50 research outputs found
ADAM SMITH'S OPTIMISTIC TELEOLOGICAL VIEW OF HISTORY
Get PDFAdam Smith's four-stage theory provides the framework for his writings on history. The fourth stage is the commercial epoch; the culmination of history in this stage is a key component in the conventional interpretation of Adam Smith as a prophet of commercialism. In two historical case studies Smith shows the capacity of commercial society to regenerate itself. This potent capacity suggests that commercial society is inevitable. At a certain point in time it also overcomes the major obstacles to its permanence. Smith's philosophy of history anticipates the end of history views of Kant and Hegel.Political Economy,
Convolution spline approximations for time domain boundary integral equations
Get PDFWe derive a new \convolution spline" approximation method for convolution Volterra integral equations. This shares some properties of convolution quadrature, but instead of being based on an underlying ODE solver is explicitly constructed in terms of basis functions which have compact support. At time step tn = nh > 0, the solution is approximated in a \backward time" manner in terms of basis functions j by u(tn􀀀t) Pnj=0 un􀀀jj(t=h) for t 2 [0; tn]. We carry out a detailed analysis for B-spline basis functions, but note that the framework is more general than this. For B-splines of degree m 1 we show that the schemes converge at the rate O(h2) when the kernel issuciently smooth. We also establish a methodology for their stability analysis and obtain new stability results for several non-smooth kernels, including the case of a highly oscillatory Bessel function kernel (in which the oscillation frequency can be O(1=h)). This is related to convergence analysis for approximation of time domain boundary integral equations (TDBIEs), and provides evidence that the new convolution spline approach could provide a useful time-stepping mechanism for TDBIE problems. In particular, usingcompactly supported basis functions would give sparse system matrices
Numerical approximation of a nonlinear partial integro-differential equation
No full textMACS studentshi
Connections between collocation and Galerkin methods
No full textWe look at the connections between collocation methods based on B-splines basis functions in time, and Galerkin methods also using B-splines, but of lower order. The collocation method can be written as a Galerkin method applied to a modified variational problem (for bounded time interval calculations). We use results introduced by Ha Duong to bound the difference between the exact solutions (if they exist) of these two variational problems. However questions remain about how to use the Ha Duong analysis to bound the difference of the approximate solutions using Galerkin.Non UBCUnreviewedAuthor affiliation: Heriot-Watt UniversityFacult
Numerical approximations of first kind Volterra convolution equations with discontinuous kernels
Get PDFThe cubic "convolution spline'" method for first kind Volterra convolution integral equations was introduced in [Convolution spline approximations of Volterra integral equations, J. Integral Equations Appl., 26:369--410, 2014]. Here we analyse its stability and convergence for a broad class of piecewise smooth kernel functions and show it is stable and fourth order accurate even when the kernel function is discontinuous. Key tools include a new discrete Gronwall inequality which provides a stability bound when there are jumps in the kernel function, and a new error bound obtained from a particular B-spline quasi-interpolant
Positivity of a weakly singular operator and approximation of wave scattering from the sphere
No full textWe investigate properties of a family of integral operators B with a weakly singular compactly supported zonal kernel function on the surface S of the unit 3D sphere. The support is over a spherical cap of height h (Formula Presented) (0, 2]. Operators like this arise in some common types of approximations of time domain boundary integral equations (TDBIE) describing the scattering of acoustic waves from the surface of the sphere embedded in an infinite homogeneous medium where h is directly related to the time step size. We show that the Legendre polynomials of degree ℓ ≥ 0 satisfy (Formula Presented) and, using spherical harmonics and the Funk–Hecke formula for the eigenvalues of B, that this is a key to unlocking positivity results for a subfamily of these operators. As well as positivity results we give detailed upper and lower bounds on the eigenvalues of B and on (Formula Presented) We give various examples of where these results are useful in numerical approximations of the TDBIE on the sphere and show that positivity of B is a necessary condition for these approximation schemes to be well defined. We also show the connection between the results for eigenvalues and the separation of variables solution of the TDBIE on the sphere. Finally we show how this relates to scattering from an infinite flat surface and Cooke’s 1937 result(Formula Presented).</p
Decoupled overlapping grids for the numerical modeling of oil wells
No full textAccurate computation of time-dependent well bore pressure is important in well test analysis - a branch of petroleum engineering where reservoir properties are estimated by comparing measured pressure responses at an oil well to results from a mathematical model. Similar methods are also used in groundwater engineering. In this paper we present the new approach of decoupled overlapping grids for accurately computing time-dependent pressure at the oil well. Our method is implemented in two stages: a global stage with a simple point or line source well approximation, and a local post-process stage with the well modeled correctly as an internal boundary. We investigate the accuracy of our method for a representative 2D problem in both homogeneous and heterogeneous isotropic domains, and compare our results with the widely used Peaceman well index solution (in the homogeneous case), and the approximate solution on locally refined grids. We also present a theoretical analysis that explains the observed O(h^2) behavior of the error in our method for the homogeneous case
On the behaviour of time discretisations of the electric field integral equation
No full textWe derive a separation of variables solution for time-domain electromagnetic scattering from a perfectly conducting in®nite ¯at plate. The time dependent part of the equations are then used as a model problem in order to study the e€ects of various time discretisations on the full scattering problem. We examine and explain how exponential and polynomial instabilities arise in the approximation schemes, and show that the time averaging which is often used in an attempt to stabilise solutions of the full problem acts to destabilise some of the schemes. Our results show that two of the time discretisations can produce good results when coupled with a space-exact approximation, and indicate that they will be useful when coupled with an accurate enough spatial approximation
Overlapping grids for the diffusion equation
No full textWe examine the use of nonmatching, overlapping grids for the approximate solution of time-dependent diffusion problems with Neumann boundary conditions. This problem arises as a model of the so-called well test analysis of oil and gas reservoirs, which has geometry modelling requirements that make overlapping grids particularly suitable. We describe the problem and the overlapping grid approximation, and then carry out a stability and convergence analysis in one space dimension (1D). We show that for suitable schemes, stability is relatively easy to establish in much more general situations. Convergence is less easy to generalise, but we demonstrate that 2D approximations appear to have the same convergence behaviour as their 1D counterparts. © The author 2006. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.</p
