Emmanuel Hanert

Advection schemes for unstructured grid ocean modelling

We study advection schemes for unstructured grid ocean models. Four linear advection schemes are investigated by solving a scalar transport equation. Schemes under consideration include continuous, nonconforming and discontinuous finite elements and finite volumes. A comprehensive derivation of the numerical schemes is presented and conservation and dispersion properties are discussed. An assessment is made by performing the test problem introduced by Hecht et al. [J. Geophys. Res. 100 (1995) 20763] in which a passive scalar field is advected through an analytical Stommel gyre. It is found that continuous finite elements and finite volumes have some difficulties to represent accurately solutions with steep gradients. As a result they are prone to generate unphysical oscillations. On the other hand, discontinuous and non-conforming finite element schemes perform better. This is due to their higher flexibility that makes them better suited to highly sheared flows

A radial basis functions method for fractional diffusion equations

One of the ongoing issues with fractional diffusion models is the design of an efficient high-order numerical discretization. This is one of the reasons why fractional diffusion models are not yet more widely used to describe complex systems. In this paper, we derive a radial basis functions (RBF) discretization of the one-dimensional space-fractional diffusion equation. In order to remove the ill-conditioning that often impairs the convergence rate of standard RBF methods, we use the RBF-QR method [1,33]. By using this algorithm, we can analytically remove the ill-conditioning that appears when the number of nodes increases or when basis functions are made increasingly flat. The resulting RBF-QR-based method exhibits an exponential rate of convergence for infinitely smooth solutions that is comparable to the one achieved with pseudo-spectral methods. We illustrate the flexibility of the algorithm by comparing the standard RBF and RBF-QR methods for two numerical examples. Our results suggest that the global character of the RBFs makes them well-suited to fractional diffusion equations. They naturally take the global behavior of the solution into account and thus do not result in an extra computational cost when moving from a second-order to a fractional-order diffusion model. As such, they should be considered as one of the methods of choice to discretize fractional diffusion models of complex systems.

A comparison of three finite elements to solve the linear shallow water equations

The purpose of the present study is to select a convenient mixed finite element formulation for ocean modelling. The finite element equivalents of Arakawas A-, B- and C-grids are investigated by using the linear shallow water equations. Numerical and analytical techniques are used to study the types of computational noise present in each element. It is shown that the P1P1 and the P1P0 element (the equivalents of the A- and B-grids respectively) allow the presence of spurious computational modes in the elevation field. For the PIP, element, these modes can be filtered out by adding a stabilizing term to the continuity equation. This method, although consistent, can lead to dissipative unphysical effects at the discrete level. The (P1P0)-P-perpendicular to element or low order Raviart-Thomas element (corresponding to the C-grid) is free of elevation noise and represents well inertia-gravity waves when the deformation radius is resolved but presents computational velocity modes. These modes are however filtered out in a more complex model in which the momentum diffusion term is not neglected

Front dynamics in fractional-order epidemic models

A number of recent studies suggest that human and animal mobility patterns exhibit scale-free, Lévy-flight dynamics. However, current reaction-diffusion epidemics models do not account for the superdiffusive spread of modern epidemics due to Lévy flights. We have developed a SIR model to simulate the spatial spread of a hypothetical epidemic driven by long-range displacements in the infective and susceptible populations. The model has been obtained by replacing the second-order diffusion operator by a fractional-order operator. Theoretical developments and numerical simulations show that fractional-order diffusion leads to an exponential acceleration of the epidemics front and a power-law decay of the fronts leading tail. Our results indicate the potential of fractional-order reaction-diffusion models to represent modern epidemics.

A Chebyshev PseudoSpectral Method to Solve the Space-Time Tempered Fractional Diffusion Equation

A Chebyshev pseudospectral method to solve the space–time tempered fractional diffusion equation

How to avoid unbounded drug accumulation with fractional pharmacokinetics

A number of studies have shown that certain drugs follow an anomalous kinetics that can hardly be represented by classical models. Instead, fractional-order pharmacokinetics models have proved to be better suited to represent the time course of these drugs in the body. Unlike classical models, fractional models can represent memory effects and a power-law terminal phase. They give rise to a more complex kinetics that better reflects the complexity of the human body. By doing so, they also spotlight potential issues that were ignored by classical models. Among those issues is the accumulation of drug that carries on indefinitely when the infusion rate is constant and the elimination flux is fractional. Such an unbounded accumulation could have important clinical implications and thus requires a solution to reach a steady state. We have considered a fractional one-compartment model with a continuous intravenous infusion and studied how the infusion rate influences the total amount of drug in the compartment. By taking an infusion rate that decays like a power law, we have been able to stabilize the amount of drug in the compartment. In the case of multiple dosing administration, we propose recurrence relations for the doses and the dosing times that also prevent drug accumulation. By introducing a numerical discretization of the model equations, we have been able to consider a more realistic two-compartment model with both continuous infusion and multiple dosing administration. That numerical model has been applied to amiodarone, a drug known to have an anomalous kinetics. Numerical results suggest that unbounded drug accumulation can again be prevented by using a drug input function that decays as a power law.

A Lévy-flight diffusion model to predict transgenic pollen dispersal

The containment of genetically modified (GM) pollen is an issue of significant concern for many countries. For crops that are bee-pollinated, model predictions of outcrossing rates depend on the movement hypothesis used for the pollinators. Previous work studying pollen spread by honeybees, the most important pollinator worldwide, was based on the assumption that honeybee movement can be well approximated by Brownian motion. A number of recent studies, however, suggest that pollinating insects such as bees perform Lévy flights in their search for food. Such flight patterns yield much larger rates of spread, and so the Brownian motion assumption might significantly underestimate the risk associated with GM pollen outcrossing in conventional crops. In this work, we propose a mechanistic model for pollen dispersal in which the bees perform truncated Lévy flights. This assumption leads to a fractional-order diffusion model for pollen that can be tuned to model motion ranging from pure Brownian to pure Lévy. We parametrize our new model by taking the same pollen dispersal dataset used in Brownian motion modelling studies. By numerically solving the model equations, we show that the isolation distances required to keep outcrossing levels below a certain threshold are substantially increased by comparison with the original predictions, suggesting that isolation distances may need to be much larger than originally thought.

Unstructured-mesh modeling of the Congo river-to-sea continuum

With the second largest outflow in the world and one of the widest hydrological basins, the Congo River is of a major importance both locally and globally. However, relatively few studies have been conducted on its hydrology, as compared to other great rivers such as the Amazon, Nile, Yangtze, or Mississippi. The goal of this study is therefore to help fill this gap and provide the first high-resolution simulation of the Congo river-estuary-coastal sea continuum. To this end, we are using a discontinuous-Galerkin finite element marine model that solves the two-dimensional depth-averaged shallow water equations on an unstructured mesh. To ensure a smooth transition from river to coastal sea, we have considered a model that encompasses both hydrological and coastal ocean processes. An important difficulty in setting up this model was to find data to parameterize and validate it, as it is a rather remote and understudied area. Therefore, an important effort in this study has been to establish a methodology to take advantage of all the data sources available including nautical charts that had to be digitalized. The model surface elevation has then been validated with respect to an altimetric database. Model results suggest the existence of gyres in the vicinity of the river mouth that have never been documented before. The effect of those gyres on the Congo River dynamics has been further investigated by simulating the transport of Lagrangian particles and computing the water age.

A fully-explicit discontinuous Galerkin hydrodynamic model for variably-saturated porous media

Groundwater flows play a key role in the recharge of aquifers, the transport of solutes through subsurface systems or the control of surface runoff. Predicting these processes requires the use of groundwater models with their applicability directly linked to their accuracy and computational efficiency. In this paper, we present a new method to model water dynamics in variablysaturated porous media. Our model is based on a fully-explicit discontinuous-Galerkin formulation of the 3D Richards equation, which shows a perfect scaling on parallel architectures. We make use of an adapted jump penalty term for the discontinuous-Galerkin scheme and of a slope limiter algorithm to produce oscillation-free exactly conservative solutions. We show that such an approach is particularly well suited to infiltration fronts. The model results are in good agreement with the reference model Hydrus-1D and seem promising for large scale applications involving a coarse representation of saturated soil.

Meshfree Approximation for Multi-Asset Options

We price multi-asset options by solving their price partial differential equations using a meshfree approach with radial basis functions under jump-diffusion and geometric Brownian motion frameworks. In the geometric Brownian motion framework, we propose an effective technique that breaks the multi-dimensional problem to multiple 3D problems. We solve the price PDEs or PIDEs with an implicit meshfree scheme using thin-plate radial basis functions. Meshfree approach is very accurate, has high order of convergence and is easily scalable and adaptable to higher dimensions and different payoff profiles. We also obtain closed form approximations for the option Greeks. We test the model on American crack spread options traded on NYMEX.