Abstract : Numerically solving hyperbolic systems of conservation and balance laws is a challenging task as their solutions may
develop extremely complicated nonsmooth structures. The number of applications in which such systems arise keeps
increasing and most of the existing methods have their restrictions and disadvantages. Therefore, it is extremely
important to develop new, highly accurate, stable, and robust numerical methods. The mini-symposium will focus on
recent developments in this field of research and will bring together researchers from different countries and provide
an opportunity for in-depth scientific discussion and exchange of ideas on the development, analysis, and applications
of modern methods.
00587 (1/2) : 2D @D401 [Chair: Alexander Kurganov]
[01485] A New Locally Divergence-Free Path-Conservative Central-Upwind Scheme for Ideal and Shallow Water Magnetohydrodynamics
Format : Talk at Waseda University
Author(s) :
Alina Chertock (North Carolina State University)
Alexander Kurganov (Southern University of Science and Technology)
Michael Redle (North Carolina State University)
Kailang Wu (Southern University of Science and Technology)
Abstract : This talk presents a new second-order unstaggered path-conservative central-upwind scheme for ideal and shallow water MHD equations. The new scheme locally preserves the divergence-free constraint, does not rely on Riemann problem solvers, and robustly produces high-resolution and non-oscillatory results. The derivation of the scheme is based on the Godunov-Powell nonconservative modifications of the studied systems and by augmenting it with the evolution equations for the corresponding derivatives of the magnetic field components.
[01517] Geometric Quasilinearization (GQL) for Bound-Preserving Schemes of Hyperbolic PDEs
Format : Talk at Waseda University
Author(s) :
Kailiang Wu (Southern University of Science and Technology)
Chi-Wang Shu (Brown University)
Abstract : Solutions to many partial differential equations satisfy certain bounds or constraints. For example, the density and pressure are positive for equations of fluid dynamics, and in the relativistic case the fluid velocity is upper bounded by the speed of light, etc. As widely realized, it is crucial to develop bound-preserving numerical methods that preserve such intrinsic constraints. Exploring provably bound-preserving schemes has attracted much attention and is actively studied in recent years. This is however still a challenging task for many systems especially those involving nonlinear constraints.
Based on some key insights from geometry, we systematically propose a novel and general framework, referred to as geometric quasilinearization (GQL), which paves a way for studying bound-preserving problems with nonlinear constraints. The essential idea of GQL is to equivalently transfer all nonlinear constraints into linear ones, through properly introducing some free auxiliary variables. We establish the fundamental principle and general theory of GQL via the geometric properties of convex regions, and propose three simple effective methods for constructing GQL. We apply the GQL approach to a variety of partial differential equations, and demonstrate its effectiveness and remarkable advantages for studying bound-preserving schemes, by diverse challenging examples and applications which cannot be easily handled by direct or traditional approaches.
[01722] An improved non-hydrostatic shallow-water type model for the simulation of landslide generated tsunamis
Format : Talk at Waseda University
Author(s) :
Manuel J Castro Diaz (University of Málaga)
Tomas Morales de Luna (Universidad de Malaga)
Cipriano Escalante Sanchez (Universidad de Málaga)
Jorge Macías Sanchez (University of Málaga)
Enrique D Fernandez Nieto (University of Sevilla)
Abstract : In this talk we present an improved version of a non-hydrostatic shallow-water system coupled with a granular landslide shallow-water type model for the numerical simulation of tsunamis generated by landslides. The system is discretized by means of a high-order WB finite volume scheme. Some numerical tests with laboratory experiments and real events will be presented to show the capabilities of the proposed model.
[01949] A New Approach for Designing Well-Balanced Schemes for the Shallow Water Equations
Format : Talk at Waseda University
Author(s) :
Remi Abgrall (University of Zurich)
Yongle Liu (University of Zurich)
Abstract : In this talk, I will introduce a new approach for constructing robust well-balanced numerical methods for one-dimensional Saint-Venant system. We combine the conservative and non-conservative formulations of the studied hyperbolic system in a natural way. The solution is globally continuous and described by a combination of point values and average values, which will be evolved by two different forms of PDEs. We demonstrate the behavior of the new scheme on a number of challenging examples.
[02636] Error analysis of finite volume methods for the Euler equations via relative energy
Format : Talk at Waseda University
Author(s) :
Maria Lukacova (University of Mainz)
Bangwei She (Capital Normal University)
Yuhuan Yuan (University of Mainz)
Abstract : We present an overview of our recent results for the error analysis of some finite volume
methods for multidimensiona Euler system. To control global error, we apply the relative energy principle and
estimate the L2 norm between a numerical solution and the strong solution.
If time permits, we will present an extension to the error analysis of the random Euler system approximated
by the Monte Carlo finite volume method.
[04782] Flux Globalization Based Well-Balanced Path-Conservative Central-Upwind Schemes
Format : Talk at Waseda University
Author(s) :
Alexander Kurganov (Southern University of Science and Technology)
Abstract : The talk will focus on numerical methods for nonconservative hyperbolic systems of balance laws. I will introduce well-balanced path-conservative central-upwind schemes, which are based on the flux globalization: both source and nonconservative product terms are incorporated into the global flux. The resulting quasi-conservative system is numerically solved using a semi-discrete central-upwind scheme with the numerical fluxes are evaluated using the path-conservative technique. Applications to several shallow water models will be demonstrated.
[01953] High order well-balanced and asymptotic preserving WENO schemes for the shallow water equations
Format : Online Talk on Zoom
Author(s) :
Yulong Xing (Ohio State University)
Abstract : Shallow water equations (SWEs) with a non-flat bottom topography have been widely used to model flows in rivers and coastal areas. In this presentation, we will talk about the applications of high-order semi-implicit well-balanced and asymptotic preserving (AP) WENO methods to this system. We consider the Froude number ranging from O(1) to 0, which in the zero Froude limit becomes the “lake equations” for balanced flow without gravity waves. We apply a well-balanced finite difference WENO reconstruction, coupled with a stiffly accurate implicit-explicit (IMEX) Runge-Kutta time discretization. The resulting semi-implicit scheme can be shown to be well-balanced, AP and asymptotically accurate at the same time. Both one- and two-dimensional numerical results are provided to demonstrate the high order accuracy, AP property and good performance of the proposed methods in capturing small perturbations of steady state solutions.
