The aim of this paper is to propose an asymptotic preserving numerical scheme for the three-temperature radiative transfer equations, which couple the radiative transfer equation with the electron and ion diffusive equations. The $H^T_N$ method is used for angular discretization, and the unified gas kinetic scheme (UGKS) is used for spatial discretization to obtain the asymptotic preserving property of the proposed scheme. Since it includes the electron-ion heat conduction, radiation-electron coupling and electron-ion coupling terms, the three-temperature model exhibits strong nonlinearity and strong coupling properties. By carefully dealing with these coupling terms implicitly and using an iteration method to solve the electron-ion diffusive system, we can adapt the $H^T_N$-based UGKS method for the grey radiative transfer equations to deal with this complex model. According to the asymptotic preserving property of the proposed scheme, it is not necessary in optically thick regimes to restrict the spatial step size smaller than the photon mean free path in order to approximate the solution of the three-temperature, two-temperature, and single-temperature diffusion limit equations under difference circumstances. The robustness and effectiveness of the current scheme are verified by several numerical experiments.

We construct a two-dimensional fourth-order space-time conservation element and solution element (CESE) schemes for solving the ideal magnetohydrodynamics (MHD) equations. In the CESE scheme, the flow variables are calculated by using the same procedure as that of the original second-order CESE scheme. The scheme preserves most favorable attributes of the original second-order CESE method. Moreover, it is simple and easy to program. The numerical example for the smooth Alfvén wave problem suggests that the scheme can achieve the fourth-order accuracy for smooth solutions. In order to verify the efficiency of the schemes, we simulate several 2D MHD problems. We find that the fourth-order scheme can capture shocks and details of complex flow structures very well, and control the magnetic divergence efficiently. Moreover, the scheme is essentially CFL number insensitive schemes. The last several complex test problems further verify the performance of proposed scheme.

Cartesian grid method and local grid refinement for discrete velocity method (DVM) are developed in this work, with the numerical flux of DVM constructed by semi-Implicit Richtmyer method. To implement the boundary condition, the interpolation approach is applied, where the distribution function at the fluid side of the boundary point is first approximated by interpolation method given the knowledge of the boundary point. Once the distribution function is interpolated, the reflected Maxwellian distribution and numerical flux at boundary point can be evaluated. Then the distribution function at the fluid point close to the boundary can be updated by finite difference formulation. However, the interpolation of the distribution function at the boundary point is a significant challenge as the breakdown of the upwind stencil will cause instability. To preserve the upwind stencil, the most effective approach is to perform interpolation along the characteristic lines. Moreover, the local grid refinement is introduced to reduce the computational cost for industrial application. To validate the proposed Cartesian grid method, some numerical examples are simulated. The results demonstrate the accuracy and stability of the present method for straight boundary with oblique and curved boundary, subsonic and supersonic flows.

We study topology optimization governed by the incompressible Navier-Stokes equations using a phase field model. Unconditional energy stability is shown for the gradient flow in continuous space. The novel generalized stabilized semi-implicit schemes for the gradient flow in first-order time discretization of Allen-Cahn and Cahn-Hilliard types are proposed to solve the resulting optimal control problem. With the Lipschitz continuity for state and adjoint variables, the energy stability for time and full discretization has been proved rigorously on the condition that the stabilized parameters are larger than specific values. The proposed gradient flow scheme can work with large time steps and exhibits a constant coefficient system in full discretization, which can be solved efficiently. Numerical examples in 2d and 3d show the effectiveness and robustness of the optimization algorithms proposed.

The article presents for the first time the use of a known solution to the problem of gas expansion into a vacuum as a test for the implementation of numerical methods. An analytical solution of the problem of gas ball expanding into vacuum is written out in an explicit form. The solution is designed for benchmarking of three-dimensional gas dynamics solvers. The test makes it possible to evaluate a spherical symmetry of a numerical solution when using a Cartesian or cylindrical coordinate system. Moreover, the actual order of approximation of an implemented numerical algorithm, and the accuracy of reproducing the solution on a moving free boundary can be estimated. We present the results of numerical solution of the problem by using three-dimensional Smoothed Particle Hydrodynamics (SPH) simulations. The simulation is performed for two methods of computing density in SPH. The algorithm for computing the dynamics of particles at gas-vacuum sharp interface is described in detail.

Designing a good preconditioner for accelerating the iterative solution of the three-dimensional multi-group radiation diffusion equations based on a cell-centered finite volume discretization has been the focus of intensive research efforts over the past few decades. In the present paper, we develop a physics-wise splitting preconditioning algorithm with selective relaxation and algebraic multigrid subsolves. The spectral distribution and the degree of the minimal polynomial of its right-preconditioned matrix together with the conditional convergence property of its iteration method are analyzed. Subsequently, we discuss its sequential implementation as well as the two-level parallelization. Lastly, the new preconditioner is applied to the experimental test cases arising from realistic simulations of the hydrodynamic instability during the deceleration phase of a laser-driven spherical implosion to illustrate the numerical robustness, computational efficiency, parallel strong and weak scalabilities, and its competitiveness with some existing monolithic and block preconditioning approaches.

This study systematically investigates the flow characteristics around an inclined rounded square cylinder in the laminar flow regime, focusing on key influential parameters. The simulations cover a broad parameter space, including incidence angles from $0^◦$ to $45^◦,$ Reynolds numbers from 45 to 170, and corner radii from 0 to 0.4. Data from direct numerical simulations, including mean force and moment coefficients, root mean square of force fluctuation coefficients, and Strouhal numbers, are meticulously analyzed and divided into two datasets for model training. The developed data-driven model exhibits exceptional predictive accuracy, mirroring a high-fidelity physical model. Based on this model, the optimization strategy also demonstrates notable accuracy. Specifically, the implementation of optimal design using the high-fidelity model allows precise control of output physical parameters, aligning them with targeted optimal conditions. This capability offers significant potential for improving the efficiency of various engineering applications.

A central-moment discrete Boltzmann method (CDBM) is proposed for reactive flows, accommodating adjustable specific heat ratios and Prandtl numbers. In the framework of CDBM, a unified set of kinetic equations is used to delineate both macroscopic quantities and higher-order central moments. Via these central moments, the nonequilibrium effects that are directly related to the thermal fluctuation beyond conventional hydrodynamic governing equations can be quantified. Moreover, the discrete Boltzmann equation of the CDBM is simpler than that of previous multiple-relaxation-time DBMs, owing to the elimination of the additional term in the DBM. Furthermore, this method is capable of modeling supersonic compressible reactive flows characterized by high Mach numbers. The model is verified through simulations encompassing sound waves, shock waves, thermal Couette flows, regular shock reflections, and supersonic reactive waves.

This paper develops the high-order entropy stable finite difference schemes for multi-dimensional compressible Euler equations with the van der Waals equation of state on adaptive moving meshes. Semi-discrete schemes are first nontrivially constructed on the newly derived high-order entropy conservative (EC) fluxes in curvilinear coordinates and scaled eigenvector matrices as well as the multi-resolution WENO reconstruction, and then the fully-discrete schemes are given by using the high-order explicit strong-stability-preserving Runge-Kutta time discretizations. The high-order EC fluxes in curvilinear coordinates are derived by using the discrete geometric conservation laws and the linear combination of the two-point symmetric EC fluxes, while the two-point EC fluxes are delicately selected by using their sufficient condition, the thermodynamic entropy and the technically selected parameter vector. The adaptive moving meshes are iteratively generated by solving the mesh redistribution equations, in which the fundamental derivative related to the occurrence of non-classical waves is involved to produce high-quality mesh. Several numerical tests are conducted to validate the accuracy, the ability to capture the classical and non-classical waves, and the high efficiency of our schemes in comparison with their counterparts on the uniform mesh.

In this paper, we extend the work [2] to establish an energy-stable parametric finite element approximation for the generalized axisymmetric Willmore flow with closed surfaces. A crucial aspect of our approach is the introduction of two novel geometric identities involving the weighted normal velocity, $\vec{x} \cdot \vec{e_1}(\vec{x_t} \cdot \vec{v})\vec{v},$ and the curvature variable, $G_S =\varkappa_S−\overline{\varkappa},$ where $\varkappa_S$ represents the mean curvature and $\overline{\varkappa}$ denotes the spontaneous curvature. We theoretically prove that the numerical method preserves the stability of the original energy. Several numerical tests are provided to demonstrate the energy stability, as well as the accuracy and efficiency of our developed numerical scheme.