Dugald B. Duncan

Averaging techniques for time-marching schemes for retarded potential integral equations

Abstract Numerical schemes for time domain retarded potential integral equations (RPIEs) are often found to be unstable, having errors which oscillate and grow exponentially with time. It is well known that averaging the solution in time can make such schemes stable by filtering out the high frequency components. However this has to be done carefully—we consider two similar averaging formulae and show that although one stabilizes the underlying scheme, the other actually destabilizes it by amplifying low spatial frequency modes. We carry out a Fourier stability analysis for each of these schemes and show that averaging the solution in space can also successfully remove unstable high spatial frequency modes. Finally we discuss the effect of the spatial approximation and quadrature formula used to approximate the integral by considering a “space-exact” model problem, and present numerical results for a scalar RPIE on both flat and curved surfaces.

Solitons on lattices

Abstract We examine a variety of numerical and approximate analytical methods to study families of solitary waves on lattices. Such waves, when they exist, travel through the lattice without loss of energy, and have approximate soliton properties on collision. Corresponding quantum problems are also briefly described.

Stability and Convergence of Collocation Schemes for Retarded Potential Integral Equations

Time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations. Solving such equations numerically is both complicated and computationally intensive, and numerical methods often prove to be unstable. Collocation schemes are easier to implement than full finite element formulations, but little appears to be known about their stability and convergence. Here we derive and analyze some new stable collocation schemes for the single layer equation for transient acoustic scattering, and use (spatial) Fourier and (temporal) Laplace transform techniques to demonstrate that such stable schemes are second order convergent.

A Mechanistic Model of Gas-Condensate Flow in Pores

Recent experimental results reported in the literature indicate that the relative permeability of gas-condensate systems increases with rate (velocity) at some conditions. To gain a better understanding of the nature of the flow and the prevailing mechanisms resulting in such behaviour flow visualisation experiments have been performed, using high pressure micromodels. The observed flow behaviour at the pore level has been employed to develop a mechanistic model describing the coupled flow of gas and condensate phases. The results of the model simulating the observed simultaneous flow of gas and condensate phases have been compared with reported core experimental results. Most features of the reported rate effect are predictable by the developed single pore model, nevertheless, its extension to include multiple pore interaction is recommended.

Coarsening in an integro-differential model of phase transitions

Coarsening of solutions of the integro-differential equationformula herewhere Ω ⊂ ℝn, J(·) [ges ] 0, e > 0 and f(u) = u3 − u (or similar bistable nonlinear term), is examined, and compared with results for the Allen–Cahn partial differential equation. Both equations are used as models of solid phase transitions. In particular, it is shown that when e is small enough, solutions of this integro-differential equation do not coarsen, in contrast to those of the Allen–Cahn equation. The special case J(·) ≡ 1 is explored in detail, giving insight into the behaviour in the more general case J(·) [ges ] 0. Also, a numerical approximation method is outlined and used on tests in both one- and two-space dimensions to verify and illustrate the main result.

Positive Effect of Flow Velocity on Gas-Condensate Relative Permeability: Network Modelling and Comparison with Experimental Results

Positive velocity dependency of relative permeability of gas–condensate systems, which has been observed in many different core experiments, is now well acknowledged. The above behaviour, which is due to two-phase flow coupling in condensing systems at low interfacial tension (IFT) conditions, was simulated using a 3D pore network model. The steady-dynamic bond network model developed for this purpose was also equipped with a novel anchoring technique, which was based on the equivalent hydraulic length concept adopted from fluid flow through pipes. The available rock data on the co-ordination number, capillary pressure, absolute permeability, porosity and one set of measured relative permeability curves were utilised to anchor the capillary, volumetric and flow characteristics of the constructed network model to those properties of the real core sample. Then the model was used to predict the effective permeability values at other IFT and velocity levels. There is a reasonable quantitative agreement between the predicted and measured relative permeability values affected by the coupling rate effect.

Approximating the Becker—Döring cluster equations

Abstract The Becker–Doring equations model the dynamics of coagulation and fragmentation of clusters of identical particles. The model is an infinite system of ordinary differential equations (ODEs) which specify the rates of change of the concentrations of r -particle clusters. For numerical computation the system is truncated at clusters of a finite size, but this might have to be prohibitively large to capture the metastable behaviour of the system. In this work we derive and investigate the properties of approximations of the Becker–Doring equations which aim to capture the metastable behaviour of the problem with a much reduced system of equations. The schemes are based on a piecewise constant flux approximation, a Galerkin method using a discrete inner product, a discretization of a PDE that in turn approximates the Becker–Doring equations and a lumped coefficients model. We establish a posteriori error estimates for three of these schemes and report on the results of a set of numerical experiments involving metastable behaviour in the solution. Three of the schemes give accurate results on a reduced set of equations, and the PDE scheme appears to be more efficient than the others for this problem.

Convolution-in-Time Approximations of Time Domain Boundary Integral Equations

We present a new temporal approximation scheme for the boundary integral formulation of time-dependent scattering problems which can be combined with either collocation or Galerkin approximation in space. It uses the backward-in-time framework introduced in [P. J. Davies and D. B. Duncan, Convolution Spline Approximations of Volterra Integral Equations, www.mathstat.strath.ac.uk/research/reports/2012 (2012)] with new temporal basis functions which share some properties with radial basis function multiquadrics. We analyze the stability and convergence properties of the new scheme for associated Volterra integral equations and perform extensive numerical tests for scattering from flat polygonal plates and open and closed cubes and spheres, which demonstrate effectiveness of this approach.

Time-marching numerical schemes for the electric field integral equation on a straight thin wire

We derive and analyse four algorithms for computing the current induced on a thin straight wire by a transient electric field. They all involve solving the thin wire electric field integral equations (EFIEs) and consist of a very accurate differential equations solver together with various schemes to approximate the vector potential integral equation. We carry out a rigorous numerical stability analysis of each of these methods. This has not previously been done for solution schemes for the thin wire EFIEs. Each scheme is shown to be stable and convergent provided the radius of the wire is small enough for the thin wire equations to be a valid model.

Solitary waves on a strongly anisotropic KP lattice

Abstract We consider a strongly anisotropic 2D atomic lattice model which in a continuum limit becomes the Kadomtsev-Petviashvili equation. Solitary waves on this lattice are studied by a variety of analytic and numerical tecniques.