Matthew E. Hubbard

A 2D numerical model of wave run-up and overtopping

A two-dimensional (2D) numerical model of wave run-up and overtopping is presented. The model (called OTT-2D) is based on the 2D nonlinear shallow water (NLSW) equations on a sloping bed, including bed shear stress. These equations are solved using an upwind finite volume technique and a hierarchical Cartesian Adaptive Mesh Refinement (AMR) algorithm. The 2D nature of the model means that it can be used to simulate wave transformation, run-up, overtopping and regeneration by obliquely incident and multi-directional waves over alongshore-inhomogeneous sea walls and complex, submerged or surface-piercing features. The numerical technique used includes accurate shock modeling, and uses no special shoreline-tracking algorithm or shoreline coordinate transformation, which means that noncontiguous flows and multiple shorelines can easily be simulated. The adaptivity of the model ensures that only those parts of the flow that require higher resolution (such as the region of the moving shoreline) receive it, resulting in a model with a high level of efficiency. The model is shown to accurately reproduce analytical and benchmark numerical solutions. Existing wave flume and wave basin datasets are used to test the ability of the model to approximate 1D and 2D wave transformation, run-up and overtopping. Finally, we study a 2D dataset of overtopping of random waves at off-normal incidence to investigate overtopping of a sea wall by long-crested waves. The data set is interesting as it has not been studied in detail before and suggests that, in some instances, overtopping at an angle can lead to more flooding than at normal incidence.

A Three-Dimensional, Adaptive, Godunov-Type Model for Global Atmospheric Flows

Abstract In this paper a Godunov-type methodology is applied to three-dimensional global atmospheric modeling. Numerical issues are addressed regarding the formulation of the tracer advection problem, the application of dimensional splitting, and the implementation of a Godunov-type scheme, based on the WAF approach, on spherical geometries. Particular attention is paid to addressing the problems that arise because of the convergence of the grid lines toward the Poles. A three-dimensional model is then built on the sphere that is based on a uniform longitude–latitude–height grid. This provides the framework within which an adaptive mesh refinement (AMR) algorithm is applied, to enhance the efficiency and accuracy with which results are obtained. These methods are not commonly used in the area of atmospheric modeling, but AMR in particular is commonly used with great success in other areas of computational fluid dynamics. The model is initially validated using a series of idealized case studies that have e...

Multiphase modelling of vascular tumour growth in two spatial dimensions.

In this paper we present a continuum mathematical model of vascular tumour growth which is based on a multiphase framework in which the tissue is decomposed into four distinct phases and the principles of conservation of mass and momentum are applied to the normal/healthy cells, tumour cells, blood vessels and extracellular material. The inclusion of a diffusible nutrient, supplied by the blood vessels, allows the vasculature to have a nonlocal influence on the other phases. Two-dimensional computational simulations are carried out on unstructured, triangular meshes to allow a natural treatment of irregular geometries, and the tumour boundary is captured as a diffuse interface on this mesh, thereby obviating the need to explicitly track the (potentially highly irregular and ill-defined) tumour boundary. A hybrid finite volume/finite element algorithm is used to discretise the continuum model: the application of a conservative, upwind, finite volume scheme to the hyperbolic mass balance equations and a finite element scheme with a stable element pair to the generalised Stokes equations derived from momentum balance, leads to a robust algorithm which does not use any form of artificial stabilisation. The use of a matrix-free Newton iteration with a finite element scheme for the nutrient reaction-diffusion equations allows full nonlinearity in the source terms of the mathematical model. Numerical simulations reveal that this four-phase model reproduces the characteristic pattern of tumour growth in which a necrotic core forms behind an expanding rim of well-vascularised proliferating tumour cells. The simulations consistently predict linear tumour growth rates. The dependence of both the speed with which the tumour grows and the irregularity of the invading tumour front on the model parameters is investigated.

Discontinuous fluctuation distribution

This paper describes a new numerical scheme for the approximation of steady state solutions to systems of hyperbolic conservation laws. It generalises the fluctuation distribution framework by allowing the underlying representation of the solution to be discontinuous. This leads to edge-based fluctuations in addition to the standard cell-based fluctuations, which are then distributed to the cell vertices in an upwind manner which retains the properties of the continuous scheme (positivity, linearity preservation, conservation, compactness and continuity). Numerical results are presented on unstructured triangular meshes in two space dimensions for linear and nonlinear scalar equations as well as the Euler equations of gasdynamics. The accuracy of the approximation in smooth regions of the flow is shown to be very similar to the corresponding continuous scheme, but the discontinuous approach improves the sharpness with which discontinuities in the flow can be captured and provides additional flexibility which will allow adaptive techniques to be applied simply to improve efficiency.

Mathematical and computational models of drug transport in tumours

The ability to predict how far a drug will penetrate into the tumour microenvironment within its pharmacokinetic (PK) lifespan would provide valuable information about therapeutic response. As the PK profile is directly related to the route and schedule of drug administration, an in silico tool that can predict the drug administration schedule that results in optimal drug delivery to tumours would streamline clinical trial design. This paper investigates the application of mathematical and computational modelling techniques to help improve our understanding of the fundamental mechanisms underlying drug delivery, and compares the performance of a simple model with more complex approaches. Three models of drug transport are developed, all based on the same drug binding model and parametrized by bespoke in vitro experiments. Their predictions, compared for a ‘tumour cord’ geometry, are qualitatively and quantitatively similar. We assess the effect of varying the PK profile of the supplied drug, and the binding affinity of the drug to tumour cells, on the concentration of drug reaching cells and the accumulated exposure of cells to drug at arbitrary distances from a supplying blood vessel. This is a contribution towards developing a useful drug transport modelling tool for informing strategies for the treatment of tumour cells which are ‘pharmacokinetically resistant’ to chemotherapeutic strategies.

A 2D extension of a Large Time Step explicit scheme ( CFL 1 ) for unsteady problems with wet/dry boundaries

A 2D Large Time Step (LTS) explicit scheme on structured grids is presented in this work. It is first detailed and analysed for the 2D linear advection equation and then applied to the 2D shallow water equations. The dimensional splitting technique allows us to extend the ideas developed in the 1D case related to source terms, boundary conditions and the reduction of the time step in the presence of large discontinuities. The boundary conditions treatment as well as the wet/dry fronts in the case of the 2D shallow water equations require extra effort. The proposed scheme is tested on linear and non-linear equations and systems, with and without source terms. The numerical results are compared with those of the conventional scheme as well as with analytical solutions and experimental data.

Nonlinear multigrid methods for second order differential operators with nonlinear diffusion coefficient

Nonlinear multigrid methods such as the Full Approximation Scheme (FAS) and Newton-multigrid (Newton-MG) are well established as fast solvers for nonlinear PDEs of elliptic and parabolic type. In this paper we consider Newton-MG and FAS iterations applied to second order differential operators with nonlinear diffusion coefficient. Under mild assumptions arising in practical applications, an approximation (shown to be sharp) of the execution time of the algorithms is derived, which demonstrates that Newton-MG can be expected to be a faster iteration than a standard FAS iteration for a finite element discretisation. Results are provided for elliptic and parabolic problems, demonstrating a faster execution time as well as greater stability of the Newton-MG iteration. Results are explained using current theory for the convergence of multigrid methods, giving a qualitative insight into how the nonlinear multigrid methods can be expected to perform in practice.

Unconditionally stable space-time discontinuous residual distribution for shallow-water flows

This article describes a discontinuous implementation of residual distribution for shallow-water flows. The emphasis is put on the space-time implementation of residual distribution for the time-dependent system of equations with discontinuity in time only. This lifts the time-step restriction that even implicit continuous residual distribution schemes invariably suffer from, and thus leads to an unconditionally stable discretisation. The distributions are the space-time variants of the upwind distributions for the steady-state system of equations and are designed to satisfy the most important properties of the original mathematical equations: positivity, linearity preservation, conservation and hydrostatic balance. The purpose of the several numerical examples presented in this article is twofold. First, to show that the discontinuous numerical discretisation does indeed exhibit all the desired properties when applied to the shallow-water equations. Second, to investigate how much the time step can be increased without adversely affecting the accuracy of the scheme and whether this translates into gains in computational efficiency. Comparison to other existing residual distribution schemes is also provided to demonstrate the improved performance of the scheme.

Non-oscillatory third order fluctuation splitting schemes for steady scalar conservation laws

This paper addresses the issue of constructing non-oscillatory, higher than second order, fluctuation splitting methods on unstructured triangular meshes. It highlights the reasons why existing approaches fail and proposes a procedure which can be applied to any high order fluctuation splitting scheme to impose positivity on it. Its success is demonstrated through application to a series of linear and nonlinear scalar problems, using a pseudo-time-stepping technique to reach steady state solutions on two-dimensional unstructured meshes.

On the accuracy of one-dimensional models of steady converging/diverging open channel flows

Shallow water flows through open channels with varying breadth are commonly modelled by a system of one-dimensional equations, despite the two-dimensional nature of the geometry and the solution. In this work steady state flows in converging/diverging channels are studied in order to determine the range of parameters (flow speed and channel breadth) for which the assumption of quasi-one-dimensional flow is valid. This is done by comparing both exact and numerical solutions of the one-dimensional model with numerical solutions of the corresponding two-dimensional flows. It is shown that even for apparently gentle constrictions, for which the assumptions from which the one-dimensional model is derived are valid, significant differences can occur. Furthermore, it is shown how the nature of the flow depends on the manner in which the boundary conditions are applied by contrasting the solutions obtained from two commonly used approaches. A brief description is also given of the numerical methods, developed recently for the solution of the one- and two-dimensional shallow water equations, and used to produce the results presented in this paper. Copyright