We elaborate a numerical method for a three-dimensional hydrodynamic multi-species plasma model described by the Euler-Maxwell equations. Our method is inspired by and extends the one-dimensional scheme from [P. Degond, F. Deluzet, and D. Savelief, Numerical approximation of the Euler-Maxwell model in the quasineutral limit, Journal of Computational Physics, 231 (4), pp. 1917–1946, 2012]. It can cope with large variations of the Debye length $λ_D$ and is robust in the quasi-neutral limit $λ_D→0$ thanks to its asymptotic-preserving (AP) property. The key ingredients of our approach are (i) a discretization of Maxwell’s equations based on primal and dual meshes in the spirit of discrete exterior calculus (DEC) also known as the finite integration technique (FIT), (ii) a finite volume method (FVM) for the fluid equations on the dual mesh, (iii) mixed implicit-explicit timestepping, (iv) special no-flux and contact boundary conditions at an outer cut-off boundary, and (v) additional stabilization in the non-conducting region outside the plasma domain based on direct enforcement of Gauss’ law. Numerical tests provide strong evidence confirming the AP property of the proposed method.

In this paper, a new class of very-high-order finite difference TENO schemes with discontinuity-resolving property for compressible Euler equations is proposed. To achieve arbitrary high-order schemes within a unified framework, a novel dual ENO-like stencil selection strategy is proposed based on the candidate stencil arrangement with incremental width. Based on a tailored coupling strategy, the first three small candidate stencils not only provide the reference to judge whether the candidate stencil with a large width is smooth or not but also determine whether the targeted cell is being crossed by genuine discontinuities or not, by combining the stencil smoothness information of several adjacent cells. Without losing generality, the typical seventh- and ninth-order TENO-DR schemes are constructed in detail and a variety of benchmark-test problems, including broadband waves, strong shock and contact discontinuities are studied. Compared to the state-of-the-art TENO schemes, the present schemes exhibit significantly improved high-resolution property and sharp shock-capturing capability, and thus are promising for more complex compressible flow simulations where the excessive numerical dissipation of existing schemes prevents the concerned flow structures from being properly resolved.

Hypersonic vehicles create high-temperature environments, thus causing air molecules to undergo chemical reactions. As a result, the nonlinearity of the mapping relationship between heat flux and inflow condition is strengthened. Accurately predicting the heat flux of real gases using machine learning methods relies on an amount of training data of real gas, but the computational cost is often unacceptable. To solve the issue stated, we propose a novel neural network named Physically Guided Neural Network based on Transfer Learning (TL-PGNN). We design a new network architecture to depict physical laws and use heat flow data of ideal gases to enhance the network’s capability to depict heat flow from only a few real gas samples. Experiments of sphere demonstrate that, compared to ordinary DNN and PGNN, applying TL-PGNN decreases the mean L1 error by 72.66% and 24.53% on the test set, respectively.

We show that the entries of the stiffness matrix, associated with the $C^0$-piecewise linear finite element discretization of the hyper-singular integral fractional Laplacian (IFL) on rectangular meshes, can be simply expressed as one-dimensional integrals on a finite interval. Particularly, the FEM stiffness matrix on uniform meshes has a block-Toeplitz structure, so the matrix-vector multiplication can be implemented by FFT efficiently. The analytic integral representations not only allow for accurate evaluation of the entries, but also facilitate the study of some intrinsic properties of the stiffness matrix. For instance, we can obtain the asymptotic decay rate of the entries, so the “dense” stiffness matrix turns out to be “sparse” with an $\mathcal{O}(h^3)$ cutoff. We provide ample numerical examples of PDEs involving the IFL on rectangular or $L$-shaped domains to demonstrate the optimal convergence and efficiency of this semi-analytical approach. With this, we can also offer some benchmarks for the FEM on general meshes implemented by other means (e.g., for accuracy check and comparison when triangulation reduces to rectangular meshes).

The momentum correction coefficient in one-dimensional blood flow models is related to the prescribed velocity profile. The exact solution of the Riemann problem for the one-dimensional blood flow equations has been previously studied only for a momentum correction coefficient equal to one (corresponding to a flat velocity profile, i.e. an inviscid fluid). In this paper we solve exactly the Riemann problem for the non-linear hyperbolic one-dimensional blood flow equations with a general constant momentum correction coefficient and a tube law that allows to describe both arteries and veins with continuous or discontinuous mechanical and geometrical properties. In the case of discontinuous properties, only the subsonic regime is considered. We propose a numerical procedure to compute the obtained exact solution and finally we validate it numerically, by comparing exact solutions to those obtained with well-known, first order, numerical schemes on a carefully designed set of test problems. A detailed knowledge about this problem will allow to determine coupling and boundary conditions arising when these models are applied on networks of vessels, ensuring full consistency with the underlying one-dimensional blood flow model without resorting to linearization techniques commonly applied when the momentum correction coefficient is assumed to be different from one.

Numerical study on compressible two-medium flows has been a hot issue in recent decades. In this study, we design quasi-conservative finite difference alternative weighted essentially non-oscillatory (AWENO) schemes up to the ninth order for the five-equation model with the stiffened gas equation of state. We propose uniformly high-order flux-based bound- and positivity-preserving (BP-P) limiters for the AWENO schemes while preserving the equilibrium solutions simultaneously. Though the BP-P limiters are used, the numerical solutions have the tendency to generate oscillations especially near strong shock and/or rarefaction waves, due to the sudden drastic scale transition of the density, pressure, etc. To resolve fine structures and the transition of different scales, the latest affine-invariant WENO (Ai-WENO) interpolation is adopted and generalized up to the ninth order. In addition, we will systematically derive CFL conditions when the Lax-Friedrichs numerical flux is applied. Moreover, we show the potential of the BP-P limiters for a variant of the five-equation model, usually suggested in finite volume and discontinuous Galerkin methods. For illustration purposes, we adopt the AWENO schemes and derive the corresponding CFL conditions. A variety of one- and two-dimensional test problems illustrate the high order of accuracy, effectiveness, and robustness of the proposed BP-P Ai-AWENO schemes.

This paper is concerned with high moment and pathwise error estimates for fully discrete mixed finite element approximations of stochastic Navier-Stokes equations with general additive noise. The implicit Euler-Maruyama scheme and standard mixed finite element methods are employed respectively for the time and space discretizations. High moment error estimates for both velocity and time-averaged pressure approximations in strong $L^2$ and energy norms are obtained, pathwise error estimates are derived by using the Kolmogorov Theorem. Unlike their deterministic counterparts, the spatial error constants grow in the order of $\mathcal{O}(k^{-\frac{1}{2}} ),$ where $k$ denotes time step size. Numerical experiments are also provided to validate the error estimates and their sharpness.

In this work, we conduct a detailed comparison between two second-order conservative Allen-Cahn (AC) models [Model A: Zheng et al., Phys. Rev. E 101, 0433202 (2020) and Model B: Mirjalili and Mani, J. Comput. Phys. 498, 112657 (2024)] for the immiscible $N$-phase flows. Mathematically, these two AC equations can be proved to be equivalent under some approximate conditions. However, the effects of these approximations are unclear from the theoretical point of view, and would be considered numerically. To this end, we propose a consistent and conservative lattice Boltzmann method for the AC models for $N$-phase flows, and present some numerical comparisons of accuracy and stability between these two AC models. The results show that both two AC models have good performances in accuracy, but the Model B is more stable for the realistic complex $N$-phase flows, although there is an adjustable parameter in the Model A.