Venue: 
AAB608, Academic and Administration Building,
Baptist University Road Campus, Hong Kong Baptist University



Keynote Speaker: 
ChiWang Shu, Brown University 



Speakers: 
Felix Kwok, Hong Kong Baptist University 
Buyang Li, The Hong Kong Polytechnic University 
Leevan Ling, Hong Kong Baptist University 
Xuelei Lin, Hong Kong Baptist University 
Weifeng Qiu, City University of Hong Kong 
Yinhua Xia, University of Science and Technology of China 
Yan Xu, University of Science and Technology of China 
Qiang Zhang, Nanjing University 
Zhiwen Zhang, University of Hong Kong 





Program  
9:3010:20 
ChiWang Shu
Entropy stable high order discontinuous Galerkin
methods for hyperbolic conservation laws
Abstracts: It is well known that semidiscrete high order discontinuous
Galerkin (DG) methods satisfy cell entropy inequalities for
the square entropy for both scalar conservation laws and
symmetric hyperbolic systems, in any space dimension and
for any triangulations. However, this property holds only for
the square entropy and the integrations in the DG methods must
be exact. It is significantly more difficult to design DG
methods to satisfy entropy inequalities for a nonsquare convex
entropy, and / or when the integration is approximated by a
numerical quadrature. In this talk, we report on our recent
development of a unified framework for designing high order DG
methods which will satisfy entropy inequalities for any given
single convex entropy, through suitable numerical quadrature
which is specific to this given entropy. Our framework applies
from onedimensional scalar cases all the way to multidimensional
systems of conservation laws. For the onedimensional case, our
numerical quadrature is based on the methodology established in
the literature, with the main ingredients being summationbyparts
(SBP) operators derived from Legendre GaussLobatto quadrature,
the entropy stable flux within elements, and the entropy stable
flux at element interfaces. We then generalize the scheme to
twodimensional triangular meshes by constructing SBP operators
on triangles based on a special quadrature rule. A local
discontinuous Galerkin (LDG) type treatment is also incorporated
to achieve the generalization to convectiondiffusion equations.
Numerical experiments will be reported to validate the accuracy
and shock capturing efficacy of these entropy stable DG methods.
This is a joint work with Tianheng Chen.

10:2010:55 
Weifeng Qiu

10:5511:10 
Coffee Break 
11:1011:45 
Qiang Zhang
Error estimates of local discontinuous
Galerkin method with generalized alternating fluxes for convection diffusion equations
Abstracts: In this talk we shall discuss the local discontinuous Galerkin methods
with the generalized upwind numerical flux and generalized alternating numerical fluxes,
when solving one and/or
twodimensional convection diffusion problems. Firstly the periodic condition is considered and the global
L^{2}norm error estimate is optimal. To do that, we will introduce the generalized GaussRadau (GGR) projection,
which is defined globally, not locally.
Secondly the singularity perturbation of convection diffusion equation with
Dirichlet condition is considered, which has a stationary outflow boundary
layer. The local L^{2}norm error estimate is doubleoptimal, namely, not only
the width of pollution domain is optimal, and also the order of L^{2}norm error
out of pollution domain is optimal. To this purpose, we will introduce suitable weight functions nearby the outflow boundary,
and carefully establish a deep investigation on the GGR projection.

11:4512:20 
Zhiwen Zhang
Compute effective diffusivity of chaotic and stochastic flows using
structure preserving schemes
Abstracts: In this talk we study the problem of computing the effective diffusivity
for a particle moving in chaotic and stochastic flows. In addition we
numerically investigate the residual diffusion phenomenon in chaotic
advection. Instead of solving the FokkerPlanck equation in the Eulerian
formulation, we compute the motion of particles in the Lagrangian
formulation, which is modelled by stochastic differential equations
(SDEs). We propose a new numerical integrator based on a stochastic
splitting method to solve the corresponding SDEs in which the
deterministic subproblem is symplectic preserving while the random
subproblem can be viewed as a perturbation. We provide rigorous error
analysis for the new numerical integrator using the backward error
analysis technique and show that our method outperforms standard
Eulerbased integrators. Numerical results are presented to demonstrate
the accuracy and efficiency of the proposed method for several typical
chaotic and stochastic flow problems of physical interests.

12:2014:00 
Lunch 
14:0014:35 
Yan Xu
Globally divergencefree discontinuous Galerkin methods for ideal magnetohydrodynamics equations
Abstracts: Ideal magnetohydrodynamic (MHD) equations are widely used in many areas in physics and engineering, and these equations have a divergencefree constraint on the magnetic field. In this paper, we propose high order globally divergencefree numerical methods to solve the ideal MHD equations. The algorithms are based on discontinuous Galerkin methods in space. The magnetic induction equations are discretized separately to approximate the normal components of the magnetic field on elements interfaces, and to extract additional information about the magnetic field when higher order accuracy is desired. This is then followed by an element by element reconstruction to obtain the globally divergencefree magnetic field. In time, strongstabilitypreserving RungeKutta methods are applied. In consideration of accuracy and stability of the methods, a careful investigation is carried out, both numerically and analytically, to study the choices of the numerical fluxes associated with the electric field at element interfaces and vertices. The resulting methods are local and the approximated magnetic fields are globally divergencefree. Numerical examples are presented to demonstrate the accuracy and robustness of the methods.

14:3515:10 
Buyang Li
Maximal L^{p}regularity of time discretization and finite element
methods
Abstracts: For a parabolic problem with maximal L^{p}regularity, we prove
that the time discretization by a linear multistep method or RungeKutta
method preserves the maximal L^{p}regularity uniformly in the stepsize if
the method is Astable (and satisfies minor additional conditions). In
particular, the implicit Euler method, the CrankNicolson method, the
secondorder backward difference formula (BDF), and the Radau IIA and
Gauss RungeKutta methods of all orders preserve maximal regularity. In
general polygons and polyhedra, possibly nonconvex, the analyticity of the
finite element heat semigroup in the L^{q} norm, 1 <= q <= \infty, and
the maximal L^{p}regularity of finite element solutions of parabolic
equations are proved.

15:1015:45 
Leevan Ling
Kernel based methods and some adaptive algorithms
Abstracts: A brief introduction of translationinvariant kernel based
methods, aka radial basis function methods, for function approximation and
PDEs will be given. These methods do not require meshes, but in return, we
must deal with highly illconditioned linear systems. In this talk, we will
introduce an adaptive algorithm that selects appropriate column subspaces
that ensure linear independency.

15:4516:00 
Coffee Break 
16:0016:35 
Yinhua Xia
Asymptotic preserving spectral deferred correction methods for hyperbolic systems with relaxation
Abstracts: In this talk, we present semiimplicit spectral deferred correction (SDC) methods for hyperbolic systems of conservation laws with stiff relaxation terms. The relaxation term is treated implicitly, and the convection terms are treated by explicit schemes.
The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit and can be constructed easily and systematically for any order
of accuracy. Weighted essentially nonoscillatory (WENO) schemes are adopted in spatial discretization to achieve high accuracy in space. After a description of the mathematical properties of the schemes, several applications will be presented to demonstrate the capability of the schemes.
This is a joint work with C. Sun.

16:3517:10 
Felix Kwok
Time parallelization of waveform relaxation methodsE
Abstracts: In this talk, we consider waveform relaxation (WR) methods for solving timedependent PDEs. WR methods are distinctive in that a typical subdomain problem is posed in both space and time; each iteration requires the parallel solution of these spacetime subproblems, followed by an exchange of interface data defined over the whole time window. An often cited advantage of WR methods is that they allow each subdomain to use a different spatial and temporal grid that is adapted to the dynamics of the local subproblem.
Here, we show that WR methods have another advantage, namely that they allow additional parallelism in time. The key observation is that in most cases, one has enough initial and interface data to start an WR iteration even before the previous iteration has completed. In other words, several iterations can run simultaneously. Based on this observation, we propose two ways of parallelizing WR methods in time. The first one uses a fixed time window and is mathematically equivalent to the original WR method. The second one chooses the timewindow size dynamically based on how many free processors are available; this leads to a method with different convergence behaviour. We demonstrate the effectiveness of both approaches by comparing their running times against those obtained from classical timestepping methods, where the same number of processors is used to parallelize in space only.

17:1017:35 
Xuelei Lin
CrankNicolson ADI scheme for Riesz spacefractional diffusion equation with nonseparable variable coefficients
Abstracts: We propose a framework for proving unconditional stability and convergence of CrankNicolson ADI schemes for twodimensional Riesz spacefractional diffusion equations with nonseparable variable coefficients in the sense of discrete L2 norm, in which the CrankNicolson discretization and an unspecified Toeplitz discretization are employed to approximate the firstorder temporal derivative and the fractional order spatial derivative, respectively. This is the first attempt to propose CrankNicolson ADI schemes for the equation with nonseparable variable coefficients. To the best of our knowledge, unconditional stability (convergence) of previous ADI schemes for the equation are established only for singlespatialvariatedependentseparable coefficients (spatiallyindependent coefficients). In contrast, we show the unconditional stability and convergence of CrankNicolson ADI schemes for the equation with nonseparable threevariates functions (one temporal variate and two spatial variates) under certain mild assumptions on the diffusion coefficients and the unspecified Toeplitz discretization. The assumptions imposed on the unspecified Toeplitz discretization are satisfied by all existing schemes. The assumptions imposed on the diffusion coefficients are related to partial Lipschitz continuity, which is weaker than Lipschitz continuity. Moreover, a splitting preconditioner is employed to precondition the onedimensional systems resulting from the propose ADI schemes, with which the condition numbers of resulting preconditioned matrices are uniformly bounded by a constant independent of discretization stepsizes. As a result, the ADI systems are solved withing an optimal arithmetic complexity. Numerical results are reported to support the theoretical analysis and to show the efficiency of the proposed method.






