• Abstract

This paper presents a three-dimensional thirteen-velocity (D3Q13) multiple-relaxation-time (MRT) lattice Boltzmann (LB) model for incompressible flows. The model can eliminate the second kind of compressibility error existing in previous D3Q13 MRT LB models. We simulate the three-dimensional steady Poiseuille flow, unsteady pulsatile flow, and lid-driven cavity flow to verify the validity of the current model. The excellent agreement between the numerical result and the analytical solution or the previous numerical result demonstrates the model's effectiveness.

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The Landau-Lifshitz-Slonczewski equation is used to describe the magnetization dynamics under the influence of a spin-polarized current in terms of a spin-velocity field in ferromagnetic materials. This equation is a strongly nonlinear evolution equation with a non-convex constraint of unit sphere. A natural method of preserving the non-convex constraint is the renormalized method. In this paper, we consider linearized backward Euler and BDF2 semi-renormalized finite element methods, where the renormalized numerical solution is not used in the discretizations of the time derivative and only used in linearized explicit parts. By using the piecewise linear finite element to make the spatial discretization, optimal $L^2$ error estimates are derived, i.e., $\mathcal O(\tau+h^2)$ for the Euler scheme and $\mathcal O(\tau^2+h^2)$ for the BDF2 scheme under some CFL type conditions. Finally, numerical results are shown to support the theoretical convergence rates.

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The paper proposes a family of novel arbitrary high-order structure preserving  exponential schemes for  the nonlinear Schrodinger  equation. First, we introduce a quadratic auxiliary variable to reformulate the original  nonlinear Schrodinger  equation into an equivalent equation with modified energy. With that, the Lawson transformation technique is applied to the equation and deduces a conservative exponential system. Then, the symplectic Runge-Kutta method approximates the exponential system in the time direction and leads to a semi-discrete conservative scheme. Subsequently, the Fourier pseudo-spectral method is applied to approximate the space of the semi-discrete to obtain a class of fully-discrete schemes. The constructed schemes are proved to inherit quadratic invariants and are stable. Some numerical examples are given to confirm the accuracy and conservation of the developed schemes at last.

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The discontinuous Galerkin (DG) method is employed to solve the auto-convolution Volterra integral equations (AVIEs). The solvability of the DG method is discussed, then it is proved that the quadrature DG (QDG) method obtained from the DG method by approximating the inner products by suitable numerical quadrature formulas, is equivalent to the piecewise discontinuous polynomial collocation method. The uniform boundedness of the DG solution is provided by defining a discrete weighted exponential norm, and the optimal global convergence order of the DG solution is obtained. In order to improve the numerical accuracy, the iterated DG method is introduced. By virtue of a projection operator, the optimal $m+1$ superconvergence order of the iterated DG solution is gained, as well as $2m$ local superconvergence order at mesh points. It is noting that both the global and local superconvergence are obtained under the same regularity assumption as that for the convergence, other than the collocation method, one has to improve the regularity of the exact solution to obtain the superconvergence of the iterated collocation method. Some numerical experiments are given to illustrate the theoretical results.

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In the paper, we consider the coupled nonlinear Schrodinger equation with high degree polynomials in the energy functional that cannot be handled by using the newly proposed quadratic auxiliary variable method. Therefore, we develop the multiple quadratic auxiliary variable approach to deal with coupled systems and construct high-accuracy structure-preserving schemes for the  equation. To fix the idea, we first apply the multiple quadratic auxiliary variable approach to the equation and obtain an equivalent system that possesses the original energy and mass. Then, a family of high-accuracy structure-preserving schemes that can conserve the mass and energy is derived by applying the Gauss collocation method and sine pseudo-spectral method to approximate the system in time and space. The given schemes have high-accuracy in time and can both inherit the mass and Hamiltonian energy of the system. Ample numerical results are given to confirm the accuracy and conservation of the developed schemes at last.

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Solving eigenvalue problems is an important subject in scientific computing. Classical numerical methods, such as finite element methods and spectral methods, have been developed with great success in applications. However, these methods struggle in the case of complex geometries. The recently developed random feature method has demonstrated its superiority in solving partial differential equations (PDEs), especially for problems with complex geometries. In this work, we develop random feature methods for solving elliptic eigenvalue problems. Our contributions include (1) using tailored separation-of-variables random feature functions to approximate eigenfunctions, (2) employing collocation points in the strong formulation or quadrature scheme (exact integration) in the weak formulation to handle the PDEs and boundary conditions, and (3) solving the generalized eigenvalue problem in which the number of conditions equals the number of unknowns. Through a series of examples from one dimension to three dimensions, we demonstrate the high accuracy and robustness of the proposed methods with respect to geometric complexity. We have made our source code publicly available at https://github.com/lingyun-2024/The-code-of-RFM-EEP.git.

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In micro-nano and even some macro-scale flows, the boundary slip cannot be ignored and has an important influence. We consider the most widely used Navier slip length model to investigate whether the solution of Stokes flow exists if a homogeneous slip is imposed as a boundary condition replacing the ordinary Dirichlet condition of velocity for some domains of simple geometries, including the two-dimensional channel, plane annulus and axisymmetric circular pipe. The Fourier series is employed to obtain the exact solutions formulated by the stream function. While a unique solution exists for most cases, it is found that yet occurs the cases of no solution under a certain slip length and infinitely multiple solutions under an infinitely discrete set of slip lengths, respectively.

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Anomalous diffusions are ubiquitous in nature, whose functional distributions are governed by the backward Feynman-Kac equation. In this paper, the local discontinuous Galerkin (LDG) method is used to solve the 2D backward Feynman-Kac equation in a rectangular domain. The spatial semi-discrete LDG scheme of the equivalent form (obtained by Laplace transform) of the original equation is established. After discussing the properties of the fractional substantial calculus, the stability and optimal convergence rates  of the semi-discrete scheme are proved by choosing an appropriate generalized numerical flux. The $L1$ scheme on the graded meshes is used to deal with the weak singularity of the solution near the initial time. Based on the theoretical results of a semi-discrete scheme, we investigate the stability and convergence of the fully discrete scheme, which shows the optimal  convergence rates. Numerical experiments are carried out to show the efficiency and accuracy of the proposed scheme. In addition, we also verify the effect of the central numerical flux on the convergence rates and the condition number of the coefficient matrix.

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In this paper, we utilize Physics-Informed Neural Networks (PINNs) without any labeled data to solve the fourth-order partial differential equation governing the bending of thin plates. We meticulously formulate and apply our framework to the bending problem of elastic thin plates. Our findings indicate that defining each solution variable through an independent network, meaning without shared network parameters, yields superior performance compared to a single-network model with multiple outputs.

During the training process, we discovered that more accurate results were achieved by adjusting the network architecture to strictly satisfy the boundary conditions, rather than incorporating them as part of the loss function. Remarkably, the PINNs are capable of obtaining relatively good results without the use of any labeled data during training, irrespective of whether soft or hard constraints are applied. In the end, the neural network's predictions for plate deflection, stress, strain, bending, and shear force all have errors less than 5\% compared to the analytical solutions. In the algorithm implementation phase, we leverage the Python library DeepXDE, which facilitates the training of PINNs by providing an efficient and expedited process.

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A nonconforming $P_2$ finite element  is constructed by enriching the conforming $P_2$ finite element space with seven $P_2$ nonconforming bubble functions (out of fifteen such bubble functions on each tetrahedron). This spacial nonconforming $P_2$ finite element, combined with the discontinuous $P_1$ finite element on general tetrahedral grids, is inf-sup stable for solving the Stokes equations. Consequently such a mixed finite element method produces optimal-order convergent solutions for solving the stationary Stokes equations. Numerical tests confirm the theory.

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The local characteristic fields of compressible Euler equations can be decomposed into linearly degenerate and genuinely nonlinear, respectively. The former is associated with a contact discontinuity (the slip lines for multi-dimensional cases) and the latter is associated with shocks and rarefaction waves (Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, 2009). The weighted essentially non-oscillatory (WENO) scheme can capture shock waves well but its resolution for short waves still needs to be improved, while one kind of new scheme constructed by the boundary variation diminishing (BVD) principle with the THINC (Tangent of Hyperbola for INterface Capturing) reconstruction can resolve the contact discontinuities with higher resolution but it may generate spurious numerical phenomena in some special cases. Combining the physical characteristics and the advantages of WENO and BVD schemes, this paper proposes a hybrid reconstruction method, in which the classical WENO scheme is used to discretize the genuinely nonlinear fields and the higher resolution BVD scheme is used to discretize the linearly degenerate field(s). Numerical experiments show that the hybrid method can preserve high accuracy near both contact discontinuities and short waves as the used higher resolution scheme while effectively avoiding spurious numerical phenomena (such as oscillations and overshoots). Compared with the BVD scheme, the hybrid method also improves computational efficiency, since it uses a more efficient WENO scheme in genuinely nonlinear fields.

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In this paper, a first-order penalty finite element method for the 2D/3D unsteady incompressible thermomicropolar fluid (UITF) equations is considered, which combines the advantage of the penalty method, first-order backward Euler scheme, and implicit or explicit iteration for the nonlinear terms to get a decoupled temporal evolution procedure, which only needs to solve a series of elliptic subproblems. Theoretically, the stability analysis and optimal error estimates of the temporal semi-discrete method are deduced. Furthermore, the classical MINI element pairs are adopted in concrete spatial discrete, and the feasibility of the method is  verified  by numerical experiments.

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In this investigation, the analysis of the nonlinear vibroacoustic and sound transmission loss behaviors of plates made of functionally graded material is presented. It is assumed that the properties of the functionally graded plates are in the form of the simple power law scheme and continuous along the thickness, under thermal load and incident oblique plane sound wave as well as the first-order shear deformation theory. For this purpose, first, using Hamilton's principle, the nonlinear partial differential equations of motion are derived by the displacement field function approach and by considering the nonlinear von K\'arm\'an strain-displacement relations. To solve the equations, using the Galerkin method, the nonlinear partial differential equations of motion lead to Duffing equation. Then, using the homotopy analysis method, the equation of the transverse movement of the plate is solved semi-analytically to obtain the nonlinear frequencies. Finally, the nonlinear vibration and acoustic response of functionally graded plates are studied by considering the variation of the important parameters such as aspect ratio, dimensionless amplitude, volume fraction power of functionally graded material, external acoustic pressure, incidence and azimuthal angles, temperature changes, phase portrait, sound transmission loss, velocity and average mean square velocity of drive point and sound power level of the functionally graded plate. Results show increasing the incidence angle leads increase in hardening effects and sound transmission loss, but growing the azimuthal angle does not have much effect on the frequency-response and sound transmission loss in the absence of the external mean flow. Also, increasing temperature changes lead to decrease in hardening effects and sound transmission loss.

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The low accuracy using unstructured meshes to solve complicated geometry problems is a significant challenge in practical engineering. It is the key to solve this issue through enhancing the accuracy of tetrahedral element. This work develops a simple strategy to enhance the accuracy of tetrahedral element with the Shepard interpolation function. In which the strain field is reconstructed by introducing the weighted strains of adjacent tetrahedral elements to central tetrahedral element. The stiffness of the discrete system is significantly softened with this simple operation, which leads to a great accuracy improvement. Furthermore, this simple modification makes linear tetrahedral element owns higher accuracy even compared with hexahedral element. The high precision tetrahedral element makes finite element analysis more acceptable for engineers. It provides a novel solution to finite element analysis using unstructured meshes for practical engineering problems with complicated geometry. In order to validate the accuracy and feasibility of present modified tetrahedral element (M-T4) in complicated geometry problems, several engineering examples are performed, including linear static analysis, modal analysis, frequency response analysis, transient response analysis, and response spectrum analysis. The remarkable performance of the M-T4 suggests its wide applicability to practical engineering problems.

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Nonlinear problems widely exist in many aspects of the natural field. The nonlinear situation makes it difficult for most existing solvers to deal with. Therefore constructing an efficient and accurate solver is a challenge. In this paper, a Legendre spectral method is developed for the nonlinear Volterra integro-differential equation. The error analysis is also provided to justify

the spectral rate of convergence for the errors of approximate solution and

approximate derivative decay exponentially in both the $L^{2}$ norm and the infinity norm.  In the end, numerical results are displayed to confirm the effectiveness of the Legendre spectral analysis.

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A class of finite difference Hermite radial basis functions weighted essentially  non-oscillatory (HRWENO) methods for solving conservation laws was presented by Abedian (Int. J. Numer. Meth. Fluids, 94 (2022), pp. 583--607).To reconstruct the fluxes in HRWENO, the common practice of reconstructing the flux functions was employed. In this follow-up research work, an alternative formulation to reconstruct the numerical fluxes is considered. First, the solution and its derivatives are directly employed to interpolate point values at interfaces of computational cells. Afterwards, the point values at interface of cell in building block are considered to obtain numerical fluxes. In this framework, arbitrary monotone fluxes can be employed, while in HRWENO the classical practice of reconstructing flux functions can be considered only to smooth flux splitting. Also, in the process of reconstruction these type of schemes consider the effectively narrower stencil of HRWENO methods. Extensive test cases such as Euler equations of compressible gas dynamics are considered to show the good performance of the methods.

• General Summary