Session s Title (if available) Tue - 17 Jan :45 ~ 16:15 Room 12 Session Chair(s): Qin SHENG (Baylor University)

Transcription

1 C9. Recent Development of High Order Numerical Methods for Hyperbolic Conservation Laws and Its Applications Chair(s): Zhen GAO (Ocean University of China, China), Guanghui HU (University of Macau, Macao). Session s Title (if available) Tue - 17 Jan :45 ~ 16:15 Room 12 Session Chair(s): Qin SHENG (Baylor University) 14:45 C9-1 (Keynote) Title of Talk: High-Order Accurate Physical-Constraints-Preserving Schemes for Special Relativistic Hydrodynamics Authors Names: Huazhong TANG Affiliation: Peking University Abstract: Relativistic hydrodynamics (RHD) plays an essential role in many fields of modern physics, e.g. astrophysics. Relativistic flows appear in numerous astrophysical phenomena from stellar to galactic scales, e.g. active galactic nuclei, super-luminal jets, core collapse super-novae, X-ray binaries, pulsars, coalescing neutron stars and black holes, micro-quasars, and gamma ray bursts, etc. The relativistic description of fluid dynamics should be taken into account if the local velocity of the flow is close to the light speed in vacuum or the local internal energy density is comparable (or larger) than the local rest mass density of the fluid. It should also be used whenever matter is influenced by large gravitational potentials, where the Einstein field theory of gravity has to be considered. The dynamics of the relativistic systems requires solving highly nonlinear equations and the analytic treatment of practical problems is extremely difficult. Hence, studying them numerically is the primary approach. We develop high-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamical (RHD) equations, built on the local Lax-Friedrich splitting, the WENO reconstruction, the physical-constraints-preserving flux limiter, and the high order strong stability preserving time discretization. They are formal extensions of the existing positivity-preserving finite difference WENO schemes for the non-relativistic Euler equations. However, developing physical-constraints-preserving methods for the RHD system becomes much more difficult than the non-relativistic case because of the strongly coupling between

2 15:15 C9-2 (Invited) the RHD equations, no explicit expressions of the conservative vector for the primitive variables and the flux vectors, and one more physical constraint for the fluid velocity in addition to the positivity of the rest-mass density and the pressure. The key is to prove the convexity and other properties of the admissible state set and discover a concave function with respect to the conservative vector replacing the pressure which is an important ingredient to enforce the positivity-preserving property for the non-relativistic case. Several one- and two-dimensional numerical examples are used to demonstrate accuracy, robustness, and effectiveness of the proposed physical-constraints-preserving schemes in solving relativistic problems with large Lorentz factor. Title of Talk: The high order positivity-preserving numerical method for compressible multi-media flow Authors Names: Chunwu WANG Affiliation: Nanjing University of Aeronautics and Astronautics Abstract: In the numerical simulations of the multi-medium flow such as blast waves or high-velocity jets, the negative density or pressure may occur, in the low density and low pressure domain, due to the numerical errors of the high order schemes. The loss of positivity of the physically positive variables may lead to nonlinear instability or blow-ups of the algorithm. In this paper, we construct high order accurate schemes which preserve positivity of density and pressure in the simulation of compressible multi-media flows. The method is base on the positivity-preserving Runge-Kutta discontinuous Galerkin (RKDG) schemes for single medium flows and the real Ghost Fluid method (RGFM) for the interface treating. The schemes are extended to the simulation of multi-media flows and the positivity-preserving limiters for pressure are modified for simplicity. The obtained limiters can be proven to keeps the property of the original limiter and be cost effective. Furthermore, we develop a positivity preserving Riemann solver to deal with the double shock approximate Riemann problem in RGFM method. Several examples are given to test robustness and efficiency of the algorithm. Numerical results show that the obtained method can maintain the positivity of pressure or density and can capture the discontinuities accurately.

3 15:45 C9-3 (Invited) Title of Talk: The Fictitious Domain Method Based on Navier-Slip Boundary Condition for Simulation of Flow-Body Interactions Authors Names: Qiaolin HE Affiliation: Sichuan University Abstract: The fictitious domain method is an effective method for simulating flow-body interaction. The method was developed by Glowinski for simulating the particulate flow with no slip boundary condition. In many applications, fluid slip at the solid surface becomes important. The generalization of the fictitious domain method to slip boundary condition is not trivial. As a step in this direction, we discuss a new least-square/fictitious domain method for Navier-Stokes problem with slip boundary conditions. The method is of the virtual control type and relies on a least-squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. Numerical results are presented. Furthermore, this method is applied to Flow-Body interaction problems with Navier-Slip boundary condition at the interface. This is a joint work with R. Glowinski and XP Wang. Session s Title (if available) Tue - 17 Jan :30 ~ 18:00 Room 12 Session Chair(s): Huazhong TANG (Peking University) 16:30 C9-4 (Keynote) Title of Talk: A Pseudo High Order Decomposition Scheme for Highly Oscillatory Optical Wave Equations Authors Names: Qin SHENG Affiliation: Baylor University Abstract: Paraxial wave equations describe the propagation of electromagnetic waves in theform of either paraboloidal waves or Gaussian beams. The partial differential equations have been used extensively in optical wave studies and computations, especially within focal regions, due to their emitting features of laser lights. Our study concerns a decomposed high order compact scheme for solving highly oscillatory paraxialwave problems in radially symmetric fields. Inspired by the decomposition strategy frequentlyused for solving singularly perturbed problems,

4 herewith we are particularly interested in a pseudohigh order multiscale approach that is highly accurate in transverse directions as well as highly efficientand effective for practical physical applications. The slit configuration is accomplished inthe transverse direction to eliminate the singularity of the differential equation in polarcoordinates. A special attention is also paid to the linear numerical stability of the proposedsplitting algorithm. It is shown that the high order method accomplished is stable under reasonableconstraints for optical applications. The method is also realistic due toits straightforward algorithmic structure. Multiple computational examples will be presented toillustrate our results and their applications in modern laser optics. 17:00 C9-5 (Invited) Title of Talk: A characteristic weak Galerkin finite element method for time-dependent convection-dominated diffusion problem Authors Names: Fuzheng GAO Affiliation: Shandong University Abstract: A characteristic weak Galerkin finite element method (WG-FEM) for time-dependent convection-dominated diffusion problems in two dimensions is developed in this paper. This method is derived by combining the WG-FEM with the characteristic technique. The error estimates are established for the corresponding characteristic WG-FEM approximations in both discrete $H^1$-norm and standard $L^2$-norm. Numerical results are presented to demonstrate the robustness, reliability, and accuracy of the characteristic WG-FEM. 17:30 C9-6 (Invited) Title of Talk: Adjoint-based an adaptive finite volume method for steady Euler equations in 2D Authors Names: Nianyu YI Affiliation: Xiangtan University

5 Abstract: We develop a high order adaptive finite volume method for the steady Euler equations in 2 dimensional. The method consists of a Newton iteration method to linearize the governing equations, and a geometrical multigrid method to solve the derived system. The non-oscillatory k-exact reconstruction with a new reconstruction patch for the element which locates on the domain boundary is proposed for improving the convergence of the steady state. The adjoint-based a posteriori error estimation is introduced for the h-adaptive method, which could deliver more efficient adaptive refinement of the mesh. The numerical tests show the effectiveness of the proposed method. Session s Title (if available) Wed - 18 Jan :15 ~ 11:45 Room 14 Session Chair(s): Leevan LING (Hong Kong Baptist University) 10:15 C9-7(Keynote) Title of Talk: Hybridization of Weighted Essentially Non-Oscillatory Finite Difference Scheme for Hyperbolic Conservation Law Authors Names: Wai Sun DON Affiliation: Ocean University of China Abstract: In this talk, I will discuss some recent development of hybridization of high order nonlinear weighted essentially non-oscillatory (WENO) finite difference scheme and classical linear (finite difference and compact) scheme and non-classical (Fourier continuation) scheme (Hybrid), together with several high order shock sensors (multi-resolution, conjugate Fourier, Radial Basis function and Quantiles) to determine the smoothness of a solution of the nonlinear hyperbolic conservation laws. The advantages and disadvantages of the various components of the Hybrid schemes in terms of accuracy and efficiency as well as several relevant critical issues of the Hybrid scheme will also be discussed and illustrated with examples. Examples, including the one dimensional shock-entropy wave interaction and two dimensional classical Riemann IVP problems, March 10 double Mach reflection problems and detonation wave problems, regarding the efficiency and accuracy of the Hybrid scheme will be discussed. 10:45 C9-8(Invited)

6 Title of Talk: Positivity/Symmetry-preserving property of the high order WENO finite difference schemes for hyperbolic conservation laws Authors Names: Zhen GAO Affiliation: Ocean University of China Abstract: High order nonlinear WENO finite difference shock capturing schemes with the global Lax-Friedrichs flux splitting, which have the capability of capturing shocks essentially non-oscillatory while resolving small scale structures efficiently, are popular for solving hyperbolic conservation laws. However, the WENO schemes still generate small numerical oscillations directly proportional to the strength of a discontinuity. Under certain extreme conditions, such as a strong shock (extreme large density and pressure ratios) and a near vacuum state (extreme low density and pressure), numerical oscillations might cause the density and pressure to become negative, thus creating a non-physical solution and possibly numerical instability, or worse, a stable but non-physical solution. In order to guarantee the positivity of density and pressure, the positivity-preserving limiters have been designed to address these problems. Numerous one- and two-dimensional examples have been used in the literature to demonstrate that the fifth order WENO scheme has failed in preserving the positivity of the solution under such challenging conditions. In this talk, we demonstrate, on the contrary, that these examples can all be successfully simulated without any need for positivity-preserving limiters. Another interesting observation is the symmetry-preserving property of the WENO scheme in several classical symmetric examples. 11:15 C9-9(Invited) Title of Talk: Adjoint-based an adaptive finite volume method for steady Euler equations Authors Names: Xucheng MENG Affiliation: University of Macau Abstract: In this work, we consider an adjoint-based adaptive finite volume method for numerically solving steady Euler equations with non-oscillatory k-exact reconstrcution. The method consists of a Newton iteration to linearize the governing equations, and a geometrical multigrid method to solve the corresponding linear system. The non-oscillatory k-exact reconstruction is applied to obtain

7 high quality solution reconstruction, and the adjoint-based a posteriori error estimation is introduced for the h-adaptive method. To improve the convergence of the steady state, we also test the new reconstruction patch for the elements which interacting with the boundary of the domain. The numerical tests are presented to demonstrate the effectiveness of the proposed method. Session s Title (if available) Wed - 18 Jan :45 ~ 16:15 Room 14 Session Chair(s): Wai Sun DON (Ocean University of China) 14:45 C9-10(invited) Title of Talk: Adaptive meshless methods and applications Authors Names: Leevan LING Affiliation: Hong Kong Baptist University Abstract: It is now commonly agreed that the global radial basis functions (GRBF) method is an attractive approach for approximating smooth functions. This superiority does not come free; one must find ways to circumvent the associated problem of ill-conditioning and the high computational cost for solving dense matrix systems. We previously proposed different variants of adaptive methods for selecting proper trial subspaces so that the instability caused by inappropriately shaped parameters were minimized. In contrast, the compactly supported radial basis functions (CSRBF) are more relaxing on the smoothness requirements. By settling with the algebraic order of convergence only, the CSRBF method, provided the support radii are properly chosen, can approximate functions with less smoothness. The reality is that end users must know the functions to be approximated a priori to decide which method to be used; this is not practical if one is solving a time-evolving partial differential equation. The solution could be smooth at the beginning but the formation of shocks may come later. In this talk, we will discuss how meshless methods should be used in the problem of shock capturing. 15:15 C9-11(invited) Title of Talk: An h-adaptive RKDG method and its applications Authors Names: Hongqiang ZHU Affiliation: Nanjing University of Posts and Telecommunications Abstract: In this talk we present an h-adaptive Runge-Kutta

8 discontinuous Galerkin method which is based on mesh refinement and coarsening. First, the framework of this method is presented, together with the implementation details. After that, we show its applications to different problems, including hyperbolic conservation laws, detonation wave simulations and Vlasov-Possion system. Numerical results of classical test problems are given to illustrate the effectiveness and the capability of this method. 15:45 C9-12(invited) Title of Talk: An efficient adaptive rescaling scheme for computing moving interface problems Authors Names: Wenjun YING Affiliation: Shanghai Jiao Tong University Abstract: In this talk, I will present an adaptive rescaling scheme for computing long-time dynamics of expanding interfaces. The main idea is to rescale the temporal and spatial variables so that the interfaces evolve logarithmically fast at early growth stage and and exponentially fast at later times. The new scales guarantee the the conservation of the area/volume enclosed by the interface. Numerical examples will be presented. This is joint work with Meng Zhao, Shuwang Li and John Lowengrub. Session s Title (if available) Wed - 18 Jan :30 ~ 18:00 Room 14 Session Chair(s): Xiaoping XIE (Sichuan University) 16:30 C9-13(invited) Title of Talk: The design of high order positivity-preserving DG scheme for radiation transfer equation Authors Names: Juan CHENG Affiliation: Institute of Applied Physics and Computational Mathematics Abstract: Numerical simulation of radiation transfer equations arises in many applications, including astrophysics, inertial confinement fusion, optical molecular imaging, shielding, and so on. The positivity-preserving property is an important and challenging issue for the numerical solution of this kind of equations. In this talk, we will introduce our recent work on high order positivity-preserving discontinuous Galerkin (DG) schemes solving steady and unsteady radiation transfer equations. The

9 properties of positivity-preserving and high order accuracy are proven rigorously. One- and two-dimensional numerical results are provided to verify the designed characteristics of the positivity-preserving schemes. This is a joint work with Chi-Wang Shu and Daming Yuan. 17:00 C9-14(invited) Title of Talk: A moving mixed finite element method for solving Navier-Stokes equations and its AMG precondition technique Authors Names: Heyu WANG Affiliation: Zhejiang University Abstract: A moving mixed finite element method based on 4P1 P1 element pair is applied to solve the unsteady Navier-Stokes equations. With a data structure of hierarchy geometry tree, the assembling procedure becomes very simple, which is actually same as P1-P1 element, but still keeps the LBB condition holding. And the linear systems of the PDE solver and numerical solution update procedure are both solved with one AMG precondition strategy. Finally, we show the effectiveness and efficiency through numerical simulations. 17:30 C9-15(invited) Title of Talk: An Efficient Multilevel Method for High-Order Moment Models with Applications to Microflow Simulation Authors Names: Zhicheng HU Affiliation: Nanjing University of Aeronautics and Astronautics Abstract: The stationary solution of Boltzmann equation has a special significance in various modern kinetic fields. While the steady state is unavailable in general via an analytical way, numerical simulation for it is also very challenging. Recently a series of high-order moment models has been derived from the Boltzmann equation with BGK-type collision term, by using the globally hyperbolic moment method. These models would give satisfactory results of macroscopic quantities of interest with a high-order convergence to those of the underlying Boltzmann equation as the model's order increases. We concentrate in this talk on efficient steady-state computation of these moment models. By using the lower-order model correction, a nonlinear multilevel moment solver, which has a unified framework for the moment model of arbitrary order, is developed. Numerical simulations for microflow verify that the resulting solver

10 improves the convergence significantly thus is able to accelerate the steady-state computation greatly. It is shown that using the lower-order model correction to accelerate the steady-state computation is also effective for the moment model with a second-order discretization. Thu - 19 Jan :45 ~ 17:45 Room 16 Session Chair(s): Juan CHENG(Institute of Applied Physics and Computational Mathematics) 15:45 C9-16(invited) Title of Talk: A robust Weak Galerkin finite element method for convection-diffusion-reaction equations Authors Names: Xiaoping XIE Affiliation: Sichuan University Abstract: We propose and analyze a weak Galerkin (WG) finite element method for 2- and 3-dimensional convection-diffusion-reaction problems on conforming or nonconforming polygon/polyhedral meshes. The WG method uses piecewise-polynomial approximations of degrees $k(k\ge 0)$ for both the scalar function and its trace on the inter-element boundaries. We show that the method is robust in the sense that the derived a priori error estimates is uniform with respect to the coefficients for sufficient smooth true solutions. Numerical experiments confirm the theoretical results. 16:15 C9-17(invited) Title of Talk: A priori error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin method for symmetrizabled system of conservation laws Authors Names: Qiang ZHANG Affiliation: Nanjing University Abstract: In this talk we present some a priori estimates in L$^2$-norm of the Runge-Kutta discontinuous Galerkin method for solving one-dimensional symmetrizable conservation laws, where the time is discretized with the third order explicit total variation diminishing Runge-Kutta algorithm. The symmetrizable property causes much more difficulties in analysis. Hence, we firstly set up a definition ofgeneralized E-fluxes to include many numerical fluxed that isused widely in practice.then we

11 introduce a numerical viscosity matrix to describe their numerical viscosity in a weak sense.by using energy techniques, we can establish the error estimates under the standard temporal-spatial condition, which is optimal in time.in general, it is quasi-optimal in space; furthermore,it is optimal in space if the upwind numerical flux is used. 16:45 C9-18(invited) Title of Talk: Weak Galerkin finite element method for Biot's consolidation problem Authors Names: Yumei CHEN, Gang CHEN, Xiaoping XIE Affiliation: China West Normal University Abstract In this paper, a fully discrete weak Galerkin finite element method is proposed to solve the Biot's consolidation problem, where weakly defined gradient and divergence operators over discontinuous functions are introduced. $\mathbb{p}_l-\mathbb{p}_l$ $(l\geq1)$ finite element combination is used for the displacement and pressure approximation in the interior of the elements, and $\mathbb{p}_{l-1}-\mathbb{p}_{l-1}$ combination for the approximation on the interfaces of the finite element partition. The stability of the fully discrete solution is derived. Optimal error estimates for the approximation of displacement and pressure in mesh-dependent norms are obtained. Numerical results are provided to confirm the theoretical results. 17:15 C9-19(invited) Title of Talk: Stability and errror estimates of implicit-explicit local discontinuous Galerkin methods Authors Names: Haijin WANG Affiliation: Nanjing University of Posts and Telecommunications Abstract : In this talk we will give the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with carefully chosen implicit-explicit (IMEX) time discretization for solving convection-diffusion equations. In the time discretization the convection term is treated explicitly and the diffusion term implicitly. By establishing an important relationship between the gradient and interface jump of the numerical solution with the independent numerical solution of the gradient in the LDG methods, we show that the IMEX-LDG

12 schemes are unconditionally stable in the sense that the time-step τ is only required to be upper-bounded by a constant which is independent of the mesh-size h, even though the convection term is treated explicitly. Also we obtain optimal error estimates in both space and time for the IMEX-LDG schemes. Numerical experiments are also given to verify the main results.