We present an efficient physics-informed neural networks (PINNs) framework, termed Adaptive Interface-PINNs (AdaI-PINNs), to improve the modeling of interface problems with discontinuous coefficients and/or interfacial jumps. This framework is an enhanced version of its predecessor, Interface PINNs or I-PINNs (Sarma et al. [1]; https://doi.org/10.1016/j.cma.2024.117135), which involves domain decomposition and assignment of different predefined activation functions to the neural networks in each subdomain across a sharp interface, while keeping all other parameters of the neural networks identical. In AdaI-PINNs, the activation functions vary solely in their slopes, which are trained along with the other parameters of the neural networks. This makes the AdaI-PINNs framework fully automated without requiring preset activation functions. Comparative studies on one-dimensional, two-dimensional, and three-dimensional benchmark elliptic interface problems reveal that AdaI-PINNs outperform I-PINNs, reducing computational costs by 2-6 times while producing similar or better accuracy.

In this paper we propose a Lattice-Boltzmann-based mesoscopic model for the three-dimensional flow of an incompressible polar fluid.
The mesoscopic model is equivalent to the usual incompressible three-dimensional Navier-Stokes equation coupled with the angular momentum equation, which describes the evolution of the angular velocity vector of the fluid particles.
The proposed model is applied to investigate the effects of the fluid polar structure on three steady flows: the steady Couette and Poiseuille flows in a square channel and the three-dimensional lid-driven cavity flow at $Re=100,$ $Re=400.$
The effects of the fluid polar structure on the above mentioned flows are investigated by varying the relevant dimensionless parameters: the coupling parameter $N$ and the geometric parameter $L.$
Results are consistent with the predictions of the theory of polar fluids. In particular, it is shown for the three-dimensional lid-driven cavity flow that the effect of the coupling parameter $N$ is to lower the effective Reynolds number and thus to increase the viscosity.

In this work, a virtual body-fitted grid is introduced and combined with the implicit velocity correction-based immersed boundary method (IBM) for simulation of incompressible flows with moving boundaries. The original implicit velocity correction-based IBM to this kind of problem requires the matrix inversion operation to be repeated at each time step, resulting in significant computational resources and time when the Gaussian elimination method is used for solution, whose computational complexity is $\mathcal{O}(M^3).$ The present method introduces a virtual body-fitted grid that moves together with the immersed boundary to overcome the above defect. As with the original implicit velocity correction-based IBM, the fractional step technique, which includes the prediction step and the correction step, is applied in the present method. Note that the correction step is implemented on the virtual grid in the present method instead of the Eulerian mesh in the original method. Since the relative positions between virtual grid points and Lagrangian points are changeless, the matrix in the correction step can be pre-calculated and stored, avoiding the need to update it at every time step. Although, within this approach, three additional steps including the marking virtual grid points and covered Eulerian points, and two interpolations between the Eulerian mesh and the virtual grid must be conducted at each time step, the computational effort is still greatly reduced as the computational complexity of these steps is $\mathcal{O}(M).$ A numerical experiment of flow around a transversely oscillating cylinder is first performed, demonstrating the improved efficiency, especially when the number of Lagrangian points is large. As validation, the flow over a flapping elliptical wing and a fluid-structure interaction (FSI) problem of vortex-induced vibrations of a circular cylinder are simulated. The numerical results are found to be in line with reference results, verifying the ability of the proposed method to simulate complex moving boundary problems.

By applying the least squares solution for the middle wave analysis inside the Riemann fan, we construct a four-state Harten-Lax-van Leer (HLL) Riemann solver for numerical simulation of magneto-hydrodynamics (MHD). First, we revisit the two-state HLL scheme and obtain the two outer intermediate states across the two out-bounding fast waves through Rankine-Hugoniot (R-H) conditions; Second, the two inner intermediate states are calculated using a geometric interpretation of the R-H conditions across the middle contact wave. This newly constructed four-state HLL solver contains different wave structures from those of the HLLD Riemann solver; namely, the two Alfvén waves are replaced as the two combination waves originated from the merging of Alfvén and slow waves inside the Riemann fan. As we tested, this solver resolves the MHD discontinuities well, and has better capture ability than the HLLD solver for the slow waves, although it appears more diffusive than the latter in the situations where the slow waves are not solely generated. Overall, the new solver has the similar accuracy as the HLLD solver, thus it is suitable for the calculation of numerical fluxes for the Godunov-type numerical simulation of MHD equations where the slow waves are expected to be resolved.

The article discusses a nonlinear system that is dependent on time and coupled by incompressible fluid and porous media flow. Treating Darcy flow as dual-mixed form, we propose a variational formulation and prove the well-posedness of weak solutions. The discretization of domain is accomplished using a triangular mesh, with the lowest order Raviart-Thomas element utilized for Darcy equations and Bernardi-Raugel element used for Navier-Stokes equations. Using the mortar method, we construct the spaces from which numerical solutions are sought. Based on backward Euler method, we establish a fully discrete algorithm. At each single time level, the first-order convergence is demonstrated through the use of the Gronwall inequality. Numerical experiments are provided to illustrate the algorithm’s effectiveness in approximating solutions.

The gas separation phenomenon of the pressure-driven binary mixture (helium–argon) flows in a microchannel is investigated numerically over a wide range of Knudsen numbers using the discrete unified gas kinetic scheme. The effects of inlet/outlet pressure ratio and Knudsen number on gas separation are studied numerically. It is found that the separation is strengthened with the increase of pressure ratio, and exhibits different trends under different rarefied conditions. The degree of gas separation changes nonlinearly with the outlet Knudsen number and exhibits a maximum in the transitional regime. In particular, we find that the relative pressure deviation and the velocity ratio between light species helium and heavy species argon can be served as indicators for gas separation under different working conditions. Finally, the phenomenon of Knudsen minimum is observed for the light species and gas mixture in the transition regime, but it does not appear for the heavy species within the considered Knudsen number range (≤10).

We describe a numerical method for the solution of acoustic exterior scattering problems based on the time-domain boundary integral representation of the solution. As the spatial discretization of the resulting time-domain boundary integral equation we use either the method of fundamental solutions (MFS) or the Galerkin boundary element method (BEM). In time we apply either a standard convolution quadrature (CQ) based on an A-stable linear multistep method or a modified CQ scheme. It is well-known that the standard low-order CQ schemes for hyperbolic problems suffer from strong dissipation and dispersion properties. The modified scheme is designed to avoid these properties. We give a careful description of the modified scheme and its implementation with differences due to different spatial discretizations highlighted. Numerous numerical experiments illustrate the effectiveness of the modified scheme and dramatic improvement with errors up to two orders of magnitude smaller in comparison with the standard scheme.

We propose a polygonal finite volume element method based on the mean value coordinates for anisotropic diffusion problems on star-shaped polygonal meshes. Because the convex cells with hanging nodes are always star-shaped, the computation on them is no longer a problem. Naturally, we apply this advantage of the new polygonal finite volume element method to construct an adaptive polygonal finite volume element algorithm. Moreover, we introduce two refinement strategies, called quadtree-based refinement strategy and polytree-based refinement strategy respectively, and they all have great performance in our numerical tests. The new adaptive algorithm allows the use of hanging nodes, and the number of hanging nodes on each edge is unrestricted in general. Finally, several numerical examples are provided to show the convergence and efficiency of the proposed method on various polygonal meshes. The numerical results also show that the new adaptive algorithm not only reduces the computational cost and the implementation complexity in mesh refinement, but also ensures the accuracy and convergence.

Models for shallow water flow often assume that the lateral velocity is constant over the water height. The recently derived shallow water moment equations are an extension of these standard shallow water equations. The extended models allow for a vertically changing velocity profile, resulting in more accuracy when the velocity varies considerably over the height of the fluid. Unfortunately, already the one-dimensional models lack global hyperbolicity, an important property of partial differential equations that ensures that disturbances have a finite propagation speed.
In this paper, cylindrical shallow water moment equations are formulated by starting from the cylindrical incompressible Navier-Stokes equations. We formulate two-dimensional axisymmetric Shallow Water Moment Equations by imposing axisymmetry in the cylindrical model. The loss of hyperbolicity is analyzed and a hyperbolic axisymmetric moment model is then derived by modifying the system matrix in analogy to the one-dimensional case, for which the hyperbolicity problem has already been observed and overcome. Numerical simulations with both discontinuous and continuous initial data in a cylindrical domain are performed using a finite volume scheme tailored to the cylindrical mesh. The newly derived hyperbolic model is clearly beneficial as it gives more stable solutions and still converges to the reference solution when increasing the number of moments.

Deep learning method is of great importance in solving partial differential equations. In this paper, inspired by the failure-informed idea proposed by Gao et al. (SIAM Journal on Scientific Computing 45(4) (2023)) and as an improvement, a new accurate adaptive deep learning method is proposed for solving elliptic problems, including interface problems and convection-dominated problems. Based on the failure probability framework, the piece-wise uniform distribution is used to approximate the optimal proposal distribution and a kernel-based method is proposed for efficient sampling. Together with the improved Levenberg-Marquardt optimization method, the proposed adaptive deep learning method shows great potential in improving solution accuracy. Numerical tests on the elliptic problems without interface conditions, on one elliptic interface problem, and on the convection-dominated problems demonstrate the effectiveness of the proposed method, as it reduces the relative errors by a factor varying from $10^2$ to $10^4$ for different cases.