phenomena.
Research Interests:
Research Interests:
Research Interests:
Abstract A discrete-time orthogonal spline collocation scheme is formulated and analyzed for a problem governing the transverse vibrations of a clamped square plate. The problem is reformulated as a Schrodinger-type system which is then... more
Abstract A discrete-time orthogonal spline collocation scheme is formulated and analyzed for a problem governing the transverse vibrations of a clamped square plate. The problem is reformulated as a Schrodinger-type system which is then approximated by a Crank–Nicolson orthogonal spline collocation scheme. This scheme is shown to be second-order accurate in time and of optimal order accuracy in space in the H 1 - and H 2 -norms.
Research Interests:
State-of-the-art numerical techniques for solving partial differential equations utilize adaptive grid techniques. In these methods, the mesh is refined based on a posteriori error estimates. However, these techniques have not been used... more
State-of-the-art numerical techniques for solving partial differential equations utilize adaptive grid techniques. In these methods, the mesh is refined based on a posteriori error estimates. However, these techniques have not been used for air pollution simulations since software is available only for certain limiting forms of equations. The aim of this study is to identify parameters that affect the accuracy of numerical solutions of the transport equations involving point sources. Based on these, a grid can be chosen a priori and used for the entire simulation. A two-dimensional finite element model using bilinear elements is presented for the advection-diffusion problem with an infinite line source. This paper is restricted to two spatial directions, with advection dominating in the x-direction and turbulent diffusion dominating in the z-direction. Numerical results for an inert plume emanating from a single infinite line source show that the ratio of the advection time scale to the turbulent diffusion time scale, KzΔxuΔz2, should be equal to or greater than one for the most accurate solutions. In addition, comparisons between the volume-averaged representation of a point source and the use of an irregular grid for point source representation demonstrate that, near the source, improved results can be obtained by placing a node at the source location.
Research Interests:
ABSTRACT We propose a geometric modeling method in R3 based on the so-called potential field (PF) modeling technique. The method is a new technique for surface reconstruction from a data set of scattered points taken on a surface. In this... more
ABSTRACT We propose a geometric modeling method in R3 based on the so-called potential field (PF) modeling technique. The method is a new technique for surface reconstruction from a data set of scattered points taken on a surface. In this method, the construction of a geometric model is based on the solution of an elliptic boundary value problem determined using the method of fundamental solutions (MFS). We consider two such problems, Laplace's equation subject to Dirichlet boundary conditions and the biharmonic Dirichlet problem. We present the results of numerical experiments which demonstrate the efficacy of these approaches. We also identify differences and potential difficulties in the application of these approaches and propose ways of addressing them.
Research Interests:
We formulate and analyze a fully-discrete approximate solution of the linear Schrödinger equation on the unit square written as a Schrödinger-type system. The finite element Galerkin method is used for the spatial discretization, and the... more
We formulate and analyze a fully-discrete approximate solution of the linear Schrödinger equation on the unit square written as a Schrödinger-type system. The finite element Galerkin method is used for the spatial discretization, and the time-stepping is done with an alternating direction implicit extrapolated Crank-Nicolson method. We demonstrate the existence and uniqueness of the approximation, and prove that the scheme is of optimal accuracy in the L2 , H1 and L∞ norms in space and second–order accurate in time. Numerical results are presented which support the theory.
Research Interests:
Abstract Frank Rizzo was a pioneer in what he called boundary integral equation methods. With his students and colleagues, he developed theory and algorithms for many problems of engineering interest. In this memorial paper, we review his... more
Abstract Frank Rizzo was a pioneer in what he called boundary integral equation methods. With his students and colleagues, he developed theory and algorithms for many problems of engineering interest. In this memorial paper, we review his life and work. A list of his publications is included.
Research Interests:
Research Interests:
The implementation of a boundary element method (BEM) is complicated by the need to provide for evaluation of integrals with singular integrand. This difficulty is normally handled by devising specialized quadrature techniques for dealing... more
The implementation of a boundary element method (BEM) is complicated by the need to provide for evaluation of integrals with singular integrand. This difficulty is normally handled by devising specialized quadrature techniques for dealing with the singularity. An alternative approach is to avoid the problem altogether by employing an auxiliary boundary. A to formulate the BEM. However, this idea has not proven very successful because accuracy of the numerical solution depends very significantly on the precise shape and location of . A and no viable algorithm for determining a suitable ∂A has, as yet, been devised.
Research Interests:
... where byis the central difference operator in the .y-direction, Um-\, um, Um+i are the values of u at the nodes (iAx,jAy,(ml)At), (iAx,jAy,mAt) and (iAx,jAy,(m + l)At) respectively (i,j = 1,2,..., Nl; m = 1,2,...) uf^+i denotes an... more
... where byis the central difference operator in the .y-direction, Um-\, um, Um+i are the values of u at the nodes (iAx,jAy,(ml)At), (iAx,jAy,mAt) and (iAx,jAy,(m + l)At) respectively (i,j = 1,2,..., Nl; m = 1,2,...) uf^+i denotes an approximation to Um+i, Ax, Ay, and At are the mesh sizes ...
Research Interests:
Research Interests:
On passe en revue les methodes d'ordre superieur a 2 pour l'integration des formulations semi-discretes des problemes paraboliques. On donne des resultats sur la stabilite et l'ordre et l'efficacite de calcul des diverses... more
On passe en revue les methodes d'ordre superieur a 2 pour l'integration des formulations semi-discretes des problemes paraboliques. On donne des resultats sur la stabilite et l'ordre et l'efficacite de calcul des diverses formules
Research Interests:
In this paper, a new method for the numerical solution of the two-dimensional parabolic equation of Tappert in horizontally-stratified media is presented. This method uses orthogonal cubic spline collocation for the semidiscretization... more
In this paper, a new method for the numerical solution of the two-dimensional parabolic equation of Tappert in horizontally-stratified media is presented. This method uses orthogonal cubic spline collocation for the semidiscretization with respect to depth, and the resulting system of differential-algebraic equations is solved using the NAG Library routine D02NNF. In the present application, orthogonal spline collocation, which has been exceedingly effective in the approximate solution of a broad class of problems, has the advantage that it systematically incorporates the requisite interface conditions. The state-of-the-art routine D02NNF, which implements backward differentiation formulas, has been used successfully in the solution of related problems.
Research Interests:
Research Interests:
The periodic initial value problem for the partial differential equation u t + u x x x + β ( u 2 ) x + γ 2 ( u 2 ) x x + ε u x x − δ u t x = 0 {u_t} + {u_{xxx}} + \beta {({u^2})_x} + \frac {\gamma }{2}{({u^2})_{xx}} + \varepsilon {u_{xx}}... more
The periodic initial value problem for the partial differential equation u t + u x x x + β ( u 2 ) x + γ 2 ( u 2 ) x x + ε u x x − δ u t x = 0 {u_t} + {u_{xxx}} + \beta {({u^2})_x} + \frac {\gamma }{2}{({u^2})_{xx}} + \varepsilon {u_{xx}} - \delta {u_{tx}} = 0 , ε \varepsilon , δ > 0 \delta > 0 , arises in fluidization models. The numerical integration of the problem is a difficult task in that many "reasonable" finite difference and finite element methods give rise to unstable discretizations. We show how to modify the standard Galerkin technique in order to stabilize it. Optimal-order error estimates are derived and the results of numerical experiments are presented. The stabilization technique suggested in the paper can be interpreted as rewriting the problem in Sobolev form and would also be useful for other equations involving terms of the form u t − δ u t x {u_t} - \delta {u_{tx}} .
Research Interests:
Research Interests:
In this article, a qualocation method is formulated and analyzed for parabolic partial integro-differential equations in one space variable. Using a new Ritz–Volterra type projection, optimal rates of convergence are derived. Based on the... more
In this article, a qualocation method is formulated and analyzed for parabolic partial integro-differential equations in one space variable. Using a new Ritz–Volterra type projection, optimal rates of convergence are derived. Based on the second-order backward differentiation formula, a fully discrete scheme is formulated and a convergence analysis is derived. Results of numerical experiments are presented which support the theoretical results.
Research Interests:
Research Interests:
An orthogonal spline collocation method (OSCM) with C1 splines of degree r ≥ 3 is analyzed for the numerical solution of singularly perturbed reaction diffusion problems in one dimension. The method is applied on a Shishkin mesh and... more
An orthogonal spline collocation method (OSCM) with C1 splines of degree r ≥ 3 is analyzed for the numerical solution of singularly perturbed reaction diffusion problems in one dimension. The method is applied on a Shishkin mesh and quasi-optimal error estimates in weighted Hm norms for m = 1, 2 and in a discrete L2-norm are derived. These estimates are valid uniformly with respect to the perturbation parameter. The results of numerical experiments are presented for C1 cubic splines (r = 3) and C1 quintic splines (r = 5) to demonstrate the efficacy of the OSCM and confirm our theoretical findings. Further, quasi-optimal a priori estimates in L2, L∞ and W 1,∞-norms are observed in numerical computations. Finally, superconvergence of order 2r − 2 at the mesh points is observed in the approximate solution and also in its first derivative when r = 5.
Research Interests:
New numerical techniques are presented for the solution of a class of linear partial integro-differential equations (PIDEs) with a positive-type memory term in the unit square. In these methods, orthogonal spline collocation (OSC) is used... more
New numerical techniques are presented for the solution of a class of linear partial integro-differential equations (PIDEs) with a positive-type memory term in the unit square. In these methods, orthogonal spline collocation (OSC) is used for the spatial discretization, and, for the time stepping, new alternating direction implicit (ADI) methods based on the backward Euler, the Crank-Nicolson, and the second order BDF methods combined with judiciously chosen quadrature rules are considered. The ADI OSC methods are proved to be of optimal accuracy in time and in the L2 norm in space. Numerical results confirm the predicted convergence rates and also exhibit optimal accuracy in the L°° and H1 norms and superconvergence phenomena.
Research Interests:
On applique la methode de Galerkin aux differences retrogrades et la methode de Crank-Nicholson-Galerkin a la solution numerique d'un probleme a valeurs initiales-limites pour une equation parabolique lineaire. On etablit les... more
On applique la methode de Galerkin aux differences retrogrades et la methode de Crank-Nicholson-Galerkin a la solution numerique d'un probleme a valeurs initiales-limites pour une equation parabolique lineaire. On etablit les conditions relatives aux donnees qui assurent l'existence de developpements asymptotiques appropries suivant les puissances du pas temporel des erreurs de discretisation
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
... A similar investigation of such conditions was carried out by Azarnia (1978). Our results cover a broader range of problems than those considered in Azarnia (1978) and include the results of that paper as a special case. ... the case... more
... A similar investigation of such conditions was carried out by Azarnia (1978). Our results cover a broader range of problems than those considered in Azarnia (1978) and include the results of that paper as a special case. ... the case considered in Azarnia (1978). ...