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We look at the periodic behaviour of the Earth’s glacial cycles and the transitions between different periodic states when either external parameters (such as $\omega $) or internal parameters (such as $d$) are varied. We model this using the PP04 model of climate change. This is a forced discontinuous Filippov (non-smooth) dynamical system. When periodically forced this has coexisting periodic orbits. We find that the transitions in this system are mainly due to grazing events, leading to grazing bifurcations. An analysis of the grazing bifurcations is given and the impact of these on the domains of attraction and regions of existence of the periodic orbits is determined under various changes in the parameters of the system. Grazing transitions arise for general variations in the parameters (both internal and external) of the PP04 model. We find that the grazing transitions between the period orbits resemble those of the Mid-Pleistocene-Transition.

In March 2020 mathematics became a key part of the scientific advice to the UK government on the pandemic response to COVID-19. Mathematical and statistical modelling provided critical information on the spread of the virus and the potential impact of different interventions. The unprecedented scale of the challenge led the epidemiological modelling community in the UK to be pushed to its limits. At the same time, mathematical modellers across the country were keen to use their knowledge and skills to support the COVID-19 modelling effort. However, this sudden great interest in epidemiological modelling needed to be coordinated to provide much-needed support, and to limit the burden on epidemiological modellers already very stretched for time. In this paper we describe three initiatives set up in the UK in spring 2020 to coordinate the mathematical sciences research community in supporting mathematical modelling of COVID-19. Each initiative had different primary aims and worked to maximise synergies between the various projects. We reflect on the lessons learnt, highlighting the key roles of pre-existing research collaborations and focal centres of coordination in contributing to the success of these initiatives. We conclude with recommendations about important ways in which the scientific research community could be better prepared for future pandemics. This manuscript was submitted as part of a theme issue on “Modelling COVID-19 and Preparedness for Future Pandemics”.

In this paper, we perform a careful analysis of the forced PP04 model for climate change, in particular the behaviour of the ice ages. This system models the transition from a glacial to an inter-glacial state through a sudden release of oceanic carbon dioxide into the atmosphere. This process can be cast in terms of a Filippov dynamical system, with a discontinuous change in its dynamics related to the carbon dioxide release. By using techniques from the theory of non-smooth dynamical systems, we give an analysis of this model in the cases of both no insolation forcing and also periodic insolation forcing. This reveals a rich, and novel, dynamical structure to the solutions of the PP04 model. In particular, we see synchronized periodic solutions with subtle regions of existence which depend on the amplitude and frequency of the forcing. The orbits can be created/destroyed in both smooth and discontinuity-induced bifurcations. We study both the orbits and the transitions between them and make comparisons with actual climate dynamics.

Funder: Isaac Newton Institute for Mathematical Sciences; Id: http://dx.doi.org/10.13039/501100005347, Funder: Wellcome Trust; Id: http://dx.doi.org/10.13039/100004440, Funder: Medical Research Council; Id: http://dx.doi.org/10.13039/501100000265, Funder: UKRI, Funder: University of Nottingham; Id: http://dx.doi.org/10.13039/501100000837, In this paper, we present work on SARS-CoV-2 transmission in UK higher education settings using multiple approaches to assess the extent of university outbreaks, how much those outbreaks may have led to spillover in the community, and the expected effects of control measures. Firstly, we found that the distribution of outbreaks in universities in late 2020 was consistent with the expected importation of infection from arriving students. Considering outbreaks at one university, larger halls of residence posed higher risks for transmission. The dynamics of transmission from university outbreaks to wider communities is complex, and while sometimes spillover does occur, occasionally even large outbreaks do not give any detectable signal of spillover to the local population. Secondly, we explored proposed control measures for reopening and keeping open universities. We found the proposal of staggering the return of students to university residence is of limited value in terms of reducing transmission. We show that student adherence to testing and self-isolation is likely to be much more important for reducing transmission during term time. Finally, we explored strategies for testing students in the context of a more transmissible variant and found that frequent testing would be necessary to prevent a major outbreak.

This report details the work carried out by the Study Group on workflow modelling of con- struction projects. Data on the progress of about a hundred projects over a single five-year planning period were provided by Heathrow Airport (the client) and their four Tier 1 construction contrac- tors. These data are mapped and analysed. Several unusual features are discovered. For example, most projects undergo several tens of adjustments in their scope and price such that while most projects are technically completed under budget, the price and duration is significantly higher than originally planned. The main question addressed was whether an optimised scheduling of the project would lead to decreased costs and more rapid completion. First, a machine learning approach is used to gain insight onto which factors are most significant in predicting the final cost and duration of each project. If more data were available, these methods could be further exploited to allow for predictions to be made on which projects are likely to over-run or go over budget and to examine connections between projects at the subcontractor level. In addition to the data-centric approach, a complementary mathematical model was de- veloped to gain a better understanding of the effect of resource constraints on cost and price extension due to resource competition of concurrent projects, ignoring the confound- ing effect of scope creep seen in the data. The model takes the form of a discrete time stochastic simulation, whose parameters are fit to the existing data. Tentative conclusions from the model indicate that better outcomes can be achieved by spreading out project start dates, and by prioritising completion of smaller projects. While more data is needed to validate the model, the results suggested that gains can be made if more thoughtful scheduling of projects is implemented, and also if the prioritisation of projects is monitored and adjusted intelligently. Our major recommendation to Heathrow Airport is to collect or retrieve more data, as outlined in the report, so that both models can be made more realistic and useful. This would allow Heathrow Airport and their contractors to develop and test strategies to make the system more efficient, ultimately saving time and money.

We make an analytic study of the diffusive, dispersive and overall errors, which arise when using semi-implicit semi-Lagrangian (SISL) finite difference methods to approximate those travelling wave solutions of the one-dimensional Burgers' equation with small diffusion, which develop sharp fronts. For the case of a fixed uniform spatial mesh, with piecewise linear interpolation, a backward error analysis approach is used to construct a precise formal analytic description of the front profile of the numerical approximation to this solution. From this description it is possible to obtain precise estimates of the front width and the front speed in terms of the spatial and temporal step size and to express the overall solution error in terms of these. These formal estimates agree closely with numerical calculations, both qualitatively and quantitatively, and display a roughly periodic behaviour as the number N x of mesh points increases, and the CFL number passes through integer values. In particular, they show that despite the otherwise poor resolution of the method, the front width is closely approximated when the CFL number is close to an integer, and the front speed is closely approximated when it is close to a half integer. The overall L 2 error also shows super-convergence for certain values of N x . This possibly motivates doing two calculations with different N x when using the SISL method on such problems to separately minimise the diffusive and dispersive errors. Similar errors in the front width and speed are observed for a number of different interpolation schemes with and without flux limiters.

Modelling the view factor, F, of a ‘grain-like’ observer near a tilted pool fire assumed to be cylindrical in shape, requires the observer (on the same ground level as the pool fire) to be as close as possible to the flame surface so that the flame surface is viewed as a plane. The orientation of the observer is to receive a maximum view, with the view factor integrated over the flame area seen by the observer. The derived expression for F was expressed in terms of standard functions and sensitive parameters: α, defined to ensure the observer receives a maximum view of the flame surface and β, the angle of inclination of the differential observer plane. Results from a planar approximation to F are compared with those of PHAST 7.2 simulated results in MATLAB. The values of the planar approximation to F was found to increase with increasing β. This suggests that the larger the value of β, the more the radiation received by a near observer. For β = 48 . 961 ∘ , horizontal distance, X = 30 m, α = 0.01 , flame length, L = 12.14 m, tilt angle, θ = 55 . 17 ∘ , and pool diameter, D = 5 m: planar and PHAST 7.2 approximations to F were found to coincide up to 6 significant digits and differ by 3.2 × 10 − 8 . The close agreement between both approximations depend heavily on the choice of flame properties and sensitive parameters. Interestingly, this result may depend sensitively on the choice of parameters, leading to prediction of planar approximation to F much higher or lower than those of PHAST 7.2 approximations. The advantage of this approach is that a knowledge of the plane where the largest concentration of radiation is located will help minimise loss of lives to fire hazards and improve the efficiency of risk safety assessment/management.

The COVID-19 pandemic has caused unprecedented disruption, particularly in retail. Where essential demand cannot be fulfilled online, or where more stringent measures have been relaxed, customers must visit shop premises in person. This naturally gives rise to some risk of susceptible individuals (customers or staff) becoming infected. It is essential to minimize this risk as far as possible while retaining economic viability of the shop. We therefore explore and compare the spread of COVID-19 in different shopping situations involving person-to-person interactions: (i) free-flowing, unstructured shopping; (ii) structured shopping (e.g. a queue). We examine which of (i) or (ii) may be preferable for minimizing the spread of COVID-19 in a given shop, subject to constraints such as the geometry of the shop; compliance of the population to local guidelines; and additional safety measures which may be available to the organizers of the shop. We derive a series of conclusions, such as unidirectional free movement being preferable to bidirectional shopping, and that the number of servers should be maximized as long as they can be well protected from infection.

We study the behavior at tipping points close to non-smooth fold bifurcations in non-autonomous systems. The focus is the Stommel-Box, and related climate models, which are piecewise-smooth continuous dynamical systems, modeling thermohaline circulation. We obtain explicit asymptotic expressions for the behavior at tipping points in the settings of both slowly varying freshwater forcing and rapidly oscillatory fluctuations. The results, based on combined multiple scale and local analyses, provide conditions for the sudden transitions between temperature-dominated and salinity-dominated states. In the context of high frequency oscillations, a multiple scale averaging approach can be used instead of the usual geometric approach normally required for piecewise-smooth continuous systems. The explicit parametric dependencies of advances and lags in the tipping show a competition between dynamic features of the model. We make a contrast between the behavior of tipping points close to both smooth Saddle–Node Bifurcations and the non-smooth systems studied on this paper. In particular we show that the non-smooth case has earlier and more abrupt transitions. This result has clear implications for the design of early warning signals for tipping in the case of the non-smooth dynamical systems which often arise in climate models.

A coupled Sun-to-Earth model is the goal for accurate forecasting of space weather. A key component of such a model is a whole atmosphere model – a general circulation model extending from the ground into the upper atmosphere – since it is now known that the lower atmosphere also drives variability and space weather in the upper atmosphere, in addition to solar variability. This objective motivates the stable extension of The Met Office’s Unified Model (UM) into the Mesosphere and Lower Thermosphere (MLT), acting as a first step towards a whole atmosphere model. At the time of performing this research, radiation and chemistry schemes that are appropriate for use in the MLT had not yet been implemented. Furthermore, attempts to run the model with existing parameterizations and a raised upper boundary led to an unstable model with inaccurate solutions. Here, this instability is examined and narrowed down to the model’s radiation scheme – its assumption of Local Thermodynamic Equilibrium (LTE) is broken in the MLT. We subsequently address this issue by relaxation to a climatological temperature profile in this region. This provides a stable extended UM which can be used as a developmental tool for further examination of the model performance. The standard vertical resolution used in the UM above 70 km is too coarse (approx. 5 km) to represent waves that are important for MLT circulation. We build on the success of the nudging implementation by testing the model at an improved vertical resolution. Initial attempts to address this problem with a 3 km vertical resolution and a 100 km lid were successful, but on increasing the resolution to 1.5 km the model becomes unstable due to large horizontal and vertical wind velocities. Increasing the vertical damping coefficient, which damps vertical velocities near the upper boundary, allows a successful year long climatology to be produced with these model settings. With the goal of a whole atmosphere model we also experiment with an increased upper boundary height. Increasing the upper model boundary to 120 and 135 km also leads to stable simulations. However, a 3 km resolution must be used and it is necessary to further increase the vertical damping coefficient. This is highly promising initial work to raise the UM into the MLT, and paves the way for the development of a whole atmosphere model.

Space weather driven atmospheric density variations affect low Earth orbit (LEO) satellites during all phases of their operational lifetime. Rocket launches, re-entry events and space debris are also similarly affected. A better understanding of space weather processes and their impact on atmospheric density is thus critical for satellite operations as well as for safety issues. The Horizon 2020 project Space Weather Atmosphere Model and Indices (SWAMI) project, which started in January 2018, aims to enhance this understanding by:Developing improved neutral atmosphere and thermosphere models, and combining these models to produce a new whole atmosphere model.Developing new geomagnetic activity indices with higher time cadence to enable better representation of thermospheric variability in the models, and improving the forecast of these indices.The project stands out by providing an integrated approach to the satellite neutral environment, in which the main space weather drivers are addressed together with model improvement. The outcomes of SWAMI will provide a pathway to improved space weather services as the project will not only address the science issues, but also the transition of models into operational services.The project aims to develop a unique new whole atmosphere model, by extending and blending the Unified Model (UM), which is the Met Office weather and climate model, and the Drag Temperature Model (DTM), which is a semi-empirical model which covers the 120–1500 km altitude range. A user-focused operational tool for satellite applications shall be developed based on this. In addition, improved geomagnetic indices shall be developed and shall be used in the UM and DTM for enhanced nowcast and forecast capability.In this paper, we report on progress with SWAMI to date. The UM has been extended from its original upper boundary of 85 km to run stably and accurately with a 135 km lid. Developments to the UM radiation scheme to enable accurate performance in the mesosphere and lower thermosphere are described. These include addition of non-local thermodynamic equilibrium effects and extension to include the far ultraviolet and extreme ultraviolet. DTM has been re-developed using a more accurate neutral density observation database than has been used in the past. In addition, we describe an algorithm to develop a new version of DTM driven by geomagnetic indices with a 60 minute cadence (denoted Hp60) rather than 3-hourlyKpindices (and corresponding ap indices). The development of the Hp60 index, and the Hp30 and Hp90 indices, which are similar to Hp60 but with 30 minute and 90 minute cadences, respectively, is described, as is the development and testing of neural network and other machine learning methods applied to the forecast of geomagnetic indices.

We introduce a novel location-based moving mesh algorithm MMSISL in which the arrival points in the Semi-Implicit Semi-Lagrangian (SISL) algorithm are located by using an equidistribution strategy. This algorithm gives a natural coupling between moving mesh methods and SISL methods. It involves little extra cost in implementation as it exploits the interpolation methods already embedded in the SISL algorithm. We apply this method to a number of partial differential equation problems in one-dimension, each of which have sharply defined features. We show that using MMSISL leads to a markedly improved performance over fixed mesh methods, with significantly reduced errors. We also show that unlike many adaptive schemes, no issues arise in the MMSISL algorithm from a CFL condition imposed restriction on the time step.

Turbochargers are a vital component for aiding engine manufacturers in meeting the latest emissions standards. However, their range of operation is limited for low mass flows by compressor surge. Operation in surge results in pressure and mass flow oscillations that are often damaging to the compressor and its installation. Since surge is a highly complex flow regime, full unsteady 3D models are generally too computationally expensive to run. The majority of current low-dimensional surge models use a cubic compressor characteristic that needs to be fitted to experimental data. Therefore, each time a compressor is studied using these models, costly experimental testing is required. In this paper, a new technique for obtaining an axisymmetric centrifugal compressor characteristic is presented. This characteristic is built using the equations of mass, momentum and energy from first principles in order to provide a more complete model than those currently obtained via experimental data. This approach enables us to explain the resulting cubic-like shape of the characteristic and hence to identify impeller inlet stall as a route into surge. The characteristic is used within a quasi-steady, map-based surge model in order to demonstrate its ability to predict the onset of surge while only providing geometric data as input. Validation is provided for this model by discussion of the qualitative flow dynamics and a good fit to experimental data, especially for low impeller speeds and pressure ratios.

In this sketch, Chris argues that maths is a highly creative subject, whose playfulness can be detected in a vast range of games and pattern spotting. He shows how arcane formulae actually underpin the completion of some of the most mundane daily tasks, such as turning on and off the light. He alludes to the mystery of maths (which some readers may feel better able to unpack than others) and rounds off his defence of maths with the flourish of a magic trick.

An equation of Monge–Ampere type has, for the first time, been solved numerically on the surface of the sphere in order to generate optimally transported (OT) meshes, equidistributed with respect to a monitor function. Optimal transport generates meshes that keep the same connectivity as the original mesh, making them suitable for r-adaptive simulations, in which the equations of motion can be solved in a moving frame of reference in order to avoid mapping the solution between old and new meshes and to avoid load balancing problems on parallel computers. The semi-implicit solution of the Monge–Ampere type equation involves a new linearisation of the Hessian term, and exponential maps are used to map from old to new meshes on the sphere. The determinant of the Hessian is evaluated as the change in volume between old and new mesh cells, rather than using numerical approximations to the gradients. OT meshes are generated to compare with centroidal Voronoi tessellations on the sphere and are found to have advantages and disadvantages; OT equidistribution is more accurate, the number of iterations to convergence is independent of the mesh size, face skewness is reduced and the connectivity does not change. However anisotropy is higher and the OT meshes are non-orthogonal. It is shown that optimal transport on the sphere leads to meshes that do not tangle. However, tangling can be introduced by numerical errors in calculating the gradient of the mesh potential. Methods for alleviating this problem are explored. Finally, OT meshes are generated using observed precipitation as a monitor function, in order to demonstrate the potential power of the technique.

We study the asymptotic behavior of sharp front solutions arising from the nonlinear diffusion equation θt=(D(θ)θx)x, where the diffusivity is an exponential function D(θ)=Doexp(βθ). This problem arises, for example, in the study of unsaturated flow in porous media where θ represents the liquid saturation. For physical parameters corresponding to actual porous media, the diffusivity at the residual saturation is D(0)=Do≪1 so that the diffusion problem is nearly degenerate. Such problems are characterized by wetting fronts that sharply delineate regions of saturated and unsaturated flow, and that propagate with a well-defined speed. Using matched asymptotic expansions in the limit of large β, we derive an analytical description of the solution that is uniformly valid throughout the wetting front. This is in contrast with most other related analyses that instead truncate the solution at some specific wetting front location, which is then calculated as part of the solution, and beyond that location, the solution is undefined. Our asymptotic analysis demonstrates that the solution has a four-layer structure, and by matching through the adjacent layers, we obtain an estimate of the wetting front location in terms of the material parameters describing the porous medium. Using numerical simulations of the original nonlinear diffusion equation, we demonstrate that the first few terms in our series solution provide approximations of physical quantities such as wetting front location and speed of propagation that are more accurate (over a wide range of admissible β values) than other asymptotic approximations reported in the literature.

In this article we give a detailed asymptotic analysis of the near critical self-similar blowup solutions to the Generalised Korteweg–de Vries equation (GKdV). We compare this analysis to some careful numerical calculations. It has been known that for a nonlinearity that has a power larger than the critical value p = 5 , solitary waves of the GKdV can become unstable and become infinite in finite time, in other words they blow up. Numerical simulations presented in Klein and Peter (2015) indicate that if p > 5 the solitary waves travel to the right with an increasing speed, and simultaneously, form a similarity structure as they approach the blow-up time. This structure breaks down at p = 5 . Based on these observations, we rescale the GKdV equation to give an equation that will be analysed by using asymptotic methods as p → 5 + . By doing this we resolve the complete structure of these self-similar blow-up solutions and study the singular nature of the solutions in the critical limit. In both the numerics and the asymptotics, we find that the solution has sech-like behaviour near the peak. Moreover, it becomes asymmetric with slow algebraic decay to the left of the peak and much more rapid algebraic decay to the right. The asymptotic expressions agree to high accuracy with the numerical results, performed by adaptive high-order solvers based on collocation or finite difference methods.

We consider the fully adaptive space–time discretization of a class of nonlinear heat equations by Rothe’s method. Space discretization is based on adaptive polynomial collocation which relies on equidistribution of the defect of the numerical solution, and the time propagation is realized by an adaptive backward Euler scheme. From the known scaling laws, we infer theoretically the optimal grids implying error equidistribution, and verify that our adaptive procedure closely approaches these optimal grids.

In moving mesh methods, the underlying mesh is dynamically adapted without changing the connectivity of the mesh. We specifically consider the generation of meshes which are adapted to a scalar monitor function through equidistribution. Together with an optimal transport condition, this leads to a Monge-Amp\`ere equation for a scalar mesh potential. We adapt an existing finite element scheme for the standard Monge-Amp\`ere equation to this mesh generation problem; this is a mixed finite element scheme, in which an extra discrete variable is introduced to represent the Hessian matrix of second derivatives. The problem we consider has additional nonlinearities over the basic Monge-Amp\`ere equation due to the implicit dependence of the monitor function on the resulting mesh. We also derive the equivalent Monge-Amp\`ere-like equation for generating meshes on the sphere. The finite element scheme is extended to the sphere, and we provide numerical examples. All numerical experiments are performed using the open-source finite element framework Firedrake., Comment: Updated following reviews, 36 pages

In 2012 the UK Government identified eight great technologies which would act as a focus for future scientific research and funding. Other governments have produced similar lists. These vary from Big Data, through Agri-Science to Energy and its Storage. Mathematics lies at the heart of all of these technologies and acts to unify them all. In this paper I will review all of these technologies and look at the math behind each of them. In particular I will look in some detail at the mathematical issues involved in Big Data and energy. Overall I will aim to show that whilst it is very important that abstract mathematics is supported for its own right, the eight great technologies really do offer excellent opportunities for exciting new mathematical research and applications.

A new discontinuity-induced bifurcation, referred to as nonsmooth Hopf-type bifurcation, observed in a nonautonomous impacting hybrid systems in \(\mathbb {R}^4\) is presented. The system studied models the bouncing motion, repeated instantaneous impacts with friction, in rotating machines with magnetic bearing support. At the nonsmooth Hopf-type bifurcation point a stable regular equilibrium and two unstable small amplitude 1-impact periodic orbits arise. The existence of this bifurcation scenario depends on a complex relationship between damping, the restitution, and the friction coefficient.

During the workshop on Climate Modeling in Nonsmooth Systems, one of the major discussions involved investigating including more realistic elements, such as fluctuations and time variation, in nonsmooth models that undergo a sudden transition, with an emphasis on conceptual climate models. A number of models were discussed, including the Stommel 1961 model, the Paillard 1997 model, the Eisenman–Wettlaufer 2009 model, and the Hogg 2008 model.

In the context of numerical solution of PDEs, dynamic mesh redistribution methods (r-adaptive methods) are an important procedure for increasing the resolution in regions of interest, without modifying the connectivity of the mesh. Key to the success of these methods is that the mesh should be sufficiently refined (locally) and flexible in order to resolve evolving solution features, but at the same time not introduce errors through skewness and lack of regularity. Some state-of-the-art methods are bottom-up in that they attempt to prescribe both the local cell size and the alignment to features of the solution. However, the resulting problem is overdetermined, necessitating a compromise between these conflicting requirements. An alternative approach, described in this paper, is to prescribe only the local cell size and augment this an optimal transport condition to provide global regularity. This leads to a robust and flexible algorithm for generating meshes fitted to an evolving solution, with minimal need for tuning parameters. Of particular interest for geophysical modelling are meshes constructed on the surface of the sphere. The purpose of this paper is to demonstrate that meshes generated on the sphere using this optimal transport approach have good a-priori regularity and that the meshes produced are naturally aligned to various simple features. It is further shown that the sphere's intrinsic curvature leads to more regular meshes than the plane. In addition to these general results, we provide a wide range of examples relevant to practical applications, to showcase the behaviour of optimally transported meshes on the sphere. These range from axisymmetric cases that can be solved analytically to more general examples that are tackled numerically. Evaluation of the singular values and singular vectors of the mesh transformation provides a quantitative measure of the mesh aniso..., Comment: Updated following reviewer comments

This paper describes a fast and reliable method for redistributing a computational mesh in three dimensions which can generate a complex three dimensional mesh without any problems due to mesh tangling. The method relies on a three dimensional implementation of the parabolic Monge-Ampere (PMA) technique, for finding an optimally transported mesh. The method for implementing PMA is described in detail and applied to both static and dynamic mesh redistribution problems, studying both the convergence and the computational cost of the algorithm. The algorithm is applied to a series of problems of increasing complexity. In particular very regular meshes are generated to resolve real meteorological features (derived from a weather forecasting model covering the UK area) in grids with over 2x10^7 degrees of freedom. The PMA method computes these grids in times commensurate with those required for operational weather forecasting.

We derive a moving mesh method based upon ideas from optimal transport theory which is suited to solving PDE problems in meteorology. In particular we show how the Parabolic Monge-Ampere method for constructing a moving mesh in two-dimensions can be coupled successfully to a pressure correction method for the solution of incompressible flows with significant convection and subject to Coriolis forces. This method can be used to resolve evolving small scale features in the flow. In this paper the method is then applied to the computation of the solution to the Eady problem which is observed to develop large gradients in a finite time. The moving mesh method is shown to work and be stable, and to give significantly better resolution of the evolving singularity than a fixed, uniform mesh.

We present a model of multilayer folding in which layers with bending stiffness EI are separated by a very stiff elastic medium of elasticity k2 and subject to a horizontal load P. By using a dynamical system analysis of the resulting fourth order equation, we show that as the end shortening per unit length E is increased, then if k2 is large there is a smooth transition from small amplitude sinusoidal solutions at moderate values of P to larger amplitude chevron folds, with straight limbs separated by regions of high curvature when P is large. The chevron solutions take the form of near heteroclinic connections in the phase-plane. By means of this analysis, values for P and the slope of the limbs are calculated in terms of E and k2.

Weather impacts all of our lives and we all take a close interest in it, with every news report finishing with a weather forecast watched by millions. Accurate weather forecasting is essential for the transport, agricultural and energy industries and the emergency and defence services. The Met Office plays a vital role by making 5-day forecasts, using advanced computer algorithms which combine numerical weather predictions (NWP) with carefully measured data (a process known as data assimilation). However, a major limitation on the accuracy of these forecasts is the sub-optimal use of this data. Adaptive methods, developed in a partnership between Bath and the Met Office have been employed to make better use of the data, thus improving the Met Office operational data assimilation system. This has lead to a significant improvement in forecast accuracy as measured by the UK Index [9] with great societal and economic impact. Forecasts, of surface temperatures, in particular, are pivotal for the OpenRoad forecasting system used by local authorities to plan road clearing and gritting when snow or ice are predicted.

We show that the four-dimensional variational data assimilation method (4DVar) can be interpreted as a form of Tikhonov regularization, a very familiar method for solving ill-posed inverse problems. It is known from image restoration problems that L1-norm penalty regularization recovers sharp edges in the image more accurately than Tikhonov, or L2-norm, penalty regularization. We apply this idea from stationary inverse problems to 4DVar, a dynamical inverse problem, and give examples for an L1-norm penalty approach and a mixed total variation (TV) L1–L2-norm penalty approach. For problems with model error where sharp fronts are present and the background and observation error covariances are known, the mixed TV L1–L2-norm penalty performs better than either the L1-norm method or the strong constraint 4DVar (L2-norm)method. A strength of the mixed TV L1–L2-norm regularization is that in the case where a simplified form of the background error covariance matrix is used it produces a much more accurate analysis than 4DVar. The method thus has the potential in numerical weather prediction to overcome operational problems with poorly tuned background error covariance matrices.

We present a method for nding solutions of large-scale binary programming problems where the calculation of derivatives is very expensive. We then apply this method to a topology optimization problem of weight minimisation subject to compliance and buckling constraints. We derive an analytic expression for the derivative of the stress stiness matrix with respect to the density of an element in the nite-element setting. Results are presented for a number of two-dimensional test problems.

In the deformation of layered materials such as geological strata, or stacks of paper, mechanical properties compete with the geometry of layering. Smooth, rounded corners lead to voids between the layers, while close packing of the layers results in geometrically induced curvature singularities. When voids are penalized by external pressure, the system is forced to trade off these competing effects, leading to sometimes striking periodic patterns. In this paper, we construct a simple model of geometrically nonlinear multi-layered structures under axial loading and pressure confinement, with non-interpenetration conditions separating the layers. Energy minimizers are characterized as solutions of a set of fourth-order nonlinear differential equations with contact-force Lagrange multipliers, or equivalently of a fourth-order free-boundary problem. We numerically investigate the solutions of this free-boundary problem and compare them with the periodic solutions observed experimentally.

In this paper we derive an obstacle problem with a free boundary to describe theformation of voids at areas of intense geological folding. An elastic layer is forced by overburdenpressure against a V-shaped rigid obstacle. Energy minimization leads to representation as a nonlinearfourth-order ordinary differential equation, for which we prove there exists a unique solution.Drawing parallels with the Kuhn–Tucker theory, virtual work, and ideas of duality, we highlight thephysical significance of this differential equation. Finally, we show that this equation scales to a singleparametric group, revealing a scaling law connecting the size of the void with the pressure/stiffnessratio. This paper is seen as the first step toward a full multilayered model with the possibility ofvoiding.

Strong constraint 4D-Variational data assimilation (4D-Var) is a method used to create an initialisation for a numerical model, that best replicates subsequent observations of the system it aims to recreate. The method does not take into account the presence of errors in the model, using the model equations as a strong constraint. This paper gives a rigorous and quantitative analysis of the errors introduced into the initialisation through the use of finite difference schemes to numerically solve the model equations. The 1D linear advection equation together with circulant boundary conditions, are chosen as the model equations of interest as they are representative of the advective processes relevant to numerical weather prediction, where 4D-Var is widely used. We consider the deterministic error introduced by finite difference approximations in the form of numerical dissipation and numerical dispersion and identify the relationship between these properties and the error in the 4D-Var initialisation. In particular, we find that a solely numerically dispersive scheme has the potential to introduce destructive interference resulting in the loss of some wavenumber components in the initialisation. Bounds for the error in the initialisation due to finite difference approximations are determined with and without observation errors. The bounds are found to depend on the smoothness of the true initial condition we wish to recover and the numerically dissipative and dispersive properties of the scheme. Numerical results are presented to demonstrate the effectiveness of the bounds. These lead to the conclusion that there exists a critical number of discretisation points when considering full sets of observations, where the effects of both the considered numerical model error and observational errors on the initialisation are minimised. The numerically dissipative and dispersive properties of the finite difference schemes also have the potential to alter the properties of the noise found in observations. Correlated noise structures may be introduced into the 4D-Var initialisation as a result. We determine when this occurs for observational errors in the form of additive white noise and find that the effect is reduced through the use of numerically non-dissipative finite difference schemes.

We use techniques from reaction-diffusion theory to study the blow-up and existence of solutions of the parabolic Monge–Ampere (M-A) equation with power source, with the following basic 2D model $$u_{t}=-|D^{2}u|+|u|^{p-1}u\quad \mathrm{in}\quad \mathbb{R}^{2}\times \mathbb{R}_{+},$$ where in two-dimensions $|D^{2}u|=u_{xx}u_{yy}-(v_{xy})^{2}$ and p > 1 is a fixed exponent. For a class of ‘dominated concave’ and compactly supported radial initial data $u_{0}(x)\geqslant0$ , the Cauchy problem is shown to be locally well posed and to exhibit finite time blow-up that is described by similarity solutions. For p ∈ (1, 2], similarity solutions, containing domains of concavity and convexity, are shown to be compactly supported and correspond to surfaces with flat sides that persist until the blow-up time. The case p > 2 leads to single-point blow-up. Numerical computations of blow-up solutions without radial symmetry are also presented. The parabolic analogy of the parabolic M-A equation in 3D for which $|D^{2}u|$ is a cubic operator is $$u_{t}=|D^{2}u|+|u|^{p-1}u\quad \mathrm{in}\quad \mathbb{R}^{3}\times \mathbb{R}_{+},$$ and is shown to admit a wider set of (oscillatory) self-similar blow-up patterns. Regional self-similarblow-up in a cubic radial model related to the fourth-order M-A equation $$u_{t}=-|D^{4}u|+u^{3}\quad \mathrm{in}\quad \mathbb{R}^{2}\times \mathbb{R}_{+},$$ where the cubic operator $|D^{4}u|$ is the catalecticant 3 × 3 determinant is also briefly discussed.

Optimal state estimation from given observations of a dynamical system by data assimilation is generally an ill-posed inverse problem. In order to solve the problem, a standard Tikhonov, or L2, regularization is used, based on certain statistical assumptions on the errors in the data. The regularization term constrains the estimate of the state to remain close to a prior estimate. In the presence of model error, this approach does not capture the initial state of the system accurately, as the initial state estimate is derived by minimizing the average error between the model predictions and the observations over a time window. Here we examine an alternative L1 regularization technique that has proved valuable in image processing. We show that for examples of flow with sharp fronts and shocks, the L1 regularization technique performs more accurately than standard L2 regularization.

In this paper, we study the dynamics of an impact oscillator with a modified reset law derived from considering a problem (the pinball machine) with an 'active impact'. Typical studies of the impact oscillator consider impacts that are governed by Newton's law of restitution where the velocity after impact is r times less than the incoming velocity. But in this paper, we consider an active impact modelling impacts, which occur in a pinball machine. In such a machine, there exist bumpers that repel the pinball at high velocity when an (even slight) impact is made, imparting an additional velocity V to the rebounding pinball. Such impacts do not obey the normal laws and in this paper, we study how to model them and the subtle dynamics that arises. The analysis proceeds by deriving a 1D map that models the impacts. This map takes the form of a piecewise linear/square-root map with a discontinuity of size proportional to V. The resulting map is similar in many aspects to a 1D 'map-with-a-gap' but also inherits features of the square-root map. We show how the subtle interplay between these two maps leads to the creation of a very large number of new period orbits, which might explain some of the complexity observed in the dynamics of a true pinball machine.

Discrete breathers are found in a nonlinear one dimensional axially loaded mechanical lattice consisting of rigid links supported laterally by linear springs. We find link centered breathers for an odd number of mechanical links and pivot centered breathers where the number of links is even. Substantial parameter regions in load-frequency parameter space are found where these breathers are linearly and nonlinearly stable. This region includes the lattice in tension, in compression, and in the unloaded state. We also find that despite the rigid nature of this mechanical system both the lateral displacement and the energy-per-link are, at least, exponentially localized in the breather core.

We consider four-dimensional variational data assimilation (4DVar) and show that it can be interpreted as Tikhonov or L2-regularisation, a widely used method for solving ill-posed inverse problems. It is known from image restoration and geophysical problems that an alternative regularisation, namely L1-norm regularisation, recovers sharp edges better than L2-norm regularisation. We apply this idea to 4DVar for problems where shocks and model error are present and give two examples which show that L1-norm regularisation performs much better than the standard L2-norm regularisation in 4DVar.

Techniques such as tomographic reconstruction may be used to provide images of electron content in the ionosphere. Models are also available which attempt to describe the dominant physical processes operating in the ionosphere, or the statistical relationships between ionospheric variables. It is sensible to try and couple model output to tomographic images with the aim of inferring the values of driver variables which best replicate some description of electron content imaged in the ionosphere, according to some criterion. This is a challenging task. The following describes an attempt to couple an ionospheric model to a tomographic reconstruction of the geomagnetic storm of 20 November 2003, along a latitudal line segment above north America. A simple model was chosen to reduce the number of input drivers that were varied. The investigation illustrates some of the issues involved in image-model coupling. The ability to make scientific deductions depends on the accuracy of the assumptions in the ionospheric model and the accuracy of the tomographic reconstruction. An ensemble technique was used to help assess confidence in the reconstruction.