Workshop on PDEs: Modelling,
Analysis and Numerical Simulation

# PDE-MANS 2020

### Granada, January 8-16

There will be poster sessions during both the school and the workshop, so let us know if you'd like to present one.

## List of posters

Energy-Dissipating, Positivity-Preserving Schemes for Kinetic Gradient Flows
Rafael Bailo

We propose fully-discrete, implicit-in-time finite-volume schemes for general non-linear non-local Fokker-Planck equations with a gradient flow structure. The schemes verify the positivity-preserving and energy-decaying properties, done conditionally by the second order scheme and unconditionally by the first order counterpart. Dimensional splitting allow for the construction of these schemes with the same properties and a reduced computational cost in any dimension. We will showcase the handling of complicated phenomena: free boundaries, meta-stability, merging, and phase transitions.

Generalised Langevin equation with simulated annealing
Martin Chak

We consider a generalised higher order Langevin equation along with simulated annealing for optimisation of non-convex functions. Under reasonable conditions, we establish convergence of the solution to the corresponding Fokker-Planck equation to a global minimum.

Horizontal gene transfer: numerical comparison between stochastic and deterministic approaches
Susely Figueroa

Horizontal Gene Transfer (HGT) denotes the transmission of genetic material between two living organisms, while the vertical transmission refers to a DNA transfer from parents to their offspring. Consistent experimental evidence report that this phenomenon plays an essential role in the evolution of certain bacterias. In particular, HGT is believed to be the main instrument of developing the antibiotic resistance. In this work, we consider several models which describe this phenomenon: a stochastic jump process (individual-based) and the deterministic nonlinear integro-differential equation obtained as a limit for large populations. We also consider a Hamilton-Jacobi equation, obtained as a limit of the deterministic model under the assumption of small mutations. The goal will be to compare these models with the help of numerical simulations. More specifically, our goal is to understand to which extent the Hamilton-Jacobi model reproduces the qualitative behavior of the stochastic model and the phenomenon of evolutionary rescue in particular.

High-order well-balanced methods for systems of balance laws: a control-based approach
Irene Gómez

In some previous works, a strategy to develop high-order numerical methods for systems of balance laws that preserve all the stationary solutions of the system has been introduced. The key ingredient of these methods is a well-balanced reconstruction operator. A strategy has been also introduced to modify any standard reconstruction operator like MUSCL, ENO, CWENO, etc. in order to be well-balanced. This strategy involves a non-linear problem at every cell at every time step that consists in finding the stationary solution whose average is the given cell value. So far this strategy has been only applied to systems whose stationary solution are known either in explicit or implicit form. The goal of this work is to present a general implementation of this technique that can be applied to any system of balance laws. To do this, the nonlinear problems to be solved in the reconstruction procedure are interpreted as control problems: they consist in finding a solution of an ODE system whose average at the computation interval is given. These problems are written in functional form and the gradient of the functional is computed on the basis of the adjoint problem. Newton's method is applied then to solve the problems. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. To test their efficiency and well-balancedness, the methods are applied to a number of systems of balance laws, ranging from easy academic systems consisting of Burgers equation with some nonlinear source terms to the shallow water equations or Euler equations of gas dynamics with gravity effects.

Flee the toxins: autotoxicity-induced traveling vegetation spots in a biomass-water-toxicity model
Annalisa Iuorio

In recent years it has become increasingly clear that one factor that can serve as an indicator to critical climate changes, and how resilient a given ecosystem is to such changes, is the dynamics of vegetation. This realization has made the understanding of the underlying mechanisms regulating these dynamics extremely important to explore. Motivated by this direction of investigation, a new ecological theory has recently emerged, which identifies the toxic compounds that are produced by the decomposition of organic material as an essential element in the behaviour of local vegetation. The introduction of a new model component modeling biomass autotoxicity induces novel spatiotemporal behaviour of vegetation patterns. In particular, autotoxicity is seen to induce movement and deformation of spot patterns. Our aim is to analytically quantify this novel phenomenon, by considering the model reduced to one spatial dimension. We use geometric singular perturbation theory to obtain an explicit expression for the corresponding asymmetric traveling pulse solution, by constructing a homoclinic orbit in the associated 5-dimensional dynamical system. The temporal stability of this pulse is determined using Evans function techniques. We find that both the plant's sensitivity to toxins and the toxin decay rate significantly influence the behaviour and shape of the biomass pulse.

Crowding and queuing at exits and bottlenecks
Gaspard Jankowiak

Experiments with pedestrians revealed that the geometry of the domain, as well as the incentive of pedestrians to reach a target as fast as possible have a strong influence on the overall dynamics. We propose and validate different mathematical models on the micro-and macroscopic level to study the influence of both effects. We calibrate the models with experimental data and compare the results on the micro-as well as macroscopic level. Our numerical simulations reproduce qualitative experimental features on both levels and indicate how geometry and motivation level influence the observed pedestrian density.

Minmax flux limiter function
Emanuele Macca

Usually, in a high order scheme flux limiters are used to avoid the spurious oscillations near shocks, that would otherwise invalidate the high order spatial reconstruction scheme. In literature, see [Toro]-[Sweby], the flux limiters approach is based on the combination between the TVD property and the hypothesis of monotonicity of the solutions to define the $\varphi.$ Our goal is to introduce a new family of Flux Limiters, based also on the TVD property, that do not use the hypothesis of monotonicity. To do this we have to extend the Sweby region [Sweby] also for negative part of $r$ and $\varphi.$

In particular, we present a method to choose the better flux limiter in a lot of practical situations.

1. [Sweby] P. K. Sweby, TVD Schemes for Inhomogeneous Conservation Laws.} In Notes on Numerical Fluid Mechanics, Vol. 24, Non–Linear Hyperbolic Equations–Theory, Computation Methods and Applications, pages 599–607. Vieweg, 1989.
2. [Toro] E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, third edition, Springer-Verlag Berlin Heidelberg, 2009.

Estimación Lineal Mínimo Cuadrática en Modelos ARH(1) Afectados por Ruido
Felícitas Doris Miranda Huaynalaya

Muchos procesos físicos, biológicos, medio ambientales y geofísicos, incluyen la variabilidad en el espacio y el tiempo. Las dificultades causadas por grandes conjuntos de datos, y el interés por representar las interacciones en el espacio y en el tiempo, así como la variación local a pequeña escala, impulsan el desarrollo de técnicas alternativas a las técnicas clásicas, dentro del contexto de la Estadística Funcional. Este trabajo ofrece una breve revisión de algunas técnicas planteadas, en el contexto de las series funcionales temporales, para abordar el problema de extrapolación espacio-temporal. Concretamente, se hace primero una breve revisión de algunos resultados recientes en el contexto de la regresión lineal funcional y la regresión espacio-temporal, a partir de la teoría de campos aleatorios. Posteriormente, se recogen los resultados sobre estimación paramétrica por máxima verosimilitud, filtrado y extrapolación, basados en el filtrado de Kalman, derivados en los trabajos de Ruiz-Medina y Salmerón (2010) y Salmerón y Ruiz-Medina (2009). Finalmente, se desarrolla, en este trabajo, una implementación alternativa del Filtrado de Kalman, basada en la estimación por mínimos cuadrados de los parámetros del modelo, mediante proyección en las bases ortogonales del operador de autocovarianza empírico del proceso Hilbertiano autorregresivo. Lo que me motivó realizar este trabajo, es la amplia área de aplicación de la estadística funcional, el modelo de regresión lineal espacial y la regresión espacio-temporal. Por lo tanto, el trabajo desarrollado, deja línea abierta para una investigación futura, que es comparar la propuesta de este trabajo (Estimación por mínimo cuadrático) y la Estimación de máxima verosimilitud-POP, terminar con lo que se avanzó en la simulación de datos con el software MatLab y posteriormente aplicar con datos reales en el área de finanza.

Well-balanced finite-volume schemes for hydrodynamic equations with general free energy
Sergio P. Pérez

The development of robust well-balanced numerical methods able to discretely preserve steady states of balance laws has attracted considerable attention since the first works nearly twenty years ago. Standard finite-volume approaches, such as the fractional-step methods, fail to accurately resolve the steady states from balance laws, in which the fluxes need to be exactly balanced with the source terms. To correct this deficiency, the well-balanced schemes are designed so as to discretely satisfy this balance when the steady state is reached. But well-balanced schemes for hydrodynamic equations with a general free energy have not been developed as of yet.

Here we outline a well-balanced finite volume scheme for a general choice of free energy, which could contain different dependencies with respect to the density and external or interaction potentials. This scheme is sufficiently flexible and can be applied to a variety of applications involving shallow water equations, cell chemotaxis, and dynamic-density functional theory, to name but a few. We show that the first and second-order schemes preserve the steady states and the nonnegativity of the density, are consistent with the original system and satisfy a cell entropy inequality.

On a class of genuinely 2D incomplete Riemann solvers for hyperbolic systems
Kleiton Andre Schneider

We propose a general class of genuinely 2D incomplete Riemann solvers for hyperbolic systems of equations. In particular, extensions of the multidimensional HLL solver proposed by Balsara [1] to 2D Polynomial Viscosity Matrix (PVM) finite volume methods are considered. The numerical flux is constructed by assembling, at each edge of the computational mesh, a one-dimensional PVM flux with two purely 2D PVM fluxes defined at corners, which take into account transversal features of the flow through the approximate solution of 2D Riemann problems. The proposed methods are applicable to general hyperbolic systems of conservation laws. A first version of the scheme was recently introduced in our work [2], where applications to magnetohydrodynamics were considered, including an efficient technique for divergence cleaning of the magnetic field based on the nonconservative form of the ideal MHD equations, which provides good results in combination with our 2D solvers. Now, we focus on applications to shallow water systems, in which the source term due the bottom topography introduces an additional difficulty. An elegant way to overcome this difficulty consists in reformulating the problem in nonconservative form. For this reason, we have extended our 2D schemes to the case of nonconservative hyperbolic systems within the framework of path-conservative schemes introduced in the work of Parés [3]. Again, the proposed schemes are applicable to general nonconservative hyperbolic systems. A number of challenging numerical experiments including genuinely 2D effects are presented to test the performances and advantages of the proposed schemes.

1. Balsara, D.S. A two-dimensional HLLC Riemann solver for conservation laws: application to Euler and magnetohydrodynamics flows. JCP, 2012; 231, 7476-7503
2. Gallardo, J.M., Schneider, K.A., Castro, M.J. On a class of two-dimensional incomplete Riemann solvers. JCP, 2019; 386, 541-567
3. Parés, C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Num Anal, 2006; 44, 300-321.

Modelling and simulation of a oscillating water column
Gastón Vergara Hermosilla

In this work we present the mathematical model and simulations of a particular wave energy converter, the so-called oscillating water column. In this device, waves governed by the one-dimensional nonlinear shallow water equations arrive from offshore, encounter a step in the bottom and then arrive into a chamber to change the volume of the air to activate the turbine. The system is reformulated as two transmission problems: one is related to the wave motion over the stepped topography and the other one is related to the wave-structure interaction at the entrance of the chamber. We finally use Riemann invariants to discretize the transmission conditions and the Lax-Friedrichs scheme to get numerical solutions.

Preliminary understandings of the Landau equation as a gradient flow
Jeremy Wu

The Landau equation is a fundamental partial differential equation in kinetic theory. We focus on the spatially homogeneous setting for a parameter gamma in [−4, 3].

We aim to give a reinterpretation of (1) as a gradient flow inspired by a similar effort from Erbar for the closely related Boltzmann equation [1]. This allows for new numerical methods to be developed for (1) [2]. Another avenue of exploration is using the notion of geodesic convexity in gradient flows to hopefully answer the open uniqueness question which is not known in full generality for γ < 0 (although local and partial results are known [3]).

1. M. Erbar, A gradient flow approach to the Boltzmann equation, arXiv:1603.00540 [math.AP] (2016).
2. J. Carrillo, J. Hu, L. Wang, and J. Wu, A particle method for the homogeneous Landau equation, arXiv:1910.03080 [math.AP] (2019).
3. N. Fournier, Uniqueness of Bounded Solutions for the Spatially Homogeneous Landau equation with a Coulomb potential, Communications in Mathematical Physics 299(3): (2010) pp. 765–782.