Structure Preserving Numerical Methods for Hyperbolic PDEs

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

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. 

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.