Jean-Christophe Nave

McGill University Department of Mathematics and Statistics

### Ph.D. UCSB 2004 - (Supervisors: Xu-Dong Liu and Sanjoy Banerjee)

### Assistant Professor of Mathematics, McGill Math and Stat Department 2010-present

### Lecturer of Applied Mathematics, MIT Math Department, 2009-2010

### Instructor of Applied Mathematics, MIT Math Department, 2005-2009

### Steering Committee of the Institut des Sciences Mathématiques 2012-present

### Applied Math. Lab. of the Centre de Recherches Mathématiques, 2010-present

### CAIMS Membership Committee

**(register [here])**, 2013-present### CNRS-Unité Mixte Internationale CNRS-UMI, 2012-present

- K. Kamrin, C. H. Rycroft, J.-C. Nave,
*Reference map technique for finite-strain elasticity and fluid-solid interaction*, J. Mech. and Phys. Solids (2012) [link] - B. Seibold, R.R. Rosales, J.-C. Nave,
*Jet Schemes for Advection Problems*, Discr. and Cont. Dynam. Syst. - B (2012) [link] - A. N. Marques, J.-C. Nave, R. R. Rosales,
*A Correction Function Method for Poisson problems with interface jump conditions*, J. Comp. Phys. (2011) 230 [link] - J.-C. Nave, R.R. Rosales, B. Seibold,
*A Gradient-Augmented Level Set Method with an Optimally Local, Coherent Advection Scheme*, J. Comp. Phys. (2010) 229 [link]

Quiz #1: Feb. 13th 2014 in class (must be taken in your section).

Quiz #2: March 13th 2014 in class (must be taken in your section).

Quiz #3: April 3rd 2014 in class (must be taken in your section).

- Dr. Andy Wan (Post Doc - Ph.D. Polytech. Montreal '14)
**[web]**

- Yoel Chetboun (Master Applied Mathematics - ENSTA)
- Cyrielle Zakini (Master Applied Mathematics - ENSTA)
- Amine Alaoui Fdili (Master Applied Mathematics - ENSTA)

- Zoe Guan (Honors B.S. in Mathematics 2015)
- Russell Milne (Honors B.S. in Mathematics 2015)
- Xi YuanYin (Honors B.S. in Mathematics 2015)
- Zhu Jiaqi (Honors B.S. in Mathematics 2015)
- Olivier Melancon (B.S. in Mathematics 2016)
- Fabio Alves (B.S. in Electrical Engineering 2016 - UFMG (Brazil))

- Prof. Francois Blanchette (UC Merced) - January 2014 to June 2014
- Dr. Alexandre Marques (Embraer) - March 2014

- Dr. David Shirokoff (Post Doc. - 2011-2014 - Ph.D. MIT '11)
**[web]**-- (co-supervised with R. Choksi) --*- Now at NJIT* - Dr. Dmitry Kolomenskiy (Post Doc - 2012-2014 Ph.D. Université de Provence '10)
**[web]** - Dr. Alexander Bihlo (Post Doc - 2012-2013 Ph.D. Vienna '11)
**[web]**--*Now at Memorial University* - Lisa Larsson (Ph.D. '14)
**[web]**-- (co-supervised with R. Choksi) --*- Now at Courant* - Olivier Mercier (Master '13)
**[web]**- Now at Universite de Montreal - CS/OR - Nelson Feyeux (Master '13)
**[web]**- Now at Université de Grenoble - Dr. Alexandre N. Marques (Ph.D. MIT '12) - Now at Embraer
- Miles Shang (B.S. in Mathematics '12) - Now at Facebook
- Sebastien Jean (B.S. in Mathematics '12) - Now at Universite de Montreal - CS/OR
- Mareike Adams (B.S. in Mathematics '12) -
**[web]**--*Now at UCSB (Geophysics)* - Geneviève Bourgeois (B.S. in Mathematics '13) -
**[web]**--*Now at Waterloo* - Simon Szatmari (B.S. in Mathematics '13) -
**[web]**--*Now at Leiden University* - Olga Yakovlenko
- Robert Gibson

Gradient-Augmented Level Set Method and Jet Schemes

###### (with B. Seibold and R.R. Rosales)

The level set approach represents surfaces implicitly, and advects
them by evolving a level set function, which is numerically defined on an Eulerian grid.
Here we present an approach that augments the level set function values
by gradient information, and evolves both quantities in a fully coupled fashion.
This maintains the coherence between function values and derivatives, while exploiting
the extra information carried by the derivatives. The method is of comparable quality to
WENO schemes, but with optimally local stencils (performing updates in time by using information
from only a single adjacent grid cell). In addition, structures smaller than the grid size can be
located and tracked, and the extra derivative information can be employed to obtain simple and accurate approximations
to the curvature. We analyze the accuracy and the stability of the new scheme, and perform benchmark tests.

Jet-Schemes are extensions of the GALS Method where higher order derivatives are used to construct the Hermite interpolation. Consequently, higher-order time stepping must used so that the overall (global convergence) order of the scheme is improved. We constructed Jet-Schemes of global order up to 5 in 2D and 3D successfully. The following papers contain details and a comparison of Jet-Schemes with WENO and DG.

**[GALSM] [JetScheme1] [JetScheme2]**

Jet-Schemes are extensions of the GALS Method where higher order derivatives are used to construct the Hermite interpolation. Consequently, higher-order time stepping must used so that the overall (global convergence) order of the scheme is improved. We constructed Jet-Schemes of global order up to 5 in 2D and 3D successfully. The following papers contain details and a comparison of Jet-Schemes with WENO and DG.

Advection of Arbitrary Sets: Characteristic Mapping Method

###### (with O. Mercier)

In this paper, we present a new numerical method for advecting
arbitrary sets in a vector field. The method computes a transformation
of the domain instead of dealing with particular sets. We propose a way of
decoupling the advection and representation steps of the computations, resulting
in significant reductions in computational times over other methods while still
guaranteeing accuracy. The decoupling also allows one to advect multiple sets at low
computational cost, and makes the method highly parallelizable. Results are presented
in two and three dimensions, and accuracy and efficiency are studied.

**[arXiv:1309.2731]**

Adaptive Gradient-Augmented Level Set Method with Multi-Resolution Error Estimation

###### (with D. Kolomenskyi and K. Schneider)

A space-time adaptive scheme is presented for solving advection equations
in two space dimensions. The gradient-augmented level set method using a semi-Lagrangian
formulation with backward time integration is coupled with a point value multiresolution
analysis using Hermite interpolation. Thus locally refined dyadic spatial grids are introduced
which are efficiently implemented with dynamic quad-tree data structures. For adaptive
time integration, an embedded Runge-Kutta method is employed. The precision of the new
fully adaptive method is analysed and speed up of CPU time and memory compression with
respect to the uniform grid discretization are reported.

**[arXiv:1401.7294]**

Poisson's Equation with Interface Jumps

###### (with A.N. Marques and R.R. Rosales)

This project has two parts:

(1) Solving Poisson's equation with jump conditions provided on a co-dim 1 interface

(2) Solving piece-wise constant coefficient Poisson's equation with interface provided on a co-dim 1 interface

In the constant coefficient case (1), we developped a technique, the Correction Function Method (CFM) which provides a correction to the RHS of the equation so that the jumps are accurately enforced. This correction is the solution to a local PDE problem (which is fast and easy to solve). Since one may compute the correction to any order of accuracy, and provided the discretization of the Laplacian is of matching order, the method is "in principle" of arbitrary order. We have constructed order 2 and order 4 schemes in 2D successfully. This is detailed in our paper.

The piece-wise constant coefficient case (2) is significantly more complicated. However, we were able to reduce it to the solving 2 CFM problems and one boundary integral (for the potential). This decomposition is close to that of described by Mayo ("The fast solution of Poisson's and the biharmonic equations on irreular regions", SIAM J. NA, 1984). Using the same ideas, we also solve Poisson's equation with arbitrary boundary conditions on an immersed boundary.

We present 2D examples of the algorithm applied to problems involving boundary conditions over complex geometries (with 4th order accuracy), and to a Poisson equation with interfaces across which the solution is discontinuous (with 3rd order accuracy).

**[CFM1] [CFM2]**

(1) Solving Poisson's equation with jump conditions provided on a co-dim 1 interface

(2) Solving piece-wise constant coefficient Poisson's equation with interface provided on a co-dim 1 interface

In the constant coefficient case (1), we developped a technique, the Correction Function Method (CFM) which provides a correction to the RHS of the equation so that the jumps are accurately enforced. This correction is the solution to a local PDE problem (which is fast and easy to solve). Since one may compute the correction to any order of accuracy, and provided the discretization of the Laplacian is of matching order, the method is "in principle" of arbitrary order. We have constructed order 2 and order 4 schemes in 2D successfully. This is detailed in our paper.

The piece-wise constant coefficient case (2) is significantly more complicated. However, we were able to reduce it to the solving 2 CFM problems and one boundary integral (for the potential). This decomposition is close to that of described by Mayo ("The fast solution of Poisson's and the biharmonic equations on irreular regions", SIAM J. NA, 1984). Using the same ideas, we also solve Poisson's equation with arbitrary boundary conditions on an immersed boundary.

We present 2D examples of the algorithm applied to problems involving boundary conditions over complex geometries (with 4th order accuracy), and to a Poisson equation with interfaces across which the solution is discontinuous (with 3rd order accuracy).

Sharp-Interface Active Penalty Method

###### (with D. Shirokoff)

The volume penalty method provides a simple, efficient approach
for solving the incompressible Navier-Stokes equations in domains with
boundaries or in the presence of moving objects. Despite the simplicity,
the method is typically limited to first order spatial accuracy. We demonstrate
that one may achieve high order accuracy by introducing an active penalty term.
One key difference from other works is that we use a sharp, unregularized mask
function. We discuss how to construct the active penalty term, and provide numerical
examples, in dimensions one and two. We demonstrate second and third order
convergence for the heat equation, and second order convergence for the
Navier-Stokes equations. In addition, we show that modifying the penalty
term does not significantly alter the time step restriction from that
of the conventional penalty method.

**[arXiv:1303.5681]**

An Iterative Algorithm for Computing the Measure of Generalized Voronoi Regions

###### (with L. Larsson and R. Choksi)

We present and analyze a fast algorithm for directly computing the measure of
a generalized Voronoi region associated with generators of arbitrary co-dimension. The algorithm
is based upon solving one Eikonal equation to generate a kernel-based operator whose iteration
accumulates \mass" along the closest generator. In particular, the algorithm does not require the
computation of the ridge set (Voronoi diagram) nor the gradient of the solution to the Eikonal
equation. The algorithm is shown to be .rst order and converge very quickly. Several illustrations
are presented including the computation of measures of in
uence associated with the Los Angeles
County highway system. The method can also be used for the fast computation of the centroid and
higher moments of the generalized Voronoi regions.

Global Analysis for Minimizers of Non-Convex Functionals: A Simple Approach via Convex Approximation

###### (with D. Shirokoff and R. Choksi)

We consider mass-constrained minimizers for a class of non-convex energy functionals
involving double-well potentials. Based upon a global convex lower bound to the energy, we introduce
a simple strategy to .nd su.cient conditions to infer whether a given critical point is a global
minimizer. We show that this strategy works well for the simplest of critical points, the constant
state. In doing so, we derive an almost optimal lower bound on the order-disorder curve for the
Ohta-Kawasaki energy. Focusing on the lamellar phase, we discuss how this strategy extends to
non-constant critical points.

**[PDF]**

Invariant Discretization Schemes Using Evolution-Projection Techniques

###### (with A. Bihlo)

Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the one-dimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for non-invariant schemes of the heat equation in computational coordinates. We propose a new methodology for handling moving discretization grids which are generally indispensable for invariant numerical schemes. The idea is to use the invariant grid equation, which determines the locations of the grid point at the next time level only for a single integration step and then to project the obtained solution to the regular grid using invariant interpolation schemes. This guarantees that the scheme is invariant and allows one to work on the simpler stationary grids. The discretization errors of the invariant schemes are established and their convergence rates are estimated. Numerical tests are carried out to shed some light on the numerical properties of invariant discretization schemes using the proposed evolution-projection strategy.

**[PDF] **

Convecting Reference Frames and Invariant Numerical Models

###### (with A. Bihlo)

In the recent paper by Bernardini et al. [J. Comput. Phys. 232 (2013), 1-6] the discrepancy in the performance of finite difference and spectral models for simulations of flows with a preferential direction of propagation was studied. In a simplified investigation carried out using the viscous Burgers equation the authors attributed the poorer numerical results of finite difference models to a violation of Galilean invariance in the discretization and propose to carry out the computations in a reference frame moving with the bulk velocity of the flow. Here we further discuss this problem and relate it to known results on invariant discretization schemes. Non-invariant and invariant finite difference discretizations of Burgers equation are proposed and compared with the discretization using the remedy proposed by Bernardini et al.

**[ArXiv:1301.5955] **

By leveraging the Subgrid accuracy of the methods presented above, we can simulate the dynamics of a droplet of water impinging on a soap film.

These simulations were performed on a regular Cartesian grid and capture quantatively the various regimes found in the experiments of T. Gilet and J. Bush**[Their Paper]**.

Using a similar approach, our method can simulate the problem of partial coalescence. The subgrid accuracy of the method allows here too to compare quantatively with experiments.

These simulations were performed on a regular Cartesian grid and capture quantatively the various regimes found in the experiments of T. Gilet and J. Bush

Using a similar approach, our method can simulate the problem of partial coalescence. The subgrid accuracy of the method allows here too to compare quantatively with experiments.

Falling Liquid Films

###### (with X.D. Liu and S. Banerjee)

Liquid films are important in many industrial applications, but also from a
fundamental point of view, they are important two-phase flow systems. In this
paper, we develop a sharp interface/level set method for the Direct Numerical
Simulation (DNS) of liquid films with large interfacial deformations, and large
density ratio between the liquid and the gas phase. We use the ghost fluid
method to capture the interface motion without smoothing properties across it,
and adopt a maximization scheme for the implicit treatment of the viscous
term in the Navier–Stokes equations. Because liquid films have very low
average depth compared to the distance between waves, several innovations are
required to handle solving the equations on grid structures of high aspect ratio.
Two-dimensional (2D) calculations for wavy films falling down a vertical wall
are presented, and good agreement is found when numerical solutions are
directly compared with the experiments of Nosoko et al. [1]. Some results are
also presented for falling liquid films transitioning naturally from 2D to 3D
surface wave structures demonstrating the potential of the method for 3D fully
coupled two-phase liquid films simulations.

**[PDF]**

Magma Flow

###### (with J. Suckale, B. Hager, L.T. Elkins-Tanton)

Concentric Fluid Shells Satability and Fiber Drawing

###### (with X. Liang, D. S. Deng, Y. Fink, S. G. Johnson)

Motivated by complex multi-fluid geometries currently being explored in fibre-device
manufacturing, we study capillary instabilities in concentric cylindrical flows of N
fluids with arbitrary viscosities, thicknesses, densities, and surface tensions in both the
Stokes regime and for the full Navier–Stokes problem. Generalizing previous work by
Tomotika (N=2), Stone and Brenner (N=3, equal viscosities) and others, we present
a full linear stability analysis of the growth modes and rates, reducing the system to
a linear generalized eigenproblem in the Stokes case. Furthermore, we demonstrate
by Plateau-style geometrical arguments that only axisymmetric instabilities need be
considered. We show that the N=3 case is already sufficient to obtain several
interesting phenomena: limiting cases of thin shells or low shell viscosity that
reduce to N=2 problems, and a system with competing breakup processes at very
different length scales. The latter is demonstrated with full three-dimensional Stokesflow
simulations. Many N > 3 cases remain to be explored, and as a first step we
discuss two illustrative N->infinity cases, an alternating-layer structure and a geometry
with a continuously varying viscosity.

**
[ShellStability]
[FiberDrawing]**

Concentrated Solar Power on Demand (CSPonD)

###### (with A. H. Slocum, D. S. Codd , J. Buongiorno, C. Forsberg, T. McKrell, C. N. Papanicolas, A. Ghobeity, C. J. Noone, S. Passerini, F. Rojas, A. Mitsos)

Self-Sustained Nonlinear Waves in Traffic Flow

###### (with M. Flynn, A. Kasimov, B. Seibold, and R.R. Rosales)

Our research is based on the observation that the considered traffic models are similar to the equations that describe detonation waves produced by explosions. Employing the theory of denotation waves, we have developed ways to analytically predict the exact shape and the speed of propagation of jamitons. Numerical simulations of the considered traffic models show that the predicted jamiton solutions are in fact achieved, if the initial traffic density is sufficiently dense. The considered jamitons can qualitatively be found both in observed real traffic as well as in experiments. The theoretical description of the jamiton solution admits a better understanding of their behavior.

**[TrafficFlow]
[Jamitons-ArXiv:0809.2828]**

We maintain a web page at MIT regarding this project:**[MIT Traffic Flow Page]**.

We maintain a web page at MIT regarding this project:

- Ryan Galagusz, David Shirokoff and Jean-Christophe Nave. A Fourier penalty method for solving the time dependent Maxwell's equations in domains with curved boundaries.

[ArXiv:1503.03914] - A. Wan, A. Bihlo, and J-C. Nave. The multiplier method to construct conservative finite-difference schemes for ordinary and partial differential equations

[ArXiv:1411.7720] - O. Mercier and J.-C. Nave. The characteristic mapping method for the linear advection of arbitrary sets.

[ArXiv:1309.2731] - A. N. Marques, J.-C. Nave, and R. R. Rosales. Fast and accurate solution of the poisson equation in an immersed setting.

[ArXiv:1401.8084] - A. Tcheng and J.-C. Nave. A low complexity algorithm for non-monotonically evolving fronts.

[PDF] - J. Crest, J.-C. Nave, S. Pabba, R. W. Cohn, and G.H. McKinley. Formation of nanofibers by capillary-driven thinning of drying visco-elastic filaments.

[PDF]

- Dmitry Kolomenskiy, J.-C. Nave, and K. Schneider. Adaptive gradient-augmented level set method with multiresolution error estimation. (Accepted)

[ArXiv:1401.7294] - D. Shirokoff, R. Choksi, and J-C. Nave. Global analysis for minimizers of non-convex functionals: A simple approach via convex approximation. (Accepted)

[PDF]

- A. Bihlo and J.-C. Nave. Convecting reference frames and invariant numerical models. J. Comp. Phys., 271:656-663 (2014)

[ArXiv:1301.5955] - D. Shirokoff and J.-C. Nave. A high-order penalization method for fluid flows with irregular boundaries. J. Sci. Comp., (April):1-25 (2014)

[ArXiv:1303.5681] - L. Larson, R. Choksi, and J.-C. Nave. An iterative algorithm for computing measures of generalized voronoi regions. SIAM J. Sci. Comp., 36(2):792-827 (2014)

[PDF]

- A. Bihlo and J.-C. Nave. Invariant discretization schemes using evolution projection techniques.
*SIGMA*, 9, (2013)

[PDF] - H. Kohno and J.-C. Nave. A mesh-less method for the advection equation and its gradient.
*Computer Physics Communications.*, (2013)

[PDF]

- P. Chidyagwal, J.-C. Nave, R.R. Rosales, and B. Seibold. A comparative study of the efficiency of jet schemes. Int. J. Num. Anal. Model. - B, (2012)

[PDF] - B. Seibold, R.R. Rosales, and J.-C. Nave. Jet schemes for advection problems. Discr. and Cont. Dynam. Syst. - B, (2012)

[PDF] - K. Kamrin, C. H. Rycroft, and J.-C. Nave. Reference map technique for finite-strain elasticity and fluid-solid interaction. J. Mech. and Phys. Solids, (2012)

[PDF]

- J. Suckale, B. Hager, L.T. Elkins-Tanton, J.-C. Nave, Reply to the comment by Mike R. James et al. on It takes three to tango: 2. Bubble dynamics in basaltic volcanoes and ramifications for modeling normal Strombolian activity. Journal of Geophysical Research, 116, B06208 (2011)

[PDF] - A. H. Slocum, D. S. Codd , J. Buongiorno, C. Forsberg, T. McKrell, J.-C. Nave, C. N. Papanicolas, A. Ghobeity, C. J. Noone, S. Passerini, F. Rojas, A. Mitsos, Concentrated solar power on demand, Solar Energy 85 (2011) 1519-1529.

[PDF] - X. Liang, D. S. Deng, J.-C. Nave, S. G. Johnson, Linear stability analysis of capillary instabilities for concentric cylindrical shells, J. Fluid Mech., 260, (2011)

[PDF] - D. S. Deng, J.-C. Nave, X. Liang, S. G. Johnson, Y. Fink, Exploration of in-fiber nanostructures from capillary instability, Optics Express , Vol. 19, No. 17, (2011).

[PDF] - A. N. Marques, J.-C. Nave, R. R. Rosales, A Correction Function Method for Poisson problems with interface jump conditions, Journal of Computational Physics 230 (2011)

[PDF]

- J.-C. Nave, S. Banerjee, X.D. Liu, "Direct Numerical Simulation of Liquid Films with Large Interfacial Deformation", Studies in Applied Mathematics, (2010) 125, 153-77

[PDF] - J. Suckale, J.-C. Nave, B. Hager, L.T. Elkins-Tanton, "Simulations of buoyancy-driven flow in the presence of large viscosity contrasts: I. Numerical approach", J. Geophys. Rev., (2010) 115, B07409, 16 pp

[PDF] - J. Suckale, B. Hager, L.T. Elkins-Tanton, J.-C. Nave, "Simulations of buoyancy-driven flow in the presence of large viscosity contrasts: II. Implications for modeling normal Strombolian-type activity", J. Geophys. Rev. (2010) 115, B07410, 17 pp.

[PDF] - J.-C. Nave, R.R. Rosales, B. Seibold, "A Gradient-Augmented Level Set Method with an Optimally Local, Coherent Advection Scheme", J. Comp. Phys., (2010) 229, 3802-27.

[PDF]

- M.R. Flynn, A.R. Kasimov, J.-C. Nave, R.R. Rosales, B. Seibold, "Self-Sustained Nonlinear Waves in Traffic Flow", Phys.Rev.E 79, 056113 (2009)

[PDF] - S. Banerjee, V. Badalassi, V. Dwivedi, J.-C. Nave, D. Hall, "The Direct Numerical Simulation of Two-Phase Flows with Interface Capturing Methods", La Houille Blanche (2005)

[PDF] - K.N. Premnath, J.-C. Nave, S. Banerjee, "Computation of multiphase flows with lattice boltzman methods", Invited Paper ASME, IMECE (2005)

[.] - S. Smolentsev, N. B. Morley, B. Freeze, R. Miraghaie, J.-C. Nave, S. Banerjee, A. Ying, M. Abdou, "Thermofluid Modeling and Experiments for Free Surface Flows of Low-Conductivity Fluid in Fusion Systems", Fusion Engineering and Design (2004).

[LINK] - J.-C. Nave, S. Banerjee, "Direct Numerical Simulation of Wavy Falling Liquid Films", 5th International Conference on Multiphase Flow, ICMF (2004)

[.]

- M.R. Flynn, A.R. Kasimov, J.-C. Nave, R.R. Rosales, B. Seibold, "On "jamitons", Self-Sustained Nonlinear Traffic Waves", available arXiv:0809.2828, (2008)

[[ArXiv:0809.2828]] - M. Shang, J.-C. Nave, A Probabilistic Method for Topological Control in Cloth Simulations (2012)

[PDF]

Department of Mathematics and Statistics

McGill University

Burnside Hall, Room 1121

805 Sherbrooke West

Montreal, QC, H3A 0B9

Canada

jcnave@math.mcgill.ca

Phone: (514) 398-3048

Fax: (514)-398-3899

- You MUST have a degree in Math.

- You should read some of my papers before contacting me.

- I will not reply to generic emails.

- Several openings are available through the CRM-ISM Postdoctoral Fellowships program.