2019/11/02-2019/11/04

**Remi Abgrall**

**Title**: Some Preliminary Results on a Kinetic
Scheme that has an Lattice Boltzmann Method Flavor

**Abstract**: In this talk, we intend to describe
one way to construct arbitrarily high order kinetic schemes on regular meshes.
The method can be arbitrarily high order in space and time, and run at CFL one.
This is a common feature with the Lattice Boltzmann Methods. However, the type
of Maxwellian we use here are different. This results in very simple and CPU
efficient methods. I will also indicate how this could be used for the
simulation of compressible multiphase and multicomponent flows.

**Alina Chertock**

**Title** : Asymptotic Preserving Methods for
Hyperbolic Conservation Laws

**Abstract**: Solutions of many hyperbolic systems
reveal a multiscale character and thus their numerical resolution presence some
major difficulties. Such problems are typically characterized by the occurrence
of a small parameter. The solutions of these problems show a nonuniform
behavior as the parameter tends to zero, for instance, the type of the limiting
solution is different in nature from that of the solutions for a fixed values
of that parameter. One of the canonical examples of such problems is low Mach
number compressible or low Froude number shallow water equations.

In these limiting regimes, the propagation speeds are very low and therefore the use of standard explicit methods would require very restrictive time and space discretization steps. This becomes rapidly too costly from a practical point of view and consequently numerical solutions for small values of the parameter may be out of reach. Moreover, standard implicit schemes, while uniformly stable, may be inconsistent with the limit problem and thus may provide a wrong solution in the zero limit. Thus, designing robust numerical algorithms, whose accuracy and efficiency is independent of the value of the small is an important and challenging task.

In the talk, we describe asymptotic preserving schemes for the shallow water equations with Corriolis forces and compressible Euler equations (based on a new hyperbolic splitting and accurate and efficient elliptic numerical solver) and show an asymptotic analysis of the schemes, which is a very important step in ensuring that our method becomes an accurate and efficient solver in the limiting regimes. A number of numerical examples will also be provided to illustrate the performance of the proposed numerical approach.

**Guoxian Chen**

**Title: **The Stability of Hydrostatic Reconstruction Method for Euler Equation with Gravity

**Abstract: **The hydrostatic reconstruction methods (HR) are a widely used method for the simulation of the shallow water simulations. due to its robustness including the well-balancing and positivity. While the entropy stability is not fully discussed. The orignal version [SISC 2004] and even our revisded version [SINUM 2017] only analysised the semi-discrete entropy inequality. Recently, Audusse et al [Math. Comput. 2016] gives a vise example while violate the full-discrete entropy inequality and proved that, if the kinetic numerical flux is used, the scheme satisfies a relaxed full-discrete iequality but with a error term which will vanish when the mesh size goes to zero. Later, Berthon and Chalons [J. Scient. Comput. 2019] modifed the HRmethod by adding an extra numerical viscosity on the approximate Riemann Solver, and proved that the modifed scheme satisfies the full discrete entropy inequality.

In this talk, we will introduce our updated result on this topic. We extend the HR scheme to the Euler equation with gravity, and prove that the scheme satisfies the full-discrete entropy inequality with an high order error term where the numerical flux only need to satisfy the full discrete entropy for the homogeneous case.

**Shumo Cui**

In some applications, the source term is significantly sensitive to the underlying solution, which is called a stiff source term. The stiffness in the source term introduces additional difficulty for designing a scheme that is both efficient and well-balanced. To this end, we will introduce a new semi-implicit Runge-Kutta time integration method that achieves both desired properties.

**Shi Jin**

**Title**: Random Batch Methods for
Interacting Particle Systems and its Applications in Consensus-based High
Dimensional Global Optimization in Machine Learning

**Abstract**: We develop random batch methods for
interacting particle systems with large number of particles. These methods use
small but random batches for particle interactions, thus the computational
cost is reduced from O(N^2) per time step to O(N), for a

system with N particles with binary interactions.

For one of the methods, we give a particle number independent error estimate under some special interactions. Then, we apply these methods to some representative problems in mathematics, physics, social and data sciences, including the Dyson Brownian motion from random matrix theory, Thomson's problem, distribution of wealth, opinion dynamics and clustering. Numerical results show that the methods can capture both the transient solutions and the global equilibrium in these problems.

We also apply this method and improve the consensus-based global optimization algorithm for high dimensional machine learning problems. This method does not require taking gradient in finding global minima for non-convex functions in high dimensions.

**Jean-Luc Guermond**

**Title:** Positive and Asymptotic Preserving Approximation of the Radiation Transport Equation

**Abstract:** We introduce a (linear) positive and asymptotic-preserving method for solving the one-group radiation transport equation. The approximation in space is discretization agnostic: the space approximation can be done with continuous or discontinuous finite elements (or finite volumes, or finite differences). The method is formally first-order accurate in space in the streaming regim. This type of accuracy is consistent with Godunov's theorem since the method is linear. The two key theoretical results are that: (1) the method is positive and (2) the method is asymptotic preserving (meaning that it is robust in the diffusion limit when the mean free path is significantly larger than the mesh size). The method is illustrated with continuous finite elements. It is observed to converge with the rate $\calO(h)$ in the $L^2$-norm on manufactured solutions, and it is $\calO(h^2)$ in the diffusion regime. Unlike other standard techniques, the proposed method does not suffer from overshoots at the interfaces of optically thin and optically thick regions

**Chieh-Sen Huang**

**Title**: Finite Volume WENO Schemes for Nonlinear Parabolic Problems with Degenerate Diffusion on Non-uniform Meshes

**Abstract**: We consider numerical approximation
of the degenerate advection-diffusion equation, which is formally parabolic but
may exhibit hyperbolic behavior. We develop both explicit and implicit
finite volume weighted essentially non-oscillatory (WENO) schemes in multiple
space dimensions on non-uniform computational meshes. The diffusion
degeneracy is reformulated through the use of the Kirchhoff transformation.
Space is discretized using WENO reconstructions with adaptive order (WENO-AO),
which have several advantages, including the avoidance of negative linear
weights and the ability to handle irregular computational meshes. A
special two-stage WENO reconstruction procedure is developed to handle
degenerate diffusion. Element averages of the solution are first
reconstructed to give point values of the solution, and these point values are
in turn used to reconstruct the Kirchhoff transform variable of the diffusive
flux. Time is discretized using the method of lines and a Runge-Kutta
time integrator. We use Strong Stability Preserving
(SSP) Runge-Kutta methods for the explicit schemes, which have a severe
parabolically scaled time step restriction to maintain stability. We also
develop implicit Runge-Kutta methods. SSP methods are only conditionally
stable, so we discuss the use of L-stable Runge-Kutta methods. We present
in detail schemes that are third order in both space and time in one and two
space dimensions using non-uniform meshes of intervals or quadrilaterals.
Efficient implementation is described for computational meshes that are
logically rectangular. Through a von Neumann (or Fourier mode) stability
analysis, we show that smooth solutions to the linear problem are
unconditionally L-stable on uniform computational meshes when using an implicit
Radau IIA Runge-Kutta method. Computational results show the ability of the
schemes to accurately approximate challenging test problems.

**Jian-Guo Liu**

**Title**: Numerical Methods for some Nonlinear Transport Equations in Biology

**Abstract**: In the first part, we introduce a
semi-discrete scheme for 2D Keller-Segel equations based on a
symmetrization reformation, which is equivalent to the convex splitting
method and is free of any nonlinear solver. We show that, this new scheme
is stable as long as the initial condition does not exceed certain
threshold, and it asymptotically preserves the quasi-static limit in the
transient regime. Furthermore, we show that the fully discrete scheme is
conservative and positivity preserving, which makes it ideal for
simulations. In the second part, we propose an accurate front capturing
scheme for a class of tumor growth models with a free boundary limit. We
show that the semi-discrete scheme naturally connects to the free boundary
limit equation. With proper spatial discretization, the fully discrete
scheme has improved stability, preserves positivity, and can be
implemented without nonlinear solvers.

**Maria Lukacova**

**Title**: Structure Preserving Schemes and K-convergence for the Euler Equations

**Abstract**: In this talk we present our recent result on the convergence analysis of the structure preserving schemes for the Euler equations. Based on the two-velocity model of Brenner and the idea of invariant domain preserving schemes of Guermond and Popov we have proposed a new finite volume method that is entropy stable, preserves positivity of density and temperature and satisfies minimum entropy principle. These structure preserving properties allow us to prove scheme's consistency and stability for multidimensional Euler equations of gas dynamics.

Taking into account ill-posedness of the latter system in the class of weak entropy solutions, the convergence analysis of numerical schemes is of fundamental importance. We will present a generalization of the Lax equivalence theorem for

nonlinear problems. Indeed, we show that numerical solutions generate a generalized

(dissipative measure-valued) solution. They also convergence strongly to the strong solutions, if the latter exist. On the other hand, if the convergence of numerical solutions is not strong then the limiting object cannot be a weak solution. Using a newly developed concept of K-convergence, we will show how to compute observable quantities,

such as the mean and variance, of the corresponding generalized solutions.

This work has been done in cooperation with E. Feireisl (Prague), B.She (Prague) and Y. Wang (Beijing).

**Yongle Liu**

**Title:** A Well-Balanced Central-Upwind Scheme for the Thermal Rotating Shallow Water Equations and its applications in the propagation of vortex

**Abstract: **We develop a well-balanced central-upwind scheme for rotating shallow water model with horizontal temperature and/or density gradients- --the thermal rotating shallow water (TRSW). The scheme is designed using the flux globalization approach: first, the source terms are incorporated into the fluxes, which results in a hyperbolic system with global fluxes; second, we apply the Riemann-problem-solver-free central-upwind scheme to the rewritten system. We ensure that the resulting method is well-balanced by switching off the numerical diffusion

when the computed solution is near (at) thermo-geostrophic equilibria. The designed scheme is successfully tested on a series of numerical examples. We then extended the numerical methods to the two dimensional case, and use the proposed scheme to study the evolution of isolated vortices in the midlatitude beta-plane and also investigate the temperature perturbations in the equatorial beta-plane.

**T. Morales de Luna**

**Title:** A General Approach for Dispersive Water Waves

**Abstract**: Shallow water type models have been successfully applied to many real life situations for the simulation of geophysical flows: river floods, sediment transport, tsunami modeling, etc. Although these models are based on an averaged process on the vertical direction, recent techniques based on a multilayer approach allows to overcome this simplification and to better describe the flow and vertical effects therein.

Nevertheless, such models are based on the assumption of hydrostatic pressure. In recent years there has been an increasing interest on including the so-called dispersive or non-hydrostatic effects into the model. Non-hydrostatic models are capable of solving many relevant features of coastal water waves that classical shallow water system can not, such as dispersion and shoaling.

We propose here to study non-hydrostatic shallow water type systems. The objective will be to present a general framework in order to preserve good dispersive properties such as phase, group velocity and shoalling for water waves. The resulting models usually require the numerical solution of an elliptic problem, which may be computationally expensive. We shall propose here a relaxed hyperbolic approximation which will allow to asymptotically preserve the dispersive relations. This technique is computationally efficient and it may be extended to arbitrary high order accuracy.

**Bojan Popov**

**Title**: Well-balanced, Invariant Domain Preserving, Explicit in Time Finite Element Approximation of the Shallow Water and the Green-Naghdi Equations

**Abstract**: We propose a new second-order,
parameter-free, well-balanced and positivity-preserving explicit approximation
technique for the approximation of the shallow water equations with topography,
dispersion effects and friction, using continuous finite elements. The
novelties of the method are the explicit treatment of the friction and
dispersive terms, the robust approximation of dry states, a commutator-based,
high-order, entropy-viscosity and a local limiting procedure based on convex
limiting. The dispersion effects are modeled as a perturbation of the
Green-Naghdi equations due to Favrie and Gavrilyuk. The new method is compatible
with dry states and is provably positivity preserving under the appropriate CFL
condition. The method is numerically validated against manufactured solutions
and in some cases compared with experimental results.

**Gabriella Puppo**:

**Title**: Cool WENO Schemes, with Applications to Gas Dynamics with Gravity Force

**Abstract**: WENO reconstructions have proved to be
extremely effective to produce reliable and robust high order schemes for
hyperbolic problems. Still, the literature shows that they can be improved, and
several patches have been proposed. One such improvement is given by
CWENO reconstructions, which combine polynomials of different degrees in order
to obtain a uniformly high order accurate reconstruction within each
cell. This characteristic is very important especially when dealing with
balance laws, and/or non uniform grids which require to compute the
reconstruction at several quadrature nodes.

In this talk, we will first introduce CWENO schemes, presenting the benefits that they yield over standard WENO algorithms.

High order reconstructions are mainly an interpolation tool. It is important to study, beside accuracy, what are their main characteristics. It is already known how non oscillatory schemes strive to find a balance between artificial dissipation and artificial dispersion. Here we will propose a new technique to evaluate these phenomena, and we will also consider a systematic study of non linear distorsive effects. This study prompted us to introduce a notion of temperature associated to each reconstruction algorithm. The numerical temperature we define measures the amplitude of spurious modes created by the non linearity of the scheme. In this framework, a scheme which has no distortion has zero temperature, and thus it is cold: any linear scheme has zero temperature. Thus, cold schemes have no distortion, but are oscillatory. To prevent spurious oscillations, a scheme must have some distorsive effects, but not too much, in other words, a scheme must be cool.

The talk ends with an application of high order reconstructions to design well balanced schemes to capture steady state solutions for gas dynamics in a gravitational field, and to construct semi-conservative schemes.

**Manuel Quezada de Luna**

**Title**:Subcell Flux Limiting for Scalar Conservation Laws

**Abstract**: Flux Corrected Transport (FCT)-like
methods impose bounds or other constraints in the solution of
convection-dominated problems. To do this, the FCT method considers a low-order
approximation that guarantees the fulfillment of the constraint via mass
lumping and artificial dissipative matrices. The accuracy of the low-order
solution is then improved by incorporating a limited anti-diffusive flux based
on a high-order discretization. These methodologies have been widely applied
within finite element methods. Nevertheless, different problems arise when the
FCT method is applied within high-order finite element spaces. In particular:

i) The low-order method is more dissipative as the degree of the polynomial space is increased (while the number of degrees of freedom is kept fixed).

ii) As the polynomial space is increased, flux limiting is applied to a larger set of degrees of freedom. The consequence of this is the presence of non-physical oscillations (even when the solution fulfills the desired constraint). See for example Figure (b).

These two problems are created (in part) by the fact that the stencil of the discretization increases with the degree of the polynomial space. In this talk, we discuss different ideas to perform subcell flux limiting within continuous and discontinuous finite element discretizations based on high-order Bernstein polynomial spaces. We present numerical studies in one- and two-dimensions.

(2) The Boundary Variation Diminishing (BVD) formulation provides a new paradigm to devise high-fidelity finite volume schemes to capture both smooth and non-smooth flow structures with superior solution quality. The BVD formulation uses multiple reconstruction functions, so-called BVD-admissible functions, for different solution structures and adaptively selects the reconstruction function from the candidates according to the principle that minimizes the jumps of the reconstructed physical variables at cell boundaries, which thus effectively reduces the dissipation errors. More profoundly, the BVD formulation provides an alternative to the conventional limiting-projection approach to eliminate numerical oscillation. With proper BVD-admissible functions and BVD algorithms, we have developed a new class of numerical schemes of great practical significance for compressible and interfacial multiphase flows. The numerical schemes have been extensively verified with various benchmark tests of single and multiphase compressible flows involving strong discontinuities and complex flow structure of broad range scales.

**Yulong Xing**

**Title**: Arbitrary Order Structure Preserving Discontinuous Galerkin Methods for the Euler Equations with Gravitation

**Abstract**: Hydrodynamical evolution in a
gravitational field arises in many astrophysical and atmospheric problems. In
this presentation, we will talk about arbitrary order structure preserving
discontinuous Galerkin finite element methods which can exactly capture the
non-trivial steady state solutions for the Euler equations under gravitational
fields, and at the same time maintain the non-negativity of some physical
quantities. Numerical tests are provided to verify the well-balanced property,
positivity-preserving property, high-order accuracy, and good resolution for
both smooth and discontinuous solutions.