**Efficient numerical schemes for non-local transport phenomena**

**Participants**

**At Inria:**

** Paola GOATIN** (PI) is Senior Researcher at Inria Sophia Antipolis – Mediterranee (SAM), leader of the ACUMES Project-Team. Expertise: analysis and numerical approximation of hyperbolic systems of conservation laws, application to traffic flow modeling.

** Regis DUVIGNEAU** is Research Scientist at Inria SAM, member of the ACUMES Project-Team. Expertise: numerical methods for simulation and optimization of complex flows.

**Felisia A. CHIARELLO** is PhD student at Inria SAM, EPI ACUMES (since Mai 2017, for 3 years). Subject: Conservation laws with non-local flux.

**At University of Bio-Bio:**

**Luis Miguel VILLADA OSORIO** (co-PI) is Professor and Program Director of Master in Mathematics at University of Bio-Bio, Chile, and External Associate Research at CI2MA University of Concepcion. Expertise: numerical methods for convection-diffusion systems, applications to traffic flows and polydisperse sedimentation.

**At Universidad de Concepcion:**

**Raimund BURGER** is Full Professor and Sub-Director at Center for Research in Mathematical Engineering (CI2MA) at Universidad de Concepcion . Expertise: mathematical and numerical analysis, scientific computing and applications of systems of conservation laws and related equations, as well as of coupled flow-transport systems, with applications in mineral processing, wastewater treatment, bioprocesses and trac modeling.

**Rafael ORDONEZ** is PhD student at CI2MA – University of Concepcion (since July 2017). Subject: Modelling, analysis and numerical solution of conservation laws with discontinuous and non-local flux arising in water resources.

**At University of Versailles:**

**Christophe CHALONS** is Professor at the University of Versailles Saint-Quentin-en-Yvelines and Director of the Laboratory of Mathematics. Expertise: Numerical methods for hyperbolic systems of conservation and non conservation laws, applications to compressible multiphase flows and free surface flows.

**Past members:**

**Elena ROSSI** was postdoc fellow at Inria SAM, EPI ACUMES (September 2017 – April 2019). Subject: Multi-class traffic flow models.

**Camilla FIORINI** was PhD student at the Laboratory of Mathematics, University of Versailles Saint-Quentin-en-Yvelines. Subject: Sensitivity analysis for hyperbolic systems with discontinuous solutions.

**Objectives**

We aim at designing more efficient numerical algorithms to accurately compute solutions of non-local conservation laws, both by a careful choice of quadrature formulas and the selection of adapted high order methods. In this perspective, we have already obtained encouraging results for scalar equations in 1 space-dimension arising in vehicular traffic and sedimentation modeling.

With this project, we want to extend the study to more general (and computationally expensive) situations of interest in applications, including systems of non-local equations and multidimensional problems. This type of problems arise for example in the modeling of multi-class vehicular traffic, each equation describing the evolution of a given class density, or in models of crowd motion.

Regarding the sensitivity analysis, we aim at extending the results obtained for classical hyperbolic systems , as discontinuous solutions of the Euler equations, to non-local problems. In particular, we will derive the governing equations of the sensitivity variables and define a correction term to be added to the sensitivity equations for these equations to be also valid across discontinuities. In a second time, we will address the construction of numerical schemes with a sharp control of the underlying numerical diffusion. This is a challenging but crucial step to obtain relevant numerical solutions for sensitivities in the context of non-local terms.

**Co-funding**

- Fondecyt-Chile Project 1181511 (2018-2021): Modelling and Numerical Analysis for Non-local Systems of Conservation Laws (PI L-M. Villada)
- Fondecyt-Chile Project 1170473 (2017-2021) (PI R. Burger)

**Exchanges**

L.-M. Villada visited Inria for 2 weeks in September 2018.

C. Chalons, P. Goatin and L.-M. Villada attended the Workshop WONAPDE in January 2019. In particular, P. Goatin and L.-M. Villada are the organizers of the mini-symposium “Numerical methods for traffic flow models”.

L.-M. Villada will visit Inria for 2 weeks in November 2019.

R. Ordonez will visit University of Versailles and Inria for 2 months in November-December 2019.

R. Burger will visit Inria for one week in December 2019.

**Results**

**2018**

During this first year, we have focused on numerical schemes for systems of non-local conservation laws in one space-dimension. In particular, we are concluding the study of Lagrangian-Remap schemes (previously proposed for classical hyperbolic systems) for a non-local multi-class traffic flow model proposed in [F.A. Chiarello, P. Goatin. Non-local multi-class traffic flow models, Netw. Heterog. Media, to appear.]. The error and convergence analysis show the effectiveness of the method, which is first order, in sharply capturing shock discontinuities and better precision with respect to other methods as Lax-Friedrichs or Godunov (even 2^{nd} order). A journal article about these results is currently in preparation: [F.A. Chiarello, P. Goatin, L.M. Villada. Lagrangian-remap schemes for non-local multi-class traffic flow models. In preparation.]

A second topic are finite volume methods for non-local multi-class traffic models based on the weighted essentially non-oscillatory (WENO) schemes proposed in [C. Chalons, P. Goatin and L. Villada, High order numerical schemes for one-dimension non-local conservation laws, SIAM J. Sci. Comput., **40 **, (2018) pp. A288-A305] in order to obtain high-order shock capturing methods for the non-local multi-class case. A talk based on this topic, titled “High Order Finite-volume Weno Schemes For Non-local Traffic Flow Models” will be presented at WONAPDE 2019 and a work to be presented to the proceedings of the HYP2018 meeting is currently in preparation: [F.A. Chiarello, P. Goatin, L.M. Villada. High Order finite volume schemes for non-local multi-class traffic flow models. In preparation.]

Numerical schemes for convection-diffusion-reaction systems of PDE with non-local flux were proposed, these efficient methods combines Weighted Essentially Non-Oscillatory (WENO) reconstruction and an implicit-explicit Runge-Kutta (IMEX-RK) method for time stepping. The obtained method avoids the restrictive time step limitation of explicit schemes. The following two journal articles about these results are currently accepted: [R. Bürger, D. Inzunza, P. Mulet and L.M. Villada. **Implicit-explicit schemes for nonlinear nonlocal equations with a gradient flow structure in one space dimension****. **to appear in *Numer. Meth. Partial Diff. Eqns.*] and [R. Bürger, G. Chowell, E. Gavilan, P. Mulet and L.M. Villada. **Numerical solution of a spatio-temporal predator-prey model with infected prey.** to appear in *Math. Biosci. Eng.*]

As far as the topic of sensitivity analysis is concerned, we have completed a couple of works devoted to the design of anti-diffusive schemes for systems of conservation laws with local closure relations, in collaboration with C. Chalons, R. Duvigneau and C. Fiorini: [Chalons C., Duvigneau R., Fiorini C. Sensitivity analysis and numerical diffusion effects for hyperbolic PDE systems with discontinuous solutions. The case of barotropic Euler equations in Lagrangian coordinates, to appear in SIAM J. Sci. Comput.] and [Chalons C., Duvigneau R., Fiorini C. Sensitivity equation method for Euler equations in presence of shocks applied to uncertainty quantification, submitted in revised form to Journal of Computational Physics]

**2019**

We have focused on numerical schemes for systems of non-local conservation laws in one space-dimension. In particular, we have developed a Lagrangian-Remap scheme) for a non-local multi-class traffic flow model, which is described and studied in [F.A. Chiarello, P. Goatin, L.M. Villada. Lagrangian-Antidiffusive Remap schemes for non-local multi-class traffic flow models, submitted.]

We also developed finite volume methods for non-local multi-class traffic models based on the weighted essentially non-oscillatory (WENO) schemes, which are proposed in [F.A. Chiarello, P. Goatin, L.M. Villada. High Order finite volume WENO schemes for non-local multi-class traffic flow models. To appear on HYP2018 Proceedings.]

**Publications**

- C. Chalons, P. Goatin and L. Villada. High order numerical schemes for one-dimension non-local conservation laws, SIAM J. Sci. Comput., 40(1) (2018), A288-A305.
- C. Chalons, R. Duvigneau and C. Fiorini. Sensitivity analysis and numerical diffusion effects for hyperbolic PDE systems with discontinuous solutions. The case of barotropic Euler equations in Lagrangian coordinates, SIAM J. Sci. Comput., 40(6) (2018), A3955–A3981. .
- R. Bürger, D. Inzunza, P. Mulet and L.M. Villada. Implicit-explicit schemes for nonlinear nonlocal equations with a gradient flow structure in one space dimension
**.***Numer. Meth. Partial Diff. Eqns.,*.*35*(3) (2019), 1008-1034 - R. Bürger, G. Chowell, E. Gavilan, P. Mulet and L.M. Villada. Numerical solution of a spatio-temporal predator-prey model with infected prey
**.***Math. Biosci. Eng.,*16(1) (2018), 438-473. - C. Chalons, R. Duvigneau and C. Fiorini. Sensitivity equation method for Euler equations in presence of shocks applied to uncertainty quantification, submitted.
- P. Goatin and E. Rossi. Well-posedness of IBVP for 1D scalar non-local conservation laws, Z. Angew. Math. Mech. (2019).
- F.A. Chiarello, P. Goatin and L.M. Villada. Lagrangian-Antidiffusive Remap schemes for non-local multi-class traffic flow models, submitted (2018).
- F.A. Chiarello, P. Goatin and L.M. Villada. High-order Finite Volume WENO schemes for non-local multi-class traffic flow models, To appear on HYP2018 Proceedings.