Invited Plenary Speaker: Jiequan Li, a Professor of Computational and Applied Mathematics at the Institute of Applied Physics and Computational Mathematics, China
High Order Temporal-Spatial Coupled Methods for
Compressible Multiphase Fluid Flow Problems
The most distinct feature of compressible multiphase flows is the presence of singularities (shocks, material interfaces, vortices and other discontinuities etc.) in flows, which arises notorious difficulties in all aspects of theoretical justification, numerical analysis, scientific computation as well as engineering applications. Just from the viewpoint of the design of numerical methods, high resolution methods have become the mainstream for several decades, however, there are still many bottleneck problems unsolved.
This lecture will address some fundamentals in this aspect, based the celebrated Lax-Wendroff method, which can be traced back traced back to Cauchy-Kowalevski’s theory in 1700's in terms of power series solution for hyperbolic problems. The irreplaceable values the Lax-Wendroff approach can be summarized as follows:
(i) It is a unique three-point second order accurate scheme. Any high order scheme should be consistent with the Lax-Wendroff method when it reduces to its second order version. Hence the Lax-Wendroff method is the reference of all high order accurate methods for compressible fluid flows.
(ii) It uses the least stencils (just three points for each time step) and is therefore most compact in the family of methods of second order both in space and time. The compactness determines the numerical dissipation of a scheme near singularities.
(iii) It is a temporal-spatial coupled method and all useful information of the governing equations are fully incorporated into the scheme. There is no need to exert extra effort even when any other physical or geometrical effects are included. The spatial-temporal coupling is a key element that is consistent with relevant physical features such as the temporal-spatial coherence in the turbulent flows.
Nevertheless, the Lax-Wendroff approach just works for smooth flows, and it should be modified to suit for capturing discontinuities. The currently-used generalized Riemann problem (GRP) method is regarded as the discontinuous version of Lax-Wendrof method, and it uses both the Cauchy-Kowalevski methodology and the singularity tracking technique. The resulting scheme is consistent directly with the corresponding physical balance laws in integral form rather than in PDE form. Hence the GRP method works well even when the flows contain very strong singularities (shocks, interfaces etc).
The lecture has four parts: (1) We will rigorously define high order/resolution methods as singularities are present in fluid flows; (2) As singularities are present, non-equilibrium effect becomes important and thus thermodynamics should be seriously taken into account. We will clarify how build the thermodynamic effect into the design of methods; (3) We will formulate a new framework to design temporal-spatial coupled high order numerical methods, in sharp contrast with the line method with Runge-Kutta type methods as representatives; (4) We will display the performance of the methods through a serious application for real engineering problems.
About the author:
Professor Jiequan Li is the distinguished professor at the Institute of Applied Physics and Computational Mathematics, Beijing, China. He is currently working in Laboratory of Computational Physics, a national-oriented key lab in China. He graduated from Institute of Mathematics, the Chinese Academy of Sciences, completed Lady Davis at the Hebrew University of Jerusalem in Israel and the Humboldt Fellowship at Magdeburg University in Germany. Also he took visiting professor positions in
Academia Sinica in Taiwan, Stanford University, the Pennsylvania State University, Mainz University, Hongkong University of Science and Technology, National University of Singapore etc. He has the expertise in the fields of computational fluid dynamics, numerical analysis and partial differential equations. His work on 2-dimensional Riemann problem for gas dynamics is internationally influential, as indicated in his book published in Longman Press. Recently he is paying more attention on the design of numerical methods for compressible fluid flows and multi-material/phase flows, which have been applied to real engineering problems. He has written more than 60 research papers in top journals and won many influential prizes in China.
Professor Li website can be found under this link
