Publications

Preprints

1. Junming Duan
   A fourth-order active flux method for parabolic problems with application to porous medium equation
   arXiv:2510.11527, Oct 2025
   Abs arXiv

   The active flux (AF) method is a compact high-order finite volume method originally proposed for solving hyperbolic conservation laws, in which cell averages and point values at cell interfaces are evolved simultaneously. This paper develops a fourth-order AF method for one- and two-dimensional parabolic problems, employing the explicit strong-stability-preserving Runge-Kutta (SSP-RK) method for time integration. The proposed method is built on a degenerate first-order system with auxiliary variables representing the derivatives of the primal variable, similar to local discontinuous Galerkin (LDG) methods, which avoids introducing pseudo-time or performing iterations within a physical time step in the existing hyperbolic formulations. The evolution of cell averages follows the standard finite volume method, ensuring conservation, while the point values of both the primal and auxiliary variables are updated using fourth-order central finite difference operators. A discrete Fourier analysis confirms the fourth-order accuracy in 1D. With the third-order SSP-RK method, the maximum CFL number for stability is 0.27 in 1D, as obtained by von Neumann analysis, larger than that of LDG methods. The proposed method is further applied to the porous medium equation, and positivity-preserving limitings are incorporated to guarantee the non-negativity of the numerical solutions. Several numerical experiments validate the theoretical results and efficacy of the method.

2. Junming Duan, Wasilij Barsukow, and Christian Klingenberg
   An asymptotic-preserving active flux scheme for the hyperbolic heat equation in the diffusive scaling
   arXiv:2508.05166, Aug 2025
   Abs arXiv

   The Active Flux (AF) method is a compact, high-order finite volume scheme that enhances flexibility by introducing point values at cell interfaces as additional degrees of freedom alongside cell averages. The method of lines is employed here for temporal discretization. A common approach for updating point values relies on the Jacobian Splitting (JS) method, which incorporates upwinding. A key advantage of the AF method over standard finite volume schemes is its structure-preserving property, motivating the investigation of its asymptotic-preserving (AP) behavior in the diffusive scaling. We show that the JS-based AF method without any modification is AP for solving the hyperbolic heat equation, in the sense that the limit scheme is a discretization of the limit heat equation. We use formal asymptotic analysis, discrete Fourier analysis, and numerical experiments to illustrate our findings.

3. Junming Duan^*, Praveen Chandrashekar, and Christian Klingenberg
   Active flux for ideal magnetohydrodynamics: A positivity-preserving scheme with the godunov-powell source term
   arXiv:2506.04857, Jun 2025
   Abs arXiv

   The Active Flux (AF) is a compact, high-order finite volume scheme that allows more flexibility by introducing additional point value degrees of freedom at cell interfaces. This paper proposes a positivity-preserving (PP) AF scheme for solving the ideal magnetohydrodynamics, where the Godunov-Powell source term is employed to deal with the divergence-free constraint. For the evolution of the cell average, apart from the standard conservative finite volume method for the flux derivative, the nonconservative source term is built on the quadratic reconstruction in each cell, which maintains the compact stencil in the AF scheme. For the point value update, the local Lax-Friedrichs (LLF) flux vector splitting is adopted for the flux derivative, originally proposed in [Duan, Barsukow, and Klingenberg, SIAM Journal on Scientific Computing, 47(2), A811–A837, 2025], and a central difference is used to discretize the divergence in the source term. A parametrized flux limiter and a scaling limiter are presented to preserve the density and pressure positivity by blending the AF scheme with the first-order PP LLF scheme with the source term. To suppress oscillations, a new shock sensor considering the divergence error is proposed, which is used to compute the blending coefficients for the cell average. Several numerical tests are conducted to verify the third-order accuracy, PP property, and shock-capturing ability of the scheme. The key role of the Godunov-Powell source term and its suitable discretization in controlling divergence error is also validated.

4. Junming Duan, Bartul Kovacic, and Jan S. Hesthaven
   Multi-GPU accelerated high-order schemes for hyperbolic conservation laws on adaptive moving meshes
   Jun 2024

Journal Articles

1. Jie Wang, Yi Zhou, Junming Duan, Zhiwei Ma, and Wei Zhang
   CLT-fx: Non-axisymmetric flexible mesh finite difference scheme for stellarator MHD simulations
   Computer Physics Communications, 316: 109776, Nov 2025
   Abs DOI

   The adaptive moving mesh CLT code is extended to be applicable for the stellarator magneto-hydrodynamic (MHD) simulations. Compared with the tokamak version, the mesh can not only be non-uniform, but can also be non-axisymmetric and strongly shaped with a concave boundary. The extra toroidal transformation from the physical domain to the computational domain has to be taken into consideration. In the computational domain, the fourth-order finite difference scheme is constructed on a Cartesian computational mesh. To verify the code, we simulate the internal kink mode in a W7-X configuration, and benchmark the linear and non-linear results with the M3D-C code. The method can also be used for different 3D equilibria, and a calculation of the resistive ballooning mode in a NCSX equilibrium is given.

2. Junming Duan^*, Wasilij Barsukow, and Christian Klingenberg
   Active flux methods for hyperbolic conservation laws – flux vector splitting and bound-preservation
   SIAM Journal on Scientific Computing, 47(2): A811–A837, Apr 2025
   Abs DOI arXiv

   The active flux (AF) method is a compact high-order finite volume method that simultaneously evolves cell averages and point values at cell interfaces. Within the method of lines framework, the existing Jacobian splitting-based point value update incorporates the upwind idea but suffers from a stagnation issue for nonlinear problems due to inaccurate estimation of the upwind direction, and also from a mesh alignment issue partially resulting from decoupled point value updates. This paper proposes to use flux vector splitting for the point value update, offering a natural and uniform remedy to those two issues. To improve robustness, this paper also develops bound-preserving (BP) AF methods for hyperbolic conservation laws. Two cases are considered: preservation of the maximum principle for the scalar case, and preservation of positive density and pressure for the compressible Euler equations. The update of the cell average is rewritten as a convex combination of the original high-order fluxes and robust low-order (local Lax-Friedrichs or Rusanov) fluxes, and the desired bounds are enforced by choosing the right amount of low-order fluxes. A similar blending strategy is used for the point value update. In addition, a shock sensor-based limiting is proposed to enhance the convex limiting for the cell average, which can suppress oscillations well. Several challenging tests are conducted to verify the robustness and effectiveness of the BP AF methods, including flow past a forward-facing step and high Mach number jets.

3. Zhihao Zhang, Huazhong Tang, and Junming Duan^*
   High-order accurate well-balanced energy stable finite difference schemes for multi-layer shallow water equations on fixed and adaptive moving meshes
   Journal of Computational Physics, 517: 113301, Nov 2024
   Abs DOI arXiv

   This paper develops high-order well-balanced (WB) energy stable (ES) finite difference schemes for multi-layer (the number of layers M\geqslant2) shallow water equations (SWEs) on both fixed and adaptive moving meshes, extending our previous works [20,51]. To obtain an energy inequality, the convexity of an energy function for an arbitrary M is proved by finding recurrence relations of the leading principal minors or the quadratic forms of the Hessian matrix of the energy function with respect to the conservative variables, which is more involved than the single-layer case due to the coupling between the layers in the energy function. An important ingredient in developing high-order semi-discrete ES schemes is the construction of a two-point energy conservative (EC) numerical flux. In pursuit of the WB property, a sufficient condition for such EC fluxes is given with compatible discretizations of the source terms similar to the single-layer case. It can be decoupled into M identities individually for each layer, making it convenient to construct a two-point EC flux for the multi-layer system. To suppress possible oscillations near discontinuities, WENO-based dissipation terms are added to the high-order WB EC fluxes, which gives semi-discrete high-order WB ES schemes. Fully-discrete schemes are obtained by employing high-order explicit SSP-RK methods and proved to preserve the lake at rest. The schemes are further extended to moving meshes based on a modified energy function for a reformulated system, relying on the techniques proposed in [51]. Numerical experiments are conducted for some two- and three-layer cases to validate the high-order accuracy, WB and ES properties, and high efficiency of the schemes, with a suitable amount of dissipation chosen by estimating the maximal wave speed due to the lack of an analytical expression for the eigenstructure of the multi-layer system.

4. Junming Duan, Qian Wang, and Jan S. Hesthaven
   Machine-learning-enhanced real-time aerodynamic forces prediction based on sparse pressure sensor inputs
   AIAA Journal, 62(7): 2601–2621, Jul 2024
   Abs DOI arXiv

   Accurate real-time prediction of aerodynamic forces is crucial for the navigation of unmanned aerial vehicles (UAVs). This paper presents a data-driven aerodynamic force prediction model based on a small number of pressure sensors located on the surface of a UAV. The model is built on a linear term that can make a reasonably accurate prediction and a nonlinear correction for accuracy improvement. The linear term is based on a reduced basis reconstruction of surface pressure, with the basis extracted from simulation data and the basis coefficients determined by solving linear pressure reconstruction equations at a set of optimal sensor locations, which are obtained by using the discrete empirical interpolation method (DEIM). The nonlinear term is an artificial neural network that is trained to bridge the gap between the DEIM prediction and the ground truth, especially when only low-fidelity simulation data are available. The model is tested on numerical and experimental dynamic stall data of a two-dimensional NACA0015 airfoil and numerical simulation data of the dynamic stall of a three-dimensional drone. Numerical results demonstrate that the machine-learning-enhanced model is accurate, efficient, and robust, even for the NACA0015 case, in which the simulations do not agree well with the wind tunnel experiments.

5. Junming Duan^* and Jan S. Hesthaven
   Non-intrusive data-driven reduced-order modeling for time-dependent parametrized problems
   Journal of Computational Physics, 497: 112621, Jan 2024
   Abs DOI arXiv

   Reduced-order models are indispensable for multi-query or real-time problems. However, there are still many challenges to constructing efficient ROMs for time-dependent parametrized problems. Using a linear reduced space is inefficient for time-dependent nonlinear problems, especially for transport-dominated problems. The non-linearity usually needs to be addressed by hyper-reduction techniques, such as DEIM, but it is intrusive and relies on the assumption of affine dependence of parameters. This paper proposes and studies a non-intrusive reduced-order modeling approach for time-dependent parametrized problems. It is purely data-driven and naturally split into offline and online stages. During the offline stage, a convolutional autoencoder, consisting of an encoder and a decoder, is trained to perform dimensionality reduction. The encoder compresses the full-order solution snapshots to a nonlinear manifold or a low-dimensional reduced/latent space. The decoder allows the recovery of the full-order solution from the latent space. To deal with the time-dependent problems, a high-order dynamic mode decomposition (HODMD) is utilized to model the trajectories in the latent space for each parameter. During the online stage, the HODMD models are first utilized to obtain the latent variables at a new time, then interpolation techniques are adopted to recover the latent variables at a new parameter value, and the full-order solution is recovered by the decoder. Some numerical tests are conducted to show that the approach can be used to predict the unseen full-order solution at new times and parameter values fast and accurately, including transport-dominated problems.

6. Jie Wang, Junming Duan, Zhiwei Ma, and Wei Zhang
   An adaptive moving mesh finite difference scheme for tokamak magneto-hydrodynamic simulations
   Computer Physics Communications, 294: 108951, Jan 2024
   Abs DOI

   An adaptive moving mesh finite difference scheme is developed for tokamak magneto-hydrodynamic (MHD) simulations, based on the CLT code (S. Wang and Z.W. Ma, Phys. Plasmas, 2015). Our numerical scheme is built on the MHD equations in curvilinear coordinates, based on a coordinate transformation from the physical domain to a computational domain. The scheme is constructed on a uniform Cartesian computational mesh that is obtained from a non-uniform adaptive moving mesh in the physical domain through the coordinate transformation. Mesh points in the physical domain in general move and concentrate in the vicinity of solutions with rapid variations by solving an adaptive mesh equation, whilst total number of mesh points remains unchanged. The local resolution can be significantly increased and computational resource is largely reduced. Comparison between results obtained with the original uniform mesh and the new adaptive moving mesh is carried out by simulation of the linear and nonlinear 2/1 tearing mode, linear and nonlinear 1/1 resistive internal kink mode. It is found that the adaptive moving mesh scheme possesses better numerical stability and convergence.

7. Zhihao Zhang, Junming Duan^*, and Huazhong Tang
   High-order accurate well-balanced energy stable adaptive moving mesh finite difference schemes for the shallow water equations with non-flat bottom topography
   Journal of Computational Physics, 492: 112451, Nov 2023
   Abs DOI arXiv

   This paper proposes high-order accurate well-balanced (WB) energy stable (ES) adaptive moving mesh finite difference schemes for the shallow water equations (SWEs) with non-flat bottom topography. To enable the construction of the ES schemes on moving meshes, a reformulation of the SWEs is introduced, with the bottom topography as an additional conservative variable that evolves in time. The corresponding energy inequality is derived based on a modified energy function, then the reformulated SWEs and energy inequality are transformed into curvilinear coordinates. A two-point energy conservative (EC) flux is constructed, and high-order EC schemes based on such a flux are proved to be WB that they preserve the lake at rest. Then high-order ES schemes are derived by adding suitable dissipation terms to the EC schemes, which are newly designed to maintain the WB and ES properties simultaneously. The adaptive moving mesh strategy is performed by iteratively solving the Euler-Lagrangian equations of a mesh adaptation functional. The fully-discrete schemes are obtained by using the explicit strong-stability preserving third-order Runge-Kutta method. Several numerical tests are conducted to validate the accuracy, WB and ES properties, shock-capturing ability, and high efficiency of the schemes.

8. Shangting Li, Junming Duan, and Huazhong Tang
   High-order accurate entropy stable adaptive moving mesh finite difference schemes for (multi-component) compressible Euler equations with the stiffened equation of state
   Computer Methods in Applied Mechanics and Engineering, 399: 115311, Sep 2022
   Abs DOI arXiv

   This paper extends the high-order entropy stable (ES) adaptive moving mesh finite difference schemes developed in Duan and Tang (2022) to the two- and three-dimensional (multi-component) compressible Euler equations with the stiffened equation of state (EOS). The two-point entropy conservative (EC) flux is first constructed in the curvilinear coordinates, which is nontrivial in the case of the stiffened EOS. With the aid of the high-order discretization of the geometric conservation laws and the linear combinations of the two-point EC fluxes, the high-order semi-discrete EC schemes are derived. The high-order semi-discrete ES schemes are constructed by adding suitable high-order dissipation term to the EC schemes such that the semi-discrete entropy inequality is satisfied and unphysical oscillations are suppressed. The high-order dissipation term is built on the multi-resolution weighted essentially non-oscillatory (WENO) reconstruction and the newly derived scaled eigenvector matrices. The explicit strong-stability-preserving Runge–Kutta methods are used for the time discretization and the mesh points are adaptively redistributed by iteratively solving the mesh redistribution equations with an appropriate monitor function, which is adapted to the multi-component flow and encodes more physical characteristics of the solutions. Several 2D and 3D numerical tests are conducted on the parallel computer system with the MPI programming to validate the accuracy and the ability to effectively capture the localized structures of the proposed schemes.

9. Junming Duan and Huazhong Tang
   An analytical solution of the isentropic vortex problem in the special relativistic magnetohydrodynamics
   Journal of Computational Physics, 456: 110903, May 2022
   Abs DOI arXiv

   The isentropic vortex problem is frequently solved to test the accuracy of numerical methods and verify corresponding code. Unfortunately, its existing solution was derived in the relativistic magnetohydrodynamics by numerically solving an ordinary differential equation. This note provides an analytical solution of the 2D isentropic vortex problem with explicit algebraic expressions in the special relativistic hydrodynamics and magnetohydrodynamics and extends it to the 3D case.

10. Junming Duan and Huazhong Tang
    High-order accurate entropy stable adaptive moving mesh finite difference schemes for special relativistic (magneto)hydrodynamics
    Journal of Computational Physics, 456: 111038, May 2022
    Abs DOI arXiv

    This paper develops high-order accurate entropy stable (ES) adaptive moving mesh finite difference schemes for the two- and three-dimensional special relativistic hydrodynamic (RHD) and magnetohydrodynamic (RMHD) equations, which is the high-order accurate extension of Duan and Tang (2021) [23]. The key point is the derivation of the higher-order accurate entropy conservative (EC) and ES finite difference schemes in the curvilinear coordinates by carefully dealing with the discretization of the temporal and spatial metrics and the Jacobian of the coordinate transformation and constructing the high-order EC and ES fluxes with the discrete metrics. The spatial derivatives in the source terms of the symmetrizable RMHD equations and the geometric conservation laws are discretized by using the linear combinations of the corresponding second-order case to obtain high-order accuracy. Based on the proposed high-order accurate EC schemes and the high-order accurate dissipation terms built on the WENO reconstruction, the high-order accurate ES schemes are obtained for the RHD and RMHD equations in the curvilinear coordinates. The mesh iteration redistribution or adaptive moving mesh strategy is built on the minimization of the mesh adaption functional. Several numerical tests are conducted to validate the shock-capturing ability and high efficiency of our high-order accurate ES adaptive moving mesh methods on the parallel computer system with the MPI communication. The numerical results show that the high-order accurate ES adaptive moving mesh schemes outperform both their counterparts on the uniform mesh and the second-order ES adaptive moving mesh schemes.

11. Junming Duan and Huazhong Tang
    High-order accurate entropy stable finite difference schemes for the shallow water magnetohydrodynamics
    Journal of Computational Physics, 431: 110136, Apr 2021
    Abs DOI arXiv

    This paper develops the high-order accurate entropy stable (ES) finite difference schemes for the shallow water magnetohydrodynamic (SWMHD) equations. They are built on the numerical approximation of the modified SWMHD equations with the Janhunen source term. First, the second-order accurate well-balanced semi-discrete entropy conservative (EC) schemes are constructed, satisfying the entropy identity for the given convex entropy function and preserving the steady states of the lake at rest (with zero magnetic field). The key is to match both discretizations for the fluxes and the non-flat river bed bottom and Janhunen source terms, and to find the affordable EC fluxes of the second-order EC schemes. Next, by using the second-order EC schemes as building block, high-order accurate well-balanced semi-discrete EC schemes are proposed. Then, the high-order accurate well-balanced semi-discrete ES schemes are derived by adding a suitable dissipation term to the EC scheme with the WENO reconstruction of the scaled entropy variables in order to suppress the numerical oscillations of the EC schemes. After that, the semi-discrete schemes are integrated in time by using the high-order strong stability preserving explicit Runge-Kutta schemes to obtain the fully-discrete high-order well-balanced schemes. The ES property of the Lax-Friedrichs flux is also proved and then the positivity-preserving ES schemes are studied by using the positivity-preserving flux limiters. Finally, extensive numerical tests are conducted to validate the accuracy, the well-balanced, ES and positivity-preserving properties, and the ability to capture discontinuities of our schemes.

12. Junming Duan and Huazhong Tang
    Entropy stable adaptive moving mesh schemes for 2D and 3D special relativistic hydrodynamics
    Journal of Computational Physics, 426: 109949, Feb 2021
    Abs DOI arXiv

    This paper develops entropy stable (ES) adaptive moving mesh schemes for the 2D and 3D special relativistic hydrodynamic (RHD) equations. They are built on the ES finite volume approximation of the RHD equations in curvilinear coordinates, the discrete geometric conservation laws, and the mesh adaptation implemented by iteratively solving the Euler-Lagrange equations of the mesh adaption functional in the computational domain with suitably chosen monitor functions. First, a sufficient condition is proved for the two-point entropy conservative (EC) flux, by mimicking the derivation of the continuous entropy identity in curvilinear coordinates and using the discrete geometric conservation laws given by the conservative metrics method. Based on such sufficient condition, the EC fluxes for the RHD equations in curvilinear coordinates are derived and the second-order accurate semi-discrete EC schemes are developed to satisfy the entropy identity for the given convex entropy pair. Next, the semi-discrete ES schemes satisfying the entropy inequality are proposed by adding a suitable dissipation term to the EC scheme and utilizing linear reconstruction with the minmod limiter in the scaled entropy variables in order to suppress the numerical oscillations of the above EC scheme. Then, the semi-discrete ES schemes are integrated in time by using the second-order strong stability preserving explicit Runge-Kutta schemes. Finally, several numerical results show that our 2D and 3D ES adaptive moving mesh schemes effectively capture the localized structures, such as sharp transitions or discontinuities, and are more efficient than their counterparts on uniform mesh.

13. Junming Duan and Huazhong Tang
    High-order accurate entropy stable nodal discontinuous Galerkin schemes for the ideal special relativistic magnetohydrodynamics
    Journal of Computational Physics, 421: 109731, Nov 2020
    Abs DOI arXiv

    This paper studies high-order accurate entropy stable nodal discontinuous Galerkin (DG) schemes for the ideal special relativistic magnetohydrodynamics (RMHD). It is built on the modified RMHD equations with a particular source term, which is analogous to the Powell‘s eight-wave formulation and can be symmetrized so that an "entropy pair" is obtained. We design an affordable "fully consistent" two-point entropy conservative flux, which is not only consistent with the physical flux, but also maintains the zero parallel magnetic component, and then construct high-order accurate semi-discrete entropy stable DG schemes based on the quadrature rules and the entropy conservative and stable fluxes. The Lax-Friedrichs flux is proven to be entropy stable when its viscosity coefficient is chosen as the speed of light, and the other two entropy stable fluxes are also derived by adding the dissipation terms to our proposed entropy conservative flux and compared to the Lax-Friedrichs flux by using numerical experiments in terms of the CPU time and accuracy. All three entropy stable numerical fluxes maintain the zero parallel magnetic component in one dimension. The resulting entropy stable DG schemes satisfy the semi-discrete "entropy inequality" for the given "entropy pair" and are integrated in time by using the high-order explicit strong stability preserving Runge-Kutta schemes to get further the fully-discrete nodal DG schemes. Extensive numerical tests are conducted to validate the accuracy and the ability to capture discontinuities of our schemes with the help of the TVB limiter. Moreover, our entropy conservative flux is compared to an existing flux through some numerical tests. The results show that the zero parallel magnetic component in the numerical flux can help to decrease the error in the parallel magnetic component in one-dimensional tests, but two entropy conservative fluxes give similar results since the error in the magnetic field divergence seems dominated in the two-dimensional tests.

14. Junming Duan and Huazhong Tang
    High-order accurate entropy stable finite difference schemes for one- and two-dimensional special relativistic hydrodynamics
    Advances in Applied Mathematics and Mechanics, 12(1): 1–29, Jan 2020
    Abs DOI arXiv

    This paper develops the high-order accurate entropy stable finite difference schemes for one- and two-dimensional special relativistic hydrodynamic equations. The schemes are built on the entropy conservative flux and the weighted essentially non-oscillatory (WENO) technique as well as explicit Runge-Kutta time discretization. The key is to technically construct the affordable entropy conservative flux of the semi-discrete second-order accurate entropy conservative schemes satisfying the semi-discrete entropy equality for the found convex entropy pair. As soon as the entropy conservative flux is derived, the dissipation term can be added to give the semi-discrete entropy stable schemes satisfying the semi-discrete entropy inequality with the given convex entropy function. The WENO reconstruction for the scaled entropy variables and the high-order explicit Runge-Kutta time discretization are implemented to obtain the fully-discrete high-order entropy stable schemes. Several numerical tests are conducted to validate the accuracy and the ability to capture discontinuities of our entropy stable schemes.

15. Junming Duan and Huazhong Tang
    An efficient ADER discontinuous Galerkin scheme for directly solving Hamilton-Jacobi equation
    Journal of Computational Mathematics, 38(1): 58–83, Jan 2020
    Abs DOI arXiv

    This paper proposes an efficient ADER (Arbitrary DERivatives in space and time) discontinuous Galerkin (DG) scheme to directly solve the Hamilton-Jacobi equation. Unlike multi-stage Runge-Kutta methods used in the Runge-Kutta DG (RKDG) schemes, the ADER scheme is one-stage in time discretization, which is desirable in many applications. The ADER scheme used here relies on a local continuous spacetime Galerkin predictor instead of the usual Cauchy-Kovalewski procedure to achieve high order accuracy both in space and time. In such predictor step, a local Cauchy problem in each cell is solved based on a weak formulation of the original equations in spacetime. The resulting spacetime representation of the numerical solution provides the temporal accuracy that matches the spatial accuracy of the underlying DG solution. The scheme is formulated in the modal space and the volume integral and the numerical fluxes at the cell interfaces can be explicitly written. The explicit formulae of the scheme at third order is provided on two-dimensional structured meshes. The computational complexity of the ADER-DG scheme is compared to that of the RKDG scheme. Numerical experiments are also provided to demonstrate the accuracy and efficiency of our scheme.

16. Dan Ling, Junming Duan, and Huazhong Tang
    Physical-constraints-preserving Lagrangian finite volume schemes for one- and two-dimensional special relativistic hydrodynamics
    Journal of Computational Physics, 396: 507–543, Nov 2019
    Abs DOI arXiv

    This paper studies the physical-constraints-preserving (PCP) Lagrangian finite volume schemes for one- and two-dimensional special relativistic hydrodynamic (RHD) equations. First, the PCP property (i.e. preserving the positivity of the rest-mass density and the pressure and the bound of the velocity) is proved for the first-order accurate Lagrangian scheme with the HLLC Riemann solver and forward Euler time discretization. The key is that the intermediate states in the HLLC Riemann solver are shown to be admissible or PCP when the HLLC wave speeds are estimated suitably. Then, the higher-order accurate schemes are proposed by using the high-order accurate strong stability preserving (SSP) time discretizations and the scaling PCP limiter as well as the WENO reconstruction. Finally, several one- and two-dimensional numerical experiments are conducted to demonstrate the accuracy and the effectiveness of the PCP Lagrangian schemes in solving the special RHD problems involving strong discontinuities, or large Lorentz factor, or low rest-mass density or low pressure, etc.

17. Junming Duan and Huazhong Tang
    A second-order accurate scheme for a kinetic equation of two-dimensional Vicsek swarming model
    Natural Science Journal of Xiangtan University, 41(1): 1–14, Jan 2019
    Abs

    This paper proposes a second-order accurate scheme for solving the two-dimensional kinetic Vicsek swarming model. The second-order Strang operator splitting technique is used to decouple the convection terms and collision terms, and the high-resolution MUSCL scheme is used to discretize the convection terms. The spectral method and the second-order implicit Runge-Kutta method are used in the velocity and time directions, respectively, for the approximation of the subproblem in the collision part. Several one- and two-dimensional numerical experiments are given to test the accuracy and effectiveness of the scheme. Numerical results It shows that the proposed scheme achieves second-order accuracy, and can resolve discontinuities in the solution to the Riemann problems better compared with the first-order scheme in [J Comput Phys, 2015, 297: 32-46].

18. Junming Duan, Yangyu Kuang, and Huazhong Tang
    Model reduction of a two-dimensional kinetic swarming model by operator projections
    East Asian Journal on Applied Mathematics, 8(1): 151–180, Jan 2018
    Abs DOI arXiv

    Hyperbolic, rotation invariant moment systems are derived for a non-linear kinetic description of two-dimensional Vicsek swarming model. The systems also preserve mass conservation, and numerical experiments show that this approach captures the key features of the model such as shock waves, contact discontinuities, rarefaction waves, vortex formations. If the system order increases, the solutions of the moment systems converge to the solution of the corresponding kinetic equation.