Abstract : Over the last few years, high-order numerical methods have found their way into the mainstream of computational sciences and are now being successfully applied in almost all areas of natural sciences and engineering. The aim of this minisymposium is to present the most recent developments in the design and theoretical analysis of high-order numerical methods, and to discuss relevant issues related to the practical implementation and applications of these methods. Topics include: theoretical aspects and numerical analysis of high-order numerical methods, non-linear problems, and applications.
[03002] High-order Structure-Preserving Schemes for Special Relativistic Hydrodynamics
Format : Talk at Waseda University
Author(s) :
Huazhong Tang (Nanchang Hangkong University and Peking University)
Abstract : Abstract: This talk mainly reviews two high-order accurate structure-preserving finite difference schemes for the special relativistic hydrodynamics (RHD). The first is the physical-constraints-preserving (PCP) scheme, which preserves the positivity of the rest-mass density and the pressure and the bounds of the fluid velocity and is built on the local Lax-Friedrichs (LxF) splitting, the WENO reconstruction, the PCP flux limiter, and the high-order strong stability preserving time discretization. The key to developing such scheme is to prove the convexity and other properties of the admissible state set and to discover a concave function with respect to the conservative vector. The second is the entropy stable (ES) scheme, whose semi-discrete version satisfies the entropy inequality. The key is to technically construct the affordable entropy conservative (EC) flux of the semi-discrete second-order accurate EC schemes satisfying the semi-discrete entropy equality for the found convex entropy pair. As soon as the EC flux is derived, the dissipation term can be added to give the semi-discrete ES schemes satisfying the semi-discrete entropy inequality. The WENO reconstruction for the scaled entropy variables and the previous time discretization are implemented to obtain the fully-discrete high-order “ES” schemes. The performance of the proposed schemes has been demonstrated by numerical experiments. By the way, we also briefly review other relative works on the structure-preserving schemes for the special RHDs. Those works have been further to the general equation of state and the special relativistic magnetohydrodynamics etc., see our papers listed below for details.
References
1. Wu, K.L. and Tang, H.Z., “High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics”, J. Comput. Phys., Vol. 298, 2015, pp. 539-564.
2. Wu, K.L. and Tang, H.Z., “Physical-constraints-preserving central discontinuous Galerkin methods for special relativistic hydrodynamics with a general equation of state”, Astrophys. J. Suppl. ser., Vol. 228, 2017, 3.
3. Wu, K.L. and Tang, H.Z., “Admissible states and physical constraints preserving numerical schemes for special relativistic magnetohydrodynamics”, Math. Mod. and Meth. Appl. Sci., Vol. 27, 2017, pp. 1871-1928.
4. Wu, K.L. and Tang, H.Z., “On physical-constraints-preserving schemes for special relativistic magnetohydrodynamics with a general equation of state”, Z. Angew. Math. Phys., Vol. 69, 2018, 84.
5. Ling, D., Duan, J.M. and Tang, H.Z., “Physical-constraints-preserving Lagrangian finite volume schemes for one- and two-dimensional special relativistic hydrodynamics”, J. Comput. Phys., Vol. 396, 2019, pp. 507-543.
6. Ling, D. and Tang, H.Z., “Genuinely multidimensional physical-constraints-preserving finite volume schemes for the special relativistic hydrodynamics”, submitted to Commun. Comput. Phys., March 4, 2023. arXiv: 2303.02686.
7. Duan, J.M. and Tang, H.Z., “High-order accurate entropy stable finite difference schemes for one- and two-dimensional special relativistic hydrodynamics”, Adv. Appl. Math. Mech., Vol. 12, 2020, pp. 1-29.
8. Duan, J.M. and Tang, H.Z., “High-order accurate entropy stable nodal discontinuous Galerkin schemes for the ideal special relativistic magnetohydrodynamics”, J. Comput. Phys., Vol. 421, 2020, 109731.
9. Duan, J.M. and Tang, H.Z., “Entropy stable adaptive moving mesh schemes for 2D and 3D special relativistic hydrodynamics”, J. Comput. Phys., Vol. 426, 2021, 109949.
[03400] High order entropy stable and positivity-preserving discontinuous Galerkin method for the nonlocal electron heat transport model
Format : Talk at Waseda University
Author(s) :
Juan Cheng (Institute of Applied Physics and Computational Mathematics)
Abstract : The nonlocal electron heat transport model in laser heated plasmas plays a crucial role in inertial confinement fusion (ICF), and it is important to solve it numerically in an accurate and robust way. In this talk, we develop a class of high-order entropy stable discontinuous Galerkin methods for the nonlocal electron heat transport model. We further design our DG scheme to have the positivity-preserving property, which is shown, by a computer-aided proof, to have no extra time step constraint than that required by L2 stability. Numerical examples are given to verify the high-order accuracy and positivity-preserving properties of our scheme. By comparing the local and nonlocal electron heat transport models, we also observe more physical phenomena such as the flux reduction and the preheat effect from the nonlocal model.
[02764] A new cut-cell interface treating method for compressible multi-medium flow
Format : Talk at Waseda University
Author(s) :
Chunwu Wang (Nanjing University of Aeronautics and Astronautics)
Abstract : In this work, a conservative sharp interface treating method is proposed for solving compressible multi-medium flow based on the cut-cell method. To overcome the small cell problem, which causes standard explicit scheme unstable, the most common approach is generating interface cells by merging small cut cells with their neighbors. We present a new and simple way to construct interface cells. Rather than considering various complex cell merging cases, we select some Cartesian cell nodes near the interface, and then connect these nodes to form the interface cells. Meanwhile, at the edges of the cell coinciding with the interface, the numerical fluxes are obtained by solving the local Riemann problem, thus the conservation of the flow variables is effectively maintained. Several numerical experiments indicate that our proposed method can capture the shock wave and material interface accurately and sharply, as well as being stable even for problems with significant density and pressure gradients.
[02690] A hybrid WENO scheme for steady Euler equations in curved geometries on Cartesian grids
Format : Talk at Waseda University
Author(s) :
Yifei Wan (University of Science and Technology of China)
Yinhua Xia (University of Science and Technology of China)
Abstract : For steady Euler equations in complex boundary domains, high-order shock-capturing schemes usually suffer not only from the difficulty of steady-state convergence but also from the problem of dealing with physical boundaries on Cartesian grids to achieve uniform high-order accuracy. In this work, we utilize a fifth-order finite difference hybrid WENO scheme to simulate steady Euler equations, and the same fifth-order WENO extrapolation methods are developed to handle the curved boundary. The values of the ghost points outside the physical boundary can be obtained by applying WENO extrapolation near the boundary, involving normal derivatives acquired by the simplified inverse Lax-Wendroff procedure. Both equivalent expressions involving curvature and numerical differentiation are utilized to transform the tangential derivatives along the curved solid wall boundary. This hybrid WENO scheme is robust for steady-state convergence and maintains high-order accuracy in the smooth region even with the solid wall boundary condition. Besides, the essentially non-oscillation property is achieved. The numerical spectral analysis also shows that this hybrid WENO scheme possesses low dispersion and dissipation errors. Numerical examples are presented to validate the high-order accuracy and robust performance of the hybrid scheme for steady Euler equations in curved domains with Cartesian grids.
[03089] Error estimates to smooth solutions of high order Runge–Kutta discontinuous Galerkin method for scalar nonlinear conservation laws with and without sonic points
Format : Talk at Waseda University
Author(s) :
Qiang Zhang (Nanjing University)
Abstract : In this talk we shall take the fourth order in time Runge--Kutta discontinuous Galerkin method, as an example of high order schemes, to establish a sharp a priori L$^2$-norm error estimates for sufficiently smooth solutions of one-dimensional scalar nonlinear conservation laws. The optimal order of accuracy in time is obtained under the standard Courant-Friedrichs-Lewy condition, and the quasi-optimal and/or optimal order of accuracy in space are achieved for many widely-used numerical fluxes, no matter whether the exact solution contains sonic points or not. Note that the convergence order in space strongly depends on the relative upwind effect of the used numerical flux, which is related to the local flowing speed and the strength of the numerical viscosity provided by the used numerical flux. Two main tools are used in this talk. One is the matrix transferring process, based on the temporal differences of stage errors. It gives a useful energy equation and help us to get the theoretical result under the acceptable temporal-spatial condition. The other is the generalized Gauss-Radau projection of the reference functions, which depends on the relative upwind effect and helps us to achieve the optimal order in space in many cases. Finally some numerical experiments are given to support the theoretical results.
[02994] Energy stable discontinuous Galerkin methods for compressible Navier–Stoles–Allen–Cahn System
Format : Talk at Waseda University
Author(s) :
Qiaolin He (Sichuan University)
Xiaoding Shi (Beijing University of Chemical Technology)
Abstract : In this work, we present a fully discrete local discontinuous Galerkin (LDG) finite element method combined with scalar auxiliary variable (SAV) approach for the compressible Navier--Stokes--Allen--Cahn (NSAC) system. We start with a linear and first order scheme for time discretization and the minimal dissipation LDG for spatial discretization, which is based on the SAV approach and is proved to be unconditionally energy stable for one dimensional case. The velocity, the density and the mass concentration of fluid mixture can be solved separately. In addition, a semi-implicit spectral deferred correction (SDC) method combined with the first order scheme is employed to improve the temporal accuracy. Due to the local properties of the LDG methods, the resulting algebraic equations at the implicit level are easy to implement. In particular, we use efficient and practical multigrid solvers to solve the resulting algebraic equations. Although there is no proof of stability for the semi-implicit SDC with LDG spatial discretization, numerical experiments of the accuracy and long time simulations are presented to illustrate the high order accuracy in both time and space, the discretized energy stability, the capability and efficiency of the proposed method. Numerical results show that the initial state determines the long time behavior of the diffusive interface for the two--phase flow, which are consistent with theoretical asymptotic stability results.
[02714] Superconvergence of LDG method for nonlinear convection-diffusion equations
Format : Talk at Waseda University
Author(s) :
Xiong Meng (Harbin Institute of Technology)
Abstract : In this talk, we present superconvergence properties of the local discontinuous Galerkin (LDG) methods for solving nonlinear convection-diffusion equations in one space dimension. The main technicality is an elaborate estimate to terms involving projection errors. By introducing a new projection and constructing some correction functions, we prove the $(2k+1)$th order superconvergence for the cell averages and the numerical flux in the discrete $L^2$ norm with polynomials of degree $k\ge 1$, no matter whether the flow direction $f'(u)$ changes or not. Superconvergence of order $k +2$ $(k +1)$ is obtained for the LDG error (its derivative) at interior right (left) Radau points, and the convergence ord er for the error derivative at Radau points can be improved to $k+2$ when the direction of the flow doesn't change. Finally, a supercloseness result of order $k+2$ towards a special Gauss-Radau projection of the exact solution is shown. The superconvergence analysis can be extended to the generalized numerical fluxes and the mixed boundary conditions. All theoretical findings are confirmed by numerical experiments.
[03001] An essentially oscillation-free discontinuous Galerkin method for hyperbolic conservation laws
Format : Talk at Waseda University
Author(s) :
Yong Liu (ICMSEC, AMSS, CAS)
Jianfang Lu (South China University of Technology)
Chi-Wang Shu (Brown University)
Abstract : In this talk, we propose a novel discontinuous Galerkin (DG) method to control the spurious oscillations when solving the scalar hyperbolic conservation laws. The spurious oscillations may be harmful to the numerical simulation, as it not only generates some artificial structures not belonging to the problems but also causes many overshoots and undershoots that make the numerical scheme less robust. To overcome this difficulty, we introduce a numerical damping term to control spurious oscillations based on the classic DG formulation. Compared to the classic DG method, the proposed DG method still maintains many good properties, such as the extremely local data structure, conservation, L2-boundedness, optimal error estimates, and superconvergence. We also extend our methods to systems of hyperbolic conservation laws. Entropy inequalities are crucial to the well-posedness of hyperbolic conservation laws, which help to select the physically meaningful one among the infinite many weak solutions. By combining with quadrature-based entropy-stable DG methods, we also developed the entropy-stable OFDG method. For time discretizations, the modified exponential Runge--Kutta method can avoid additional restrictions of time step size due to the numerical damping. Extensive numerical experiments are shown to demonstrate our algorithm is robust and effective.
[03431] Structure-preserving methods for Boltzmann continuous slowing down equations
Format : Talk at Waseda University
Author(s) :
Vincent Bosboom (University of Twente)
Herbert Egger (Johannes Kepler Universität)
Matthias Schlottbom (University of Twente)
Abstract : We discuss the linearized Boltzmann equation (LBE) describing the transport of particles under the influence of the Lorentz force and (inelastic) scattering. For this purpose, we consider the equation in the continuous slowing down approximation (CSDA).
We present a high-order discretization of the equation based on a mixed finite element $P_N$ approximation that preserves the energy-conserving/dissipating nature of the different physical processes.
Additionally, we provide stability estimates of our discretization and present some numerical examples.
[03359] Positivity-preserving high-order DG method for weakly compressible two-phase flows
Format : Talk at Waseda University
Author(s) :
Fan Zhang (University of Science and Technology Beijing)
Abstract : This work focuses on the development of a high-order DG method for solving a three-equation model of weakly compressible two-phase flows. A novel WENO limiter and a positivity-preserving limiter are designed and applied in the numerical simulations. Moreover, we prove that the proposed method satisfies the uniform-pressure-velocity criterion which is a necessary condition for maintaining an oscillation-free phase interface.
[03094] Arbitrary high-order fully-decoupled numerical schemes for phase-field models of two-phase incompressible flows
Format : Talk at Waseda University
Author(s) :
Ruihan Guo (Zhengzhou University)
Abstract : Due to the coupling between the hydrodynamic equation and the phase-field equation in two-phase incompressible flows, it is desirable to develop efficient and high-order accurate numerical schemes that can decouple these two equations. One popular and efficient strategy is adding an explicit stabilizing term to the convective velocity in the phase-field equation to decouple them. The resulting numerical methods are only first-order accurate in time, and it seems extremely difficult to generalize the idea of stabilization to the second-order version or higher. In this talk, we employ the spectral deferred correction method to improve the temporal accuracy, based on the first-order decoupled and energy stable scheme constructed by the stabilization idea. The novelty lies in how decoupling and linear implicit properties are maintained to improve efficiency. Within the framework of the spatially discretized local discontinuous Galerkin method, the resulting numerical schemes are fully decoupled, efficient, and high-order accurate in both time and space. Numerical experiments are performed to validate the high order accuracy and efficiency of the methods for solving phase-field models of two-phase incompressible flows.
[03096] Accuracy-enhancement of discontinuous Galerkin methods for PDEs containing high-order spatial derivatives
Format : Talk at Waseda University
Author(s) :
Qi Tao (Beijing University of Technology)
Liangyue Ji (Milton Keynes College)
Jennifer K. Ryan (KTH Royal Institute of Technology in Stockholm)
Yan Xu (University of Science and Technology of China)
Abstract : In this talk, we shall first introduce the accuracy-enhancement of discontinuous Galerkin (DG) methods for solving PDEs with high-order spatial derivatives. It is well known that there are highly oscillatory errors for finite element approximations to PDEs that contain hidden superconvergence points. To exploit this information, a Smoothness-Increasing Accuracy Conserving (SIAC) filter is used to create a superconvergence filtered solution. This is accomplished by convolving the DG approximation against a B-spline kernel. We then present theoretical error estimates in the negative-order norm for the local DG (LDG) and ultra-weak local DG (UWLDG) approximations to PDEs containing high order spatial derivatives. Numerical results will be shown to confirm the theoretical results.
[03105] A discontinuous Galerkin method for the Camassa-Holm-Kadomtsev-Petviashvili type equations
Format : Talk at Waseda University
Author(s) :
Qian Zhang (Harbin Institute of Technology, Shenzhen )
Yan Xu (University of Science and Technology of China)
Yue Liu (The University of Texas at Arlington)
Abstract : This paper develops a high-order discontinuous Galerkin (DG) method for the Camassa-Holm-Kadomtsev-Petviashvili (CH-KP) type equations on Cartesian meshes. The significant part of the simulation for the CH-KP type equations lies in the treatment for the integration operator $\partial^{-1}$. Our proposed DG method deals with it element by element, which is efficient and applicable for most solutions. Using the instinctive energy of the original PDE as a guiding principle, the DG scheme can be proved as an energy stable numerical scheme. In addition, the semi-discrete error estimates results for the nonlinear case are derived without any priori assumption. Several numerical experiments demonstrate the capability of our schemes for various types of solutions.
