Special but not exclusive topics are uncertainty quantification, MultiLevelMonte Carlo methods,
measured valued solutions, statistical solutions, high performance computing. 


Venue: 
SWT501, Shaw Tower,
Shaw Campus, Hong Kong Baptist University



Organizers: 
Rolf Jeltsch, ETH Zurich 
Leevan Ling, Hong Kong Baptist University 



Speakers: 
Harish Kumar, Indian Institute of Technology, Delhi 
Alexander Kurganov, Southern University of Science and Technology and Tulane University 
Filippo Leonardi, ETH Zurich 
Kjetil Olsen Lye, ETH Zurich 
Dekang Mao, Shanghai University 
KehMing Shyue, National Taiwan University 
Jonas Šukys, Eawag: Swiss Federal Institute of Aquatic Science and Technology 
Kun Xu, Hong Kong University of Science and Technology 



Program  (Abstracts) 
Friday, 10 November 2017 
09:3010:00 
Welcome, Opening and Photos 
10:0010:45 
Alexander Kurganov
Centralupwind schemes for shallow water models
Abstracts: In the first part of the talk, I will describe a general framework for designing finitevolume methods (both upwind and central) for hyperbolic systems of conservation laws. I will focus on Riemannproblemsolverfree nonoscillatory central schemes and, in particular, on centralupwind schemes that belong to the class of central schemes, but has some upwind features that help to reduce the amount of numerical diffusion typically present in staggered central schemes such as, for example, the firstorder LaxFriedrichs and secondorder NessyahuTadmor scheme.

10:4511:30 
Jonas Šukys
Uncertainty quantification using parallel multilevel Monte Carlo: applications to shallow water, Euler, magnetohydrodynamics, and multiphase cavitation flows
Joint work with: C. Linares, U. Rasthofer, P. Hadjidoukas, F. Wermelinger, S. Mishra,
Ch. Schwab, M. Castro, and P. Koumoutsakos
Abstract: Complex liquid, gas, and plasma flow problems can be modeled in terms of nonlinear conservation laws. Examples include shallow water equations for lakes, rivers, earthquake or landslide generated tsunamis, multiphase Euler equations for cavitating vapor cloud collapses, and Darcy's law for porous subsurface flows. Many of the above nonlinear dynamical systems exhibit strong dependence on uncertain input data, such as initial data, sources and model coefficients. In this talk I will present a mathematical setting as well as two numerical methods for nonintrusive uncertainty quantification and propagation in such flows, namely finite volume method for the spatiotemporal discretization and the multilevel Monte Carlo statistical sampling technique, which accelerates standard Monte Carlo method by several orders of magnitude using clever variance reduction obtained from simulations with coarser spatiotemporal resolutions as control variates. Efficient implementation of such hierarchical discretization schemes relies on several more advanced mathematical concepts such as multilevel aliasfree representation for random bathymetry and unbiased spectral parallel FFT generation of random porosity fields. Numerical experiments using inhouse developed ALSVIDUQ, CubismMPCF, and PyMLMC software up to one trillion mesh elements and 500'000 cores will be presented, illustrating the efficiency of the methods and pushing be boundaries of current scientific knowledge in this field.

11:3012:00 
Discussion 
12:0013:30 
Lunch 
13:3014:15 
Kun Xu
Importance of time accurate flux function on construction of highorder schemes
Abstract: The higher order CFD methods for compressible flow are mostly based on WENO and DG formulations, where the exact or approximate Riemann solver is used for the flux evaluation. The use of the 1storder Riemann flux function may be the barrier for the further development of higherorder accurate, robust, and efficient methods. In CFD community, the necessity of using highorder flux function, such as those based on the generalized Riemann problem and gaskinetic scheme, has not been fully recognized. In this talk, we are going to demonstrate the importance of highorder time accurate flux function, and its usage in the development of higherorder schemes. With the support of numerical examples, the superior advantages of the newly developed higherorder gas kinetic schemes over the existing WENO and DG methods will be presented.

14:1515:00 
KehMing Shyue
An operator splitting method for dispersive wave problems
Abstract: Our aim in this talk is to describe a simple operatorsplitting approach for the efficient numerical simulation of dispersive wave problems. The algorithm uses the IordanskiKogarkoWijngaarden model (cf. [1, 2, 4]) for pressure waves in bubbly liquids as the basis, and reformuate it into a hyperbolicelliptic system so that higherorder derivatives terms modelling dispersive effects of solutions can be handled straightforwardly by the method. Sample numerical results are shown to demonstrate the feasibility of the proposed method for a class of benchmark problems in bubbly liquids, and also for problems modeled by the GreenNaghdi equations of the longwave shallow water flow [3].
References
[1] S.V. Iordanski. On the equations of motion of the liquid containing gas bubbles. Z. Prik. Mekh. Tekhn. Fiziki, N3:102111, 1960 (in Russian).
[2] B.S. Kogarko. On the model of cavitating liquid. Dokl. AN SSSR, 137:13311333, 1961 (in Russian).
[3] O. Le Metayer, S. L. Gavrilyuk, and S. Hank. A numerical scheme for the GreenNaghdi model. J. Comput. Phys., 229:20342045, 2010.
[4] L. van Wijngaarden. On the equations of motion for mixtures of liquid and gas bubbles. J. Fluid Mech., 33:465474, 1968.

15:0015:30 
Break 
15:3016:15 
Harish Kumar
Positivitypreserving Highorder Discontinuous Galerkin Schemes for Tenmoment Gaussian Closure Equations
Joint work with: Dr. Praveen Chandrashekar (TIFRCAM, Bangalore) and Ms. Asha Meena (IIT Delhi)
Abstract: Euler equations for compressible flows treats pressure as a scalar quantity. However, for several applications this description of pressure is not suitable. Many extended model based on the higher moments of Boltzmann equations are considered to overcome this issue. One such model is Tenmoment Gaussian closure equations, which treats pressure as symmetric tensor.
In this work, we develop a higherorder, positivity preserving Discontinuous Galerkin (DG) scheme for Tenmoment Gaussian closure equations. The key challenge is to preserve positivity of density and symmetric pressure tensor. This is achieved by constructing a positivity limiter. In addition to preserve positivity, the scheme also ensures the accuracy of the approximation for smooth solutions. The theoretical results are then verified using several numerical experiments.
In the second part of the talk, I will discuss how centralupwind schemes can be extended to hyperbolic systems of balance laws, such as the SaintVenant system and related shallow water models. The main difficulty in this extension is preserving a delicate balance between the flux and source terms. This is especially important in many practical situations, in which the solutions to be captured are (relatively) small perturbations of steadystate solutions. The other crucial point is preserving positivity of the computed water depth (and/or other quantities, which are supposed to remain nonnegative). I will present a general approach of designing wellbalanced positivity preserving centralupwind schemes and illustrate their performance on a number of shallow water models.

16:1517:00 
Kjetil Olsen Lye
Efficient MonteCarlo Methods for Statistical Solutions of Hyperbolic Conservation Laws
Joint work with: Siddhartha Mishra and Ulrik Fjordholm
Abstract: An open question in the field of hyperbolic conservation laws is the question of wellposedness. Recent theoretical and numerical evidence have indicated that multidimensional systems of hyperbolic conservation laws exhibit random behavior, even with deterministic initial data. We use the framework of statistical solutions to model this inherit randomness. We review the theory of statistical solutions for conservation laws.
Afterwards, we introduce a convergent numerical method for computing the statistical solution of conservation laws, and prove that it converges in the Wasserstein distance through narrow convergence for the case of scalar conservation laws. For the scalar case, we validate our theory by computing the structure functions of the Burgers' equation with random initial data. We especially focus on Brownian initial data, and the measurement of the scalings of the structure functions. The results agree well with the theory, and we get the expected convergence rate. We furthermore show that we can get faster computations using Multilevel MonteCarlo for computing the statistical solutions of scalar conservation laws.
In the case of systems of equations, we test our theory against the compressible Euler equations in two space dimensions. We check our numerical algorithm against two illbehaved initial data, the KelvinHelmholtz instability and the RichtmeyerMeshkov instability, and compute the corresponding structure functions. We furthermore show that in the case of these illbehaved initial data, Multilevel MonteCarlo cannot improve upon the MonteCarlo algorithm in computing the statistical solutions.

17:0017:30 
Discussion 

Dinner 

Saturday, 11 November 2017 
09:3010:15 
Dekang Mao
Numerical dissipations in NavierStokeslike form for conservative fronttracking method
Abstract: In simulations of interfacial instabilities Euler system of fluid dynamics is taken as the governing equations, and in doing so the smallscale physical dissipations, mass diffusion, viscosity and heat condition, are ignored. In capturing simulations, smallscale rollups on the interfaces occur in early times and are believed to be the artifacts caused by numerical dissipations of the capturing methods. Fronttracking simulations eliminate the numerical dissipations across the interfaces; however, they lack proper mechanism to stabilize the interface and the situation will get worse. This talk presents numerical dissipations designed for a previously developed conservative fronttracking method, which stabilize the tracked interfaces and prevent the earlytime rollups. These numerical dissipations simulate in someway the missing physical dissipations in the Euler system on the interfaces. The designed numerical dissipations are of the order of mesh size; therefore, they tend to zero along with the grid refinement. Numerical examples are presented to show the efficiency and effectiveness of these numerical dissipations.

10:1511:00 
Filippo Leonardi
Approximating ensembles of incompressible flows
Abstract: Ensembles of solutions arise when considering the flow as a statistical quantity rather than a deterministic one. The main benefits of considering ensembles of solutions are two. On one hand, it allows us to obtain (statistical and "in the large") information about the fluid behaviour, information that would otherwise be impossible to gather by looking at the individual solution. On the other hand, it allows us to incorporate the notion of uncertainty quantification within the same framework, at no extra cost.
We will discuss two frameworks for ensembles of solutions: statistical solutions, i.e. solutions as pushforward measures on function spaces, and measure valued solutions, i.e. spacetime parametrised Young measures. We will focus on the efficient approximation of these types of solutions and discuss implementations of those approximations.

11:0011:30 
Discussion and closing 





