This article describes developing and improving sixth-order characteristic-wise Weighted Essentially Non-Oscillatory (WENO) finite difference schemes. These schemes are specially designed to solve scalar and system hyperbolic conservation laws with high accuracy/resolution and robustness. The schemes have been enhanced by using a new reference global smoothness indicator, which ensures the optimal order of accuracy for smooth solutions. The schemes also incorporate affine-invariant nonlinear Ai-weights that are independent of the scaling of solution and the choice of sensitivity parameter. The improved nonlinear weights enhance the essentially non-oscillatory (ENO) capturing of discontinuities and minimize the numerical dissipation, especially for long-time simulations. The study also introduces the positivity-preserving limiter to ensure that the numerical solution of Euler equations is physically valid. The effectiveness of improved schemes is demonstrated through one- and two-dimensional benchmark shock-tube problems, such as the Sod, Lax, and Woodward-Colella problems. The improved schemes are compared with other WENO schemes in terms of accuracy, resolution, ENO, and robustness.

In this paper, we propose an absolutely stabilized mixed virtual element method (mixed VEM) for the incompressible Stokes equations. We employ the $C^0$-conforming virtual element space for the velocity and discontinuous polynomial space for the pressure. The velocity-pressure pair contains "arbitrary-order" polynomials, and stability is guaranteed by the "absolutely stabilized finite element formulation" originally introduced in [21]. We establish error estimates in the energy norm for both velocity and pressure, as well as an $L^2$-error estimate for the velocity. Several numerical experiments validate the theoretical analysis.

We study the strong convergence of the general one-step method for neutral stochastic delay differential equations with a variable delay. First, we give the notions of C-stability and B-consistency, and then establish a fundamental theorem of strong convergence for the general one-step method solving the nonlinear neutral stochastic delay differential equations, where the corresponding diffusion coefficient with respect to the non-delay variables is highly nonlinear. Then, we construct the split-step backward Euler method which is a special implicit one-step method, and prove that it is C-stable, B-consistent, and strongly convergent of order 1/2. Finally, we give some numerical experiments to support the obtained results.

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.

The thermal lattice Boltzmann equation (TLBE) has superior numerical stability as an explicit algorithm. However, its applications on nonuniform meshes are complicated. This paper clarifies the intrinsic mechanism for stabilizing computations in TLBE and proposes two solvers that combine the numerical stability of TLBE and flexible finite difference/volume schemes for nonuniform meshes. Through a brief review of the lattice Boltzmann method, it is concluded that the entropy increase of the collision operator is essential for numerical stability. This paper first proposes a macroscopic entropy-increasing (MEI) model for convection-diffusion problems by combining the MEI process and TLBE. The von Neumann stability analysis proves that the MEI model has no upper limit for mesh Fourier number. However, the accuracy of the MEI model is found to be sensitive to higher-order deviation terms. Therefore, a hybrid model that combines the MEI and the equilibrium-moment-based models is proposed to solve the problem. The von Neumann stability analysis demonstrates that the hybrid model can completely recover the numerical stability of TLBE. Numerical investigations validate the good stability and accuracy of the hybrid model. Most importantly, it can be easily applied to nonuniform meshes, whereas implementing TLBE on nonuniform meshes is relatively complicated.

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.

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.

The conventional method of polynomial particular solutions is only applicable to partial differential equations on the real number field. Building on this method, this paper proposes two novel approaches for numerical simulation of the time-dependent Schrödinger equation. Under the assumption of treating the time variable as an ordinary spatial variable, the first approach approximates the real and imaginary parts of the equation using two different sets of linear combinations, which are then substituted into the corresponding original governing equations to form a coupled differential equation. The second approach is derived based on the basic form of complex coefficient partial differential equations and involves polynomial particular solutions with imaginary terms. Using the same numerical examples, compared to conventional finite difference method, Fourier spectral method and radial basis function collocation method, this algorithm's stability is not limited by the grid ratio of the time step and spatial step. It is not only simple and feasible but also suitable for solving high-dimensional problems.

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.

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.

This paper mainly focuses on the discontinuous Galerkin (DG) method for solving the semi-explicit index-1 integro-differential algebraic equation (IDAE), which is a coupled system of Volterra integro-differential equations (VIDEs) and second-kind Volterra integral equations (VIEs). The DG approach is applied to both the VIDE and VIE components of the system. The global convergence respectively in the $L^2$-norm and $L^\infty$-norm is established, and the local superconvergence for VIDE component is obtained. Furthermore, numerical examples are presented to validate the theoretical convergence and superconvergence results.