Prof Matthew Hubbard

Development and application of computational algorithms for approximating multidimensional systems of nonlinear conservation laws. These systems of partial differential equations typically represent fluid flow (e.g. Euler equations, shallow water equations) and are predominantly hyperbolic in nature, which necessitates the use of upwind algorithms. My focus is on the development of "clever" algorithms, for which the numerical approximation inherits the fundamental properties of the underlying mathematical model, a goal which is particularly challenging in multiple space dimensions.

• Residual distribution schemes were proposed in the 1980s specifically to apply upwinding (i.e. the numerical approximation respects the direction in which information travels in the mathematical model) in a genuinely multidimensional manner. My PhD applied them to the steady shallow water equations [GHP95, HB97] in combination with simple, but effective, mesh movement techniques [BH98]. Subsequent work studied the application of boundary conditions [H01], and created a multidimensional "limiting" procedure [HR00], a more sophisticated approach to mesh movement [BLH02] and higher order accuracy [HL05, H07, HM09]. Most recent developments involve a discontinuous representation [H08a, H08b] which, when combined with a space-time discretisation, allows the creation of unconditionally positive, second order accurate schemes for time-dependent problems [HR11, SHR12, SH13]. Current work is focussing on improving the efficiency of these schemes [WHR12].


• Finite volume schemes are currently the most commonly used for approximation advection-dominated flows. I have applied these schemes to a range of problems, including shallow water flows in channels [HG00], wave run-up and overtopping [HD02], atmospheric dynamics [H02, HN03] and tumour growth [HB13], as well as outlining a framework for multidimensional slope limiting [H99].


• Well-balanced schemes consider situations where the fluxes of the conservation law are balanced by additional terms. I have helped to develop upwind finite volume schemes which preserve hydrostatic balance exactly for shallow water flows over variable bed topography [HG00, HD02, SHR12] and geostrophic balance approximately for shallow water flows in a rotating frame of reference [SH13].

Development and application of techniques which improve the efficiency of computational algorithms by modifying the mesh on which the approximation is carried out.

• Adaptive mesh refinement is a commonly-used technique by which the resolution of the computational mesh is adapted locally to the features of the approximation by refinement and derefinement. I have applied it to coastal modelling [HD02], and atmospheric dynamics [H02, HN03] as well as supervising work on applying it to electromechanical cardiac dynamics [KBHG11].


• Mesh movement is an alternative approach to adapting the computational mesh in which the nodes are moved to resolve the important features of the solution. I have helped to apply it to steady state fluid flow problems in combination with residual distribution schemes [BH98, BLH02] and supervised work on applying it to electromechanical cardiac dynamics on a deforming domain using finite element schemes [KBHG11].


• Moving mesh finite elements [BHJ05a, BHJ05b, BHJ11] provide a new approach to approximating multidimensional moving boundary problems in which the mesh node and boundary velocities are derived from a conservation principle. It can be improved using scale invariance properties of the underlying equations [BHJJ06], strongly imposing Dirichlet boundary conditions in a conservative manner [HBJ09] and using different monitor functions to guide the mesh movement [MHJ11], and has been used to track internal interfaces without any artificial smoothing [BHJM09].

Computational simulation is growing in importance in biomedical modelling, though it plays a slightly different role to that in its more traditional fields of application. Instead of being used to seek progressively more accurate approximations in larger and more complex geometries to better understand phenomena supported by generally agreed mathematical models, the models themselves are still under development. It is therefore essential that the algorithms used do not introduce spurious behaviour which might be interpreted as a feature of the mathematical model. The nature of biomedical applications also means that multiscale modelling, using both discrete and continuous approaches, is important.

• Cardiac modelling is one of the biomedical applications in which computational simulation is best established. I have been involved in the development of an adaptive computational model of cardiac dynamics in which the electrical activity in the cells is coupled with mechanical response of the tissue [KBHG11]. This has subsequently been used to investigate end-stage heart disease.


• Cancer growth is an area in which computational simulation has only recently become commonplace, allowing analysis of the behaviour of more complex representations of the cell and tissue interactions. I have developed a multidimensional, multiphase model of vascular tumour growth [HB13] which couples finite element and finite volume schemes.


• Drug delivery to tumours is a crucial aspect of cancer therapy which is not well understood. I have recently become involved in a project which is developing mathematical and computational models for the delivery of doxorubicin alongside in vitro experiments for validation. I will also shortly start work on a similar project working on combination therapy (temozolomide and radiation therapy) for glioma, a particularly dangerous form of brain tumour.


• Angiogenesis (the creation and growth of new blood vessels) is a biological process which is fundamental to the growth of cancer and therefore a target for many therapeutic strategies. I have been involved in investigating the relationship between different ways of modelling angiogenesis [HJS09].

The systems of equations that arise from discretising partial differential equations in multiple space dimensions are typically very large (in 3D it is common to have tens or hundreds of millions of equations to solve for the same number of unknowns. It is therefore essential to have efficient methods for solving these linear and nonlinear systems.

• Multigrid is a multilevel iterative technique which is capable of solving linear and nonlinear systems of equations with optimal efficiency, i.e. the work required for each iteration is proportional to the size of the system and the number of iterations is independent of the size of the system. I am involved in two projects working on the mathematical and empirical analysis of nonlinear multigrid algorithms and their application to complex systems of partial differential equations.