The accuracy and efficiency of numerical methods are hot topics in computational fluid dynamics. In the previous work [J. Comput. Phys. 355 (2018) 385] of Du et al., a two-stage fourth order ($S_2O_4$) numerical scheme for hyperbolic conservation laws is proposed, which is based on dimension-by-dimensional HWENO5 and WENO5 reconstructions and GRP solver, and uses a $S_2O_4$ time-stepping framework. In this paper, we aim to design a new type of $S_2O_4$ finite volume scheme, to further improve the compactness and efficiency of the numerical scheme. We design an improved $S_2O_4$ framework for two-dimensional compressible Euler equations, and develop nonlinear compact Hermite reconstructions to avoid oscillations near discontinuities. The new two-stage fourth order numerical schemes based on the above nonlinear compact Hermite reconstructions and GRP solver are high-order, stable, compact, efficient and essentially non-oscillatory. In addition, the above reconstructions and the corresponding numerical schemes are extended to eighth-order accuracy in space, and can be theoretically extended to any even-order accuracy. Finally, we present a large number of numerical examples to verify the excellent performance of the designed numerical schemes.

Many interesting physical problems described by systems of hyperbolic conservation laws are stiff, and thus impose a very small time-step because of the restrictive CFL stability condition. In this case, one can exploit the superior stability properties of implicit time integration which allows to choose the time-step only from accuracy requirements, and thus avoid the use of small time-steps. We discuss an efficient framework to devise high order implicit schemes for stiff hyperbolic systems without tailoring it to a specific problem. The nonlinearity of high order schemes, due to space- and time-limiting procedures which control nonphysical oscillations, makes the implicit time integration difficult, e.g. because the discrete system is nonlinear also on linear problems. This nonlinearity of the scheme is circumvented as proposed in (Puppo et al., Comm. Appl. Math. & Comput., 2023) for scalar conservation laws, where a first order implicit predictor is computed to freeze the nonlinear coefficients of the essentially non-oscillatory space reconstruction, and also to assist limiting in time. In addition, we propose a novel conservative flux-centered a-posteriori time-limiting procedure using numerical entropy indicators to detect troubled cells. The numerical tests involve classical and artificially devised stiff problems using the Euler’s system of gas-dynamics.

The success of deep learning in various applications has generated a growing interest in understanding its theoretical foundations. This paper presents a theoretical framework that explains why over-parameterized neural networks can perform well. Our analysis begins from the perspective of approximation theory and argues that over-parameterized deep neural networks with bounded norms can effectively approximate the target. Additionally, we demonstrate that the metric entropy of such networks is independent of the number of network parameters. We utilize these findings to derive consistency results for over-parameterized deep regression and the deep Ritz method, respectively. Furthermore, we prove convergence rates when the target has higher regularity, which, to our knowledge, represents the first convergence rate for over-parameterized deep learning.

We introduce a novel phase-field model designed for ternary Cahn-Hilliard (CH) dynamics, incorporating contact angle boundary conditions within complex domains. In this model, we utilize a fixed phase field variable to accurately represent intricate domains within the ternary CH system. Simultaneously, the remaining two phase field variables are employed to simulate CH dynamics effectively. The contact angle term is derived from Young’s equality and the hyperbolic tangent profile of the equilibrium interface. To ensure compliance with the hyperbolic tangent property at the interface, a fidelity term is incorporated into the original CH model. This addition reduces mass loss for each phase and improves the accuracy of the contact angle effect. Moreover, we implement a finite difference scheme along with a nonlinear multigrid method to solve the corrected ternary CH model. A series of numerical experiments is conducted in both two- and three-dimensional spaces to demonstrate the efficiency and robustness of the proposed model.

In this paper, we develop and analyze an efficient discontinuous Galerkin method for stochastic Allen-Cahn equation driven by multiplicative noise. The proposed method is realized by symmetric interior penalty discontinuous Galerkin finite element method for space domain and implicit Euler method for time domain. Several new estimates and techniques are developed. Under some suitable regularity assumptions, we rigorously establish strong convergence results for the proposed fully discrete numerical scheme and obtain optimal convergence rates in both space and time. Numerical experiments are also carried out to validate our theoretical results and demonstrate the effectiveness of the proposed method.

In this study, we propose a blend of the average of THINC-EM and MUSCL (ATM) methods based on the AUSMD scheme for solving detonation wave problems. It is well known that the simulation of the detonation problems can produce incorrect shock information or strong spurious due to the stiff source term. Accurate simulation of detonation problems plays a crucial role in the design of detonation engines. The proposed ATM method combines the MUSCL and THINC-EM methods with different weighting functions, the optimized parameters of which are determined by the Q-learning method in order to accurately capture detonation waves, shock waves, and expansion fans. To validate the proposed numerical method, one and two-dimensional shock tube and the detonation tube and nozzles are chosen as benchmark test cases. Our numerical results show that the proposed the ATM type AUSMD scheme has great potential for handling more complex detonation problems and pulse detonation engine flow problems.

Algebraic Multigrid (AMG) is one of the most widely used iterative algorithms for solving large sparse linear equations $Ax=b.$ In AMG, the coarse grid is a key component that affects the efficiency of the algorithm, the construction of which relies on the strong threshold parameter $θ.$ This parameter is generally chosen empirically, with a default value in many current AMG solvers of 0.25 for 2D problems and 0.5 for 3D problems. However, for many practical problems, the quality of the coarse grid and the efficiency of the AMG algorithm are sensitive to $θ;$ the default value is rarely optimal, and sometimes is far from it. Therefore, how to choose a better $θ$ is an important question. In this paper, we propose a deep learning based auto-tuning method, AutoAMG($θ$) for multiscale sparse linear equations, which are common in practical problems. The method uses Graph Neural Network (GNN) to extract matrix features, and a Multilayer Perceptron (MLP) to build the mapping between matrix features and the optimal $θ,$ which can adaptively predict $θ$ values for different matrices. Numerical experiments show that AutoAMG($θ$) can achieve significant speedup compared to the default $θ$ value.

In this paper, we develop and analyze two stabilized mixed virtual element schemes for the Stokes problem based on the lowest-order velocity-pressure pairs (i.e., a piecewise constant approximation for pressure and an approximation with an accuracy order $k = 1$ for velocity). By applying local pressure jump and projection stabilization, we ensure the well-posedness of our discrete schemes and obtain the corresponding optimal $H^1$- and $L^2$-error estimates. The proposed schemes offer a number of attractive computational properties, such as, the use of polygonal/polyhedral meshes (including non-convex and degenerate elements), yielding a symmetric linear system that involves neither the calculations of higher-order derivatives nor additional coupling terms, and being parameter-free in the local pressure projection stabilization. Finally, we present the matrix implementations of the essential ingredients of our stabilized virtual element methods and investigate two- and three-dimensional numerical experiments for incompressible flow to show the performance of these numerical schemes.

A moving mesh finite element method is studied for the numerical solution of Bernoulli free boundary problems. The method is based on the pseudo-transient continuation with which a moving boundary problem is constructed and its steady-state solution is taken as the solution of the underlying Bernoulli free boundary problem. The moving boundary problem is solved in a split manner at each time step: the moving boundary is updated with the Euler scheme, the interior mesh points are moved using a moving mesh method, and the corresponding initial-boundary value problem is solved using the linear finite element method. The method can take full advantages of both the pseudo-transient continuation and the moving mesh method. Particularly, it is able to move the mesh, free of tangling, to fit the varying domain for a variety of geometries no matter if they are convex or concave. Moreover, it is convergent towards steady state for a broad class of free boundary problems and initial guesses of the free boundary. Numerical examples for Bernoulli free boundary problems with constant and non-constant Bernoulli conditions and for nonlinear free boundary problems are presented to demonstrate the accuracy and robustness of the method and its ability to deal with various geometries and nonlinearities.

AFEPack is a general-purpose C++ library for numerical solutions of partial differential equations. With over two decades development, AFEPack has been successfully applied for scientific and engineering computational problems in a variety of areas such as computational fluid dynamics, electronic structure calculations, computational micromagnetics. In this paper, design philosophy of the library, algorithms and data structures used in the discretization of governing equations, numerical linear algebra for the discretized system, as well as the pre-processing and post-processing of the simulations, will be described systematically for the AFEPack. The realization of two main features of the library, i.e., adaptive mesh methods and parallel computing, will be introduced in detail. The potential of the library for large scale scientific/engineering problems would be demonstrated by several examples. The future works on developing the library will also be discussed.