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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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This paper proposes an explicit method for the coupled forward backward stochastic differential equations (FBSDEs). Our method combines skillfully a weak second order stochastic Runge-Kutta method for solving forward equations with a two-step method for solving backward equations. We give a convergence theorem for the proposed method when the FBSDEs is weakly coupled (the forward equations are independent of the variable $Z$). Finally, some numerical results are presented, and the numerical results show that our method still works well even if the forward equations depend on $Z$.

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The discretization of the fractional regime-switching option pricing model leads to the linear system with the block diagonal and Kronecker product structure. A fast matrix splitting iteration method is presented to solve the discrete system. Theoretical analyses prove the convergence of the proposed iteration method. Numerical experiments show that the new method is efficient and outperforms the existing methods in terms of the number of iteration steps and the elapsed CPU time.

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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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This paper presents ALM-PINN, an adaptive physical informed neural network algorithm optimized by Levenberg-Marquardt. ALM-PINN is tailored to overcome challenges for solving singular perturbation problems (SPP). Traditional neural network-based solvers reframe solving differential equations task as a multi-objective optimization problem involving residual or Ritz error. However, significant disparities in the magnitudes of loss functions and their gradients frequency result in suboptimal training and convergence challenges. Addressing these issues, ALM-PINN introduces a learnable parameter for the perturbation parameter and constructs a two-terms loss function. The first loss term emphasizes approximating the governing equation, while the second term minimizes the difference between perturbation and learnable parameters. This adaptive learning strategy not only mitigates convergence issues in directly solving SPP but also alleviates the computational burden with asymptotic iteration from a large initial value. For one-dimensional tasks, ALM-PINN enhances training efficiency and reduces complexity by enforcing hard constraints on boundary conditions, streamlining the loss function sub-terms. The efficacy of ALM-PINN is evaluated on five SPPs, demonstrating its ability to capture sharp changes in physical quantities within the boundary layer, even with small perturbation coefficients. Furthermore, ALM-PINN exhibits reduced errors in both $L_\infty$ and $L_2$ norms, coupled with improved convergence speed and stability.

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In this paper, we present the design of a posteriori error estimator for the plasmon phenomenon on the graphene surface and propose a method to achieve local high-precision numerical calculations when plasmon phenomena occur on the graphene surface. We provide a lower bound estimate for the posteriori error estimator, along with a proof of convergence. Specifically, the constructed posterior error estimator enables local refinement in regions where the error is significant at the graphene interface. Firstly, we outline the construction of the posterior error estimator and provide the proof of its lower bound. Secondly, we establish the convergence of the Adaptive Edge Finite Element Method  (AEFEM). Finally, we present numerical results that validate the effectiveness of the error estimator.

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In this paper we study the nonconforming Crouzeix-Raviart element and the enriched Crouzeix-Raviart element methods firstly for the transmission eigenvalue problem of inhomogeneous media and the modified transmission eigenvalue problem in inverse scattering. Using $\mathbb{T}$-coercivity method, we prove the convergence and the a priori error estimate of approximate eigenpair for the transmission eigenvalue problem, and based on the obtained results we prove the a priori error estimate for the modified transmission eigenvalue problem  by the $\mathbb{T}$-coercivity method and G{\aa}rding inequality, and further prove that the discrete eigenvalues for the problem with metamaterial background approximate the exact eigenvalue from above. We also carry out numerical experiments to validate the theoretical findings and the efficiency of the proposed methods.

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Dynamic stall poses a significant aerodynamic challenge for vertical axis wind turbines (VAWTs) and serves as a key impediment to enhancing their overall efficiency. Consequently, to elucidate the underlying mechanism of dynamic stall and facilitate a clearer understanding of the entire dynamic stall process, a numerical investigation is conducted over the blades of a VAWT system. A two-dimensional numerical simulation is conducted by employing unsteady Reynolds-averaged Navier-Stokes (RANS) calculations with a Reynolds stress turbulence model. Four real-time key indicators are generalized to quantitatively characterize dynamic stall over the VAWTs. These indicators have been demonstrated to accurately predict the initiation and detachment times of the dynamic stall vortex (DSV), providing deeper insights into the underlying mechanism of DSV development. Additionally, this study explores the impact of blade thickness on the dynamic stall of VAWTs. Three airfoils from the NACA 4-digit series, each featuring distinct ratios of thickness to chord length, are examined. As the blade cross-section transitions from NACA 0012 to NACA 0015 and NACA 0018, heightened blade thickness results in a delayed onset of dynamic stall, shifting from leading edge stall to trailing edge stall. Moreover, the blade thickness significantly influences the efficiency of VAWT system, with increased airfoil thickness contributing to an improved utilization of wind energy.

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In this paper, we propose a new inexact iterative subspace projection method for real symmetric positive definite generalized eigenvalue problems based on the WYD (an abbreviation of the initials of the proposers: Wilson, Yuan, and Dickens [1]) method. Firstly an analysis of the convergence condition of the approximate eigenvectors is given when ${\bf Ar}_{k+1}=\lambda{\bf Br}_k$ is inexactly solved by the Krylov subspace methods. Then the inexact iterative WYD (IIWYD) method is constructed, which utilizes the approximate Ritz subspace generated by the WYD method with an inexact solver as the main search space. The IIWYD method improves the quality of the search space during the iterative process, significantly reduces the number of iteration steps, and improves the overall computational efficiency and stability. The results of numerical experiments show that the IIWYD method is more efficient and stable compared to the locally optimal block preconditioned conjugate gradient (LOBPCG) method and the Jacobi-Davidson (JD) method. In addition, we also discuss the effects of the refined strategy and the conjugate strategy in our method.

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Based on the generalized scalar auxiliary variable approach and vector penalty projection method, some fully discrete schemes with first- and second-order accuracy in time direction are constructed for solving the incompressible magnetohydrodynamic model. It is a combination of mixed finite element approximation for spatial discretization and first-order backward Euler/second-order backward differential formula for temporal discretization. The proposed schemes own several features: it decouples unknown physical variables and linearizes the nonlinear terms, then it only needs to solve some linear equations at each temporal level; although the divergence of numerical velocity is not exactly equal to zero, it can approximately meet the mass conservation when one takes small penalty parameter; while the computation of the velocity and pressure are decoupled, numerical results show that the velocity and pressure can reach second-order accuracy in time. The resulting schemes are supported by numerical analysis and simulation.

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In this paper, we develop through a careful selection of the auxiliary variables and numerical fluxes an energy conservative local discontinuous Galerkin (LDG) method based on a hybrid form of the general Euler-Korteweg (EK) equations with a variable capillarity coefficient. This energy conservative LDG discretization is of optimal order of accuracy for alternating numerical fluxes, but not for central numerical fluxes which result in the reduction of one order of accuracy when odd degree polynomial basis functions are used. Also, a relatively simple energy conservative LDG discretization for the EK-equations with an irrotational velocity field is presented. Due to the presence of a highly nonlinear third-order spatial derivative term, which originates from the divergence of the Korteweg stress tensor, we employ the novel semi-implicit spectral deferred correction (SDC) method as temporal discretization. The SDC method can be applied to highly nonlinear ordinary differential equations (ODEs) without separating stiff and non-stiff components and is numerically stable for a time step proportional to the mesh size. Numerical experiments, including ones with adaptive meshes, are performed to illustrate the accuracy and capability of the proposed methods to solve the EK-equations.

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A new extrapolation cascadic multigrid (EXCMG) method is developed to solve large sparse symmetric positive definite systems resulting from the classical 25-point finite-difference discretizations of the three-dimensional (3D) biharmonic equation on rectangular domains. We accomplish this by designing a quartic interpolation-based prolongation operator and using the symmetric successive over-relaxation (SSOR) preconditioned CG method as the multigrid smoother. For the new prolongation operator, quartic interpolations are used for the finite difference solutions on coarse and fine grids twice and once so that two approximations can be obtained on the next finer grid,  and then the completed Richardson extrapolation is used for these two approximations to obtain an excellent initial guess on the next finer grid. The proposed EXCMG method with the new prolongation operator is easier to implement than the original EXCMG method. Numerical experiments demonstrate that the new EXCMG is a highly efficient solver for the 3D biharmonic equation and is considerably faster than the original EXCMG method and the aggregation-based algebraic multigrid (AGMG) method developed by Y. Notay. The proposed EXCMG method can solve discrete 3D biharmonic equations with more than 100 million unknowns in dozens of seconds.

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The mean field games (MFG) theory has broad application in mathematical modeling of social phenomena. The Mean Field Games System (MFGS) is the key to the MFG theory. This is a system of two nonlinear parabolic partial differential equations with two opposite directions of time $t\in (0,T). $ The topic of Coefficient Inverse Problem (CIPs) for the MFGS is a newly emerging one. A CIP for the MFGS is studied. The input data are Dirichlet and Neumann boundary conditions either on a part of the lateral boundary (incomplete data) or on the whole lateral boundary (complete data). In addition to the initial conditions at $\left\{t=0\right\}, $ terminal conditions at $\left\{t=T\right\} $ are given. The terminal conditions mean the final overdetermination. The necessity of assigning all these input data is explained. H\"{o}lder and Lipschitz stability estimates are obtained for the cases of incomplete and complete data respectively. These estimates imply uniqueness of the CIP.

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In this paper, a pointwise goal-oriented residual-based a posteriori error estimator is proposed for linear elliptic equations with restricted source terms. The pointwise error is directly estimated by introducing the dual problem with a Dirac delta source term instead of using classical mollification technique. The goal-oriented error estimator is proved to be the upper bound of the pointwise error. Numerical experiments show the advantage of the adaptive finite element method (AFEM) based on this error estimator, which can preserve the monotonicity of the pointwise error, compared with the goal-oriented AFEM using the mollification technique.

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This paper proposes a direct Eulerian generalized Riemann problem (GRP) scheme for two-layer shallow water equations.  The model takes into account the distinctions between different densities and velocities, and is obtained by taking the vertical averaging across the layer depth. The source terms generated from the mass and momentum exchange prevent us from solving the Riemann problem analytically. We consider an equivalent conservative two-layer model which describes the horizontal velocity with two degrees of freedom. The rarefaction wave and the shock wave are analytically resolved by using the Riemann invariants and Rankine-Hugoniot condition, respectively. Numerical simulations are also given on some typical problems in order to verify the good performance of the GRP method.

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