John M. Stockie

Jeffrey K. Wiens, John M. Stockie, JF Williams

We investigate a model for traffic flow based on the Lighthill-Whitham-Richards model that consists of a hyperbolic conservation law with a discontinuous, piecewise-linear flux. A mollifier is used to smooth out the discontinuity in the flux function over a small distance epsilon << 1 and then the analytical solution to the corresponding Riemann problem is derived in the limit as epsilon goes to 0. For certain initial data, the Riemann problem can give rise to zero waves that propagate with infinite speed but have zero strength. We propose a Godunov-type numerical scheme that avoids the otherwise severely restrictive CFL constraint that would arise from waves with infinite speed by exchanging information between local Riemann problems and thereby incorporating the effects of zero waves directly into the Riemann solver. Numerical simulations are provided to illustrate the behaviour of zero waves and their impact on the solution. The effectiveness of our approach is demonstrated through a careful convergence study and comparisons to computations using a third-order WENO scheme.

Maurizio Ceseri, John M. Stockie

We develop a mathematical model for a three-phase free boundary problem in one dimension that involves the interactions between gas, water and ice. The dynamics are driven by melting of the ice layer, while the pressurized gas also dissolves within the meltwater. The model incorporates a Stefan condition at the water-ice interface along with Henry's law for dissolution of gas at the gas-water interface. We employ a quasi-steady approximation for the phase temperatures and then derive a series solution for the interface positions. A non-standard feature of the model is an integral free boundary condition that arises from mass conservation owing to changes in gas density at the gas-water interface, which makes the problem non-self-adjoint. We derive a two-scale asymptotic series solution for the dissolved gas concentration, which because of the non-self-adjointness gives rise to a Fourier series expansion in eigenfunctions that do not satisfy the usual orthogonality conditions. Numerical simulations of the original governing equations are used to validate the series approximations.

Jane Shaw MacDonald, Rafael Ordoñez Cardales, John M. Stockie

Nordic skiing provides fascinating opportunities for mathematical modelling studies that exploit methods and insights from physics, applied mathematics, data analysis, scientific computing and sports science. A typical ski course winds over varied terrain with frequent changes in elevation and direction, and so its geometry is naturally described by a three-dimensional space curve. The skier travels along a course under the influence of various forces, and their dynamics can be described using a nonlinear system of ordinary differential equations (ODEs) that are derived from Newton's laws of motion. We develop an algorithm for solving the governing equations that combines Hermite spline interpolation, numerical quadrature and a high-order ODE solver. Numerical simulations are compared with measurements of skiers on actual courses to demonstrate the effectiveness of the model. Throughout, we aim to illustrate how elementary concepts from undergraduate courses in calculus and scientific computing can be applied to study real problems in sport, which we hope will provide stimulating examples for both instructors and students. At the same time, we demonstrate how these concepts are capable of providing novel insights into skiing that should also be of interest to sport scientists.

M. Alamgir Hossain, Sam Pimentel, John M. Stockie

We implement a data assimilation framework for integrating ice surface and terminus position observations into a numerical ice-flow model. The model uses the well-known shallow shelf approximation (SSA) coupled to a level set method to capture ice motion and changes in the glacier geometry. The level set method explicitly tracks the evolving ice-atmosphere and ice-ocean boundaries for a marine outlet glacier. We use an Ensemble Transform Kalman Filter to assimilate observations of ice surface elevation and lateral ice extent by updating the level set function that describes the ice interface. Numerical experiments on an idealized marine-terminating glacier demonstrate the effectiveness of our data assimilation approach for tracking seasonal and multi-year glacier advance and retreat cycles. The model is also applied to simulate Helheim Glacier, a major tidewater-terminating glacier of the Greenland Ice Sheet that has experienced a recent history of rapid retreat. By assimilating observations from remotely-sensed surface elevation profiles we are able to more accurately track the migrating glacier terminus and glacier surface changes. These results support the use of data assimilation methodologies for obtaining more accurate predictions of short-term ice sheet dynamics.

Bamdad Hosseini, John M. Stockie

We propose a numerical algorithm for solving the atmospheric dispersion problem with elevated point sources and ground-level deposition. The problem is modelled by the 3D advection-diffusion equation with delta-distribution source terms, as well as height-dependent advection speed and diffusion coefficients. We construct a finite volume scheme using a splitting approach in which the Clawpack software package is used as the advection solver and an implicit time discretization is proposed for the diffusion terms. The algorithm is then applied to an actual industrial scenario involving emissions of airborne particulates from a zinc smelter using actual wind measurements. We also address various practical considerations such as choosing appropriate methods for regularizing noisy wind data and quantifying sensitivity of the model to parameter uncertainty. Afterwards, we use the algorithm within a Bayesian framework for estimating emission rates of zinc from multiple sources over the industrial site. We compare our finite volume solver with a Gaussian plume solver within the Bayesian framework and demonstrate that the finite volume solver results in tighter uncertainty bounds on the estimated emission rates.

Bamdad Hosseini, Nilima Nigam, John M. Stockie

In this article we consider regularizations of the Dirac delta distribution with applications to prototypical elliptic and hyperbolic partial differential equations (PDEs). We study the convergence of a sequence of distributions $\mathcal{S}_H$ to a singular term $\mathcal{S}$ as a parameter $H$ (associated with the {support size} of $\mathcal{S}_H$) shrinks to zero. We characterize this convergence in both the weak-$\ast$ topology of distributions, as well as in a weighted Sobolev norm. These notions motivate a framework for constructing regularizations of the delta distribution that includes a large class of existing methods in the literature. This framework allows different regularizations to be compared. The convergence of solutions of PDEs with these regularized source terms is then studied in various topologies such as pointwise convergence on a deleted neighborhood and weighted Sobolev norms. We also examine the lack of symmetry in tensor product regularizations and effects of dissipative error in hyperbolic problems.

Ali Reza Soheili, John M. Stockie

We propose a moving mesh adaptive approach for solving time-dependent partial differential equations. The motion of spatial grid points is governed by a moving mesh PDE (MMPDE) in which a mesh relaxation time τis employed as a regularization parameter. Previously reported results on MMPDEs have invariably employed a constant value of the parameter τ. We extend this standard approach by incorporating a variable relaxation time that is calculated adaptively alongside the solution in order to regularize the mesh appropriately throughout a computation. We focus on singular problems involving self-similar blow-up to demonstrate the advantages of using a variable relaxation ime over a fixed one in terms of accuracy, stability and efficiency.