• Abstract

In this article, we study a parabolic equation with both nonlinear time-derivative term and nonlinear diffusion term by the finite volume element method. The optimal error estimate in $H^{1}$-norm is proved for fully discrete scheme. The suboptimal error estimate in $L^{2}$-norm is proved both for semi-discrete scheme and fully discrete scheme. We prove the existence of solution for the fully discrete scheme. Numerical results show the effectiveness of our method.

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Numerical simulation of a three-dimensional semiconductor device is a fundamental problem in information science. The mathematical model is defined by a nonlinear system of initial-boundary problem including four partial differential equations: an elliptic equation for electrostatic potential, two convection-diffusion equations for electron concentration and hole concentration, a heat conduction equation for temperature. The electrostatic potential appears within the concentration equations and heat conduction equation, and the electric field strength controls the concentrations and the temperature. The electric field potential is solved by the conservative block-centered method, and the order of the accuracy is improved by the electric potential. The concentrations and temperature are computed by the upwind block-centered multistep method, where three different numerical methods are involved. The multistep method is adopted to approximate the time derivative. The block-centered method is used to discretize the diffusion. The upwind scheme is applied to approximate the convection to avoid numerical dispersion and nonphysical oscillation. The block-centered difference simulates diffusion, concentrations, temperature, and the adjoint vector functions simultaneously. It has the local conservation of mass, which is an important nature in numerical simulation of a semiconductor device. By using the variation, energy estimates, induction hypothesis, embedding theorem and the technique of a priori estimates of differential equations, convergence of the optimal order is obtained. Numerical examples are provided to show the effectiveness and viability. This method provides a powerful tool for solving the challenging benchmark problem.

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In this paper, we develop a new class of conservative characteristic finite difference methods for solving convection-dominated diffusion problems with fourth-order accuracy in both time and space. Specifically, the method of characteristics is utilized to handle the convection term, which allows for greater flexibility in the choice of time step sizes. To achieve high-order temporal accuracy, we propose characteristics-based optimal implicit strong stability preserving (SSP) Runge-Kutta methods implemented along the streamline. Furthermore, a conservative interpolation is employed to calculate values at the tracking points. By introducing diverse fourth-order approximation operators on the uniform Eulerian and irregular Lagrangian meshes, we can deal with the diffusion term with high accuracy while preserving the conservation property. The mass conservation for our proposed method is theoretically proved, and is verified through numerical experiments. Moreover, the numerical tests demonstrate that our scheme achieves temporal and spatial fourth-order accuracy and generates non-oscillatory solutions, even with large time step sizes.

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A  high-order unified solution method is presented for simulating the coupled fluid flow and heat transfer phenomena in solid materials. The  method integrates the energy transfer processes of fluid and solid by transforming the compressible Navier-Stokes equations into a dimensionless system through a double-time-scale approach. A time scaling factor is introduced  into the system to bridge and expedite the energy transfer processes between the mediums, thereby accelerating convergence towards a global steady state. To ensure consistent accuracy across material interfaces, high-order formulations are introduced to obtain the interface temperature and heat flux. The effectiveness of the methods are  verified  and validated through numerical experiments with a three-dimensional discontinuous Galerkin flow solver. Numerical results demonstrate the globally high-order accuracy of the unified method for fluid-solid conjugate heat transfer problems, enabling rapid and robust convergence with large CFL numbers of up to $10^8$.

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Biot's model is a multiphysics model that describes the interaction of a poroelastic material with its interstitial fluid flow. In this study, we focus on investigating the convergence behavior of a global-in-time iterative decoupled algorithm based on a three-field formulation. During each iteration, the algorithm involves solving a reaction-diffusion subproblem across the entire temporal domain, followed by resolving a Stokes subproblem over the same time interval. This algorithm is recognized for its "partially parallel-in-time" property, enabling the implementation of a parallel procedure when addressing the Stokes subproblem. We establish its global convergence with a new technique by confirming that the limit of the sequence of numerical solutions of the global-in-time algorithm is the numerical solution of the fully coupled algorithm. Numerical experiments validate the theoretical predictions and underline the efficiency gained by implementing the parallel procedure within the proposed global-in-time algorithm.

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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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In extreme scenarios, classical high-order WENO schemes may result in non-physical states. The Positivity-Preserving Limiter (PP-Limiter) is often used to ensure positivity if $\CFL \leq 0.5$ with a third-order TVD Runge-Kunta (RK3) scheme. This study proposes two novel conservative WENO-Z methods: AT-PP and AO-PP to improve efficiency with $0.5 < \CFL < 1$ if desired. The AT-PP method detects negative cells after each RK3 stage posteriori and computes a new solution with the PP-Limiter ($\CFL < 0.5$) for that step. The AO-PP method progressively lowers the WENO operator's order and terminates with the first-order HLLC solver, proven positivity-preserving with $\CFL<1$, only at negative cells at that RK3 stage. A single numerical flux enforces conservation at neighboring interfaces. Extensive 1D and 2D shock-tube problems were conducted to illustrate the performance of AT-PP and AO-PP with $\CFL = 0.9$. Both methods outperformed the classical PP-Limiter in accuracy and resolution, while AO-PP performed better computationally in some cases. The AO-PP method is globally conservative and accurate, adaptiveness, and robustness while resolving fine-scale structures in smooth regions, capturing strong shocks and gradients with ENO-property, improving computational efficiency, and preserving the positivity, all without imposing a restrictive limit on the CFL condition.

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In this paper we propose a continuous downscaling data assimilation algorithm for solving reaction-diffusion equations with a critical parameter. For the spatial discretization we consider the finite element methods. Two backward differentiation formulae (BDF), a backward Euler method and a two-step backward differentiation formula, are employed for the time discretization. Employing the dissipativity property of the underlying reaction-diffusion equation, under suitable conditions on the relaxation (nudging) parameter and the critical parameter, we obtain uniform-in-time error estimates for all the methods for the error between the fully discrete approximation and the reference solution corresponding to the measurements given on a coarse mesh by an interpolation operator. Numerical experiments verify and complement our theoretical results.

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An investigation into postbuckling behavior of thin and ultra-thin plates/shells is first performed in this paper. The present formulation is established upon a shell theory using the von Karman assumption and isogeometric analysis (IGA). In the present approach, the bifurcation point in eigenvalue buckling analysis and limit points can be detected simply via the modified Riks method. Hence, eigenvalue buckling analysis and postbuckling analysis are unified in the most simple flowchart. The above advantages allow buckling and postbuckling analyses of ultra-thin, thin and moderate thick plates/shells can be performed directly and conveniently. The obtained results are verified with the reference ones. In addition, the results show that buckling shapes and postbuckling responses of thin structures and ultra-thin structures are significantly different. Effects of slenderness, aspect ratios and curvature on postbuckling response of ultra-thin plates/shells are investigated particularly. Especially, some new results of ultra-thin plates/shells are first provided in this paper.

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This paper focuses on an energy-stable local discontinuous Galerkin (LDG) method for a binary compressible flow model. Since the densities and the momentum are highly coupled in the equations,  and the test and basis functions in LDG discretizations have to be in the same finite element space, it is difficult to obtain stable LDG discretizations for the binary compressible flow model. To tackle this issue, we take the mass average velocity ${\bf v}$ and its square as auxiliary variables. These auxiliary variables are chosen in the stability analysis as the test functions for the momentum and density balance equations, respectively. Using the Crank-Nicolson (CN) time integration method, we can prove then the stability of the LDG-CN discretization. Computations are provided to demonstrate the accuracy, efficiency and capabilities of the numerical method.

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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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In this paper, we develop an immersed Petrov-Galerkin finite element method for solving two-dimensional Stokes interface problems. The proposed method do not require the solution mesh to align with the fluid interface. We utilize the $\text{iso}P_2$-$P_0$ element, which adopts piecewise linear approximation for velocity on fine elements and piecewise constant approximation for pressure on coarse elements. The vector-valued solution map is constructed to approximate the velocity and pressure based on the  jump conditions across the interface. Several numerical experiments demonstrate that the proposed method maintain the optimal convergence rate in the $L_2$-norm and the $H_1$-norm for the velocity and the $L_2$-norm for the pressure.

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The lattice Boltzmann method is a relatively modern method. It has been widely applied to industrial flows (for small scales) governed by the Navier Stokes equations, but little applied to large-scale flows, such as geophysical flows. In this paper, we then attempt to apply the lattice Boltzmann method to coastal flows governed by the Saint-Venant equations (also called Shallow water equations)  via an application to the real case of the Nador lagoon (Moroccan Mediterranean coast).

The establishment of the equations of the model and their discretization is given here. The introduction of turbulence by the large eddy simulations was also presented. The verification of the developed code concerns two test cases in hydraulic engineering. The first verification test deals with complex flow in a $180 ^\circ$ bent channel. For this case, the model has faithfully reproduced all the characteristics of this flow (pressure/depression zone and the recirculation zone after the bend). Similarly, the results of this test case have been successfully compared to measurements available in the literature.  The second verification test analyzes the flow around a spur-dike. Finally, the developed  code has been adapted for its application to the real case of the Nador lagoon whose flow is generated by the superposition of the tidal waves $M_2$ and $S_2$. The model reproduced with good accuracy the dynamics of the flow and more particularly the two vortices observed on both sides of the lagoon inlet. The founding results are successfully compared with the results of previous studies published for the lagoon.

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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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This paper develops a robust three-level time split high-order Leapfrog/ Crank-Nicolson technique for solving the two-dimensional unsteady Sobolev and regularized long wave equations arising in fluid mechanics. A deep analysis of the stability and error estimates of the proposed approach is considered using the $L^{\infty}(0,T;H^{2})$-norm. Under a suitable time step requirement, the theoretical studies indicate that the constructed numerical scheme is strongly stable (in the sense of $L^{\infty}(0,T;H^{2})$-norm), temporal second-order accurate and convergence of order $\mathcal{O}(h^{8/3})$ in space, where $h$ denotes the grid step. This result suggests that the proposed algorithm is less time consuming, faster and more efficient than a broad range of numerical methods widely discussed in the literature for the considered problem. Numerical experiments confirm the theory and demonstrate the efficiency and utility of the three-level time split high-order formulation.

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A well-known experimental setup for the study of segregation  by size in a dry granular medium consists of two layers of spheres composed of large and small rigid spheres. These layers are contained within an annular region of concentric cylinders covered above and below by plates. One of the cylinders is rotated and thereby applies shear to the granular mixture. The spheres will then mix and the large ones rise while the small ones settle in vertical direction. This phenomenon can be modelled by  a conservation law whose flux involves a piecewise constant or smooth coefficient [L. May, M. Shearer, and K. Daniels, J. Nonlin. Sci., 20 (2010), pp. 689--707] that describes dependence of the shear rate on depth. This model is solved  by the hyperfast front tracking method adapted to a conservation law with discontinuous flux. In this way the  coefficient can efficiently be identified from experimental observations. Numerical examples are presented.

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