This paper proposes a high-order maximum-principle-preserving (MPP) conservative scheme for convection-dominated diffusion equations. For high-order spatial discretization, we first use the fifth-order weighted compact nonlinear scheme (WCNS5) for the convection term and the sixth-order central difference scheme for the diffusion term. Owing to the nonphysical oscillations caused by the high-order scheme, we further adopt a parameterized MPP flux limiter by modifying a high-order numerical flux toward a lower-order monotone numerical flux to achieve the maximum principle. Subsequently, the resulting spatial scheme is combined with third-order strong-stability-preserving Runge-Kutta (SSPRK) temporal discretization to solve convection-dominated diffusion problems. Several one-dimension (1D) and two-dimension (2D) numerical experiments show that the proposed scheme maintains up to fifth-order accuracy and strictly preserves the maximum principle. The results indicate the proposed scheme’s strong potential for solving convection-dominated diffusion and incompressible flow problems.

A simple weighted essentially non-oscillatory (SWENO) scheme for solving the convection-diffusion equation is proposed in this paper, where a seventh-order SWENO method and the third-order strong stability preserving (SSP) Runge-Kutta method are adopted for discretizing the space and time, respectively. Then the hyperbolic and diffusive part can achieve the seventh- and sixth-order accuracy, respectively. The proposed method has the following advantages. Firstly, negative linear weights are avoided. Secondly, one reconstruction with one stencil is required for the computation of convective and diffusive fluxes. Finally, the new method does not require the transformation while the diffusion coefficients are degenerate. Numerical examples demonstrate that the new method can achieve sixth-order accuracy in the smooth region and guarantee non-oscillatory properties for the discontinuous problems for one- and two-dimensional cases.

A class of efficient multistep methods for forward backward stochastic differential equations (FBSDEs) are proposed and studied in this paper. According to the characteristic of our multistep methods and replacing the corresponding Brownian motion increments with appropriate two-point distributed random variables, we obtain a very efficient algorithm to approximate the conditional expectations involved in the multistep methods. It is thanks to this efficient algorithm that we can obtain a class of efficient fully discrete form of high-order methods for the FBSDEs. Finally, some numerical results are presented to illustrate the efficiency of our multistep methods.

In this paper, the proposed method utilizes finite difference and Fourier series methods to calculate the mean curvature vectors and normalized curvature weights at the vertices of manifold triangular meshes. Specifically, this stable method achieves the $L^2$ convergence of the mean curvature vector. Furthermore, by comparing the method proposed in this paper with previously proposed classical methods, the results show that this method effectively balances precision and stability, and significantly reduces the larger errors observed on triangular meshes with small angles (approximately $0^◦$).

This paper develops a class of robust weak Galerkin methods for stationary incompressible convective Brinkman-Forchheimer equations. The methods adopt piecewise polynomials of degrees $m (m ≥ 1)$ and $m−1$ respectively for the approximations of velocity and pressure variables inside the elements and piecewise polynomials of degrees $k $$(k=m−1, m),$ and $m$ respectively for their numerical traces on the interfaces of elements, and are shown to yield globally divergence-free velocity approximation. Existence and uniqueness results for the discrete schemes, as well as optimal a priori error estimates, are established. A convergent linearized iterative algorithm is also presented. Numerical experiments are provided to verify the performance of the proposed methods.

Deep learning algorithms have been successfully applied to numerically solve linear Kolmogorov partial differential equations (PDEs). A recent research shows that if the initial functions are bounded, the empirical risk minimization (ERM) over clipped ReLU networks generalizes well for solving the linear Kolmogorov PDE. In this paper, we propose to use a truncation technique to extend the generalization results for polynomially growing initial functions. Specifically, we prove that under an assumption, the sample size required to achieve an generalization error within $\varepsilon$ with a confidence level $\varrho$ grows polynomially in the size of the clipped neural networks and $(\varepsilon^{−1}, \varrho^{−1}),$ which means that the curse of dimensionality is broken. Moreover, we verify that the required assumptions hold for Black-Scholes PDEs and heat equations which are two important cases of linear Kolmogorov PDEs. For the approximation error, under certain assumptions, we establish approximation results for clipped ReLU neural networks when approximating the solution of Kolmogorov PDEs. Consequently, we establish that the ERM over artificial neural networks indeed overcomes the curse of dimensionality for a larger class of linear Kolmogorov PDEs.

In the paper, we introduce two new classes of orthogonal functions, new log orthogonal functions (NLOFs) and generalized new log orthogonal functions (GNLOFs), which are based on generalized Laguerre polynomials. We construct basis approximations analysis for the new orthogonal functions and apply them to solve several typical differential equations. Our errors analysis and numerical results show that our methods based on the new orthogonal functions are particularly suitable for functions which have weak singularities at one endpoint, and can obtain exponential convergence rate, as opposed to low algebraic rates if usual orthogonal polynomials are used.

A third-order accurate implicit-explicit Runge-Kutta time marching numerical scheme is proposed and implemented for the Landau-Lifshitz-Gilbert equation, which models magnetization dynamics in ferromagnetic materials, with arbitrary damping parameters. This method has three remarkable advantages: (1) only a linear system with constant coefficients needs to be solved at each Runge-Kutta stage, which greatly reduces the time cost and improves the efficiency; (2) the optimal rate convergence analysis does not impose any restriction on the magnitude of damping parameter, which is consistent with the third-order accuracy in time for 1-D and 3-D numerical examples; (3) its unconditional stability with respect to the damping parameter has been verified by a detailed numerical study. In comparison with many existing methods, the proposed method indicates a better performance on accuracy and efficiency, and thus provides a better option for micromagnetics simulations.

The numerical simulation of fluid dynamics problems plays an important role in the studies of laser inertial confinement fusion (ICF) and magnetic confinement fusion (MCF). Due to the complexity of physical processes and the large deformations of flow field, the numerical simulation of these problems has considerable difficulty. Taking the advantages of the discontinuous Galerkin (DG) method and the Lagrangian scheme, a second-order Lagrangian type scheme for solving the compressible Euler equations is developed on unstructured triangular meshes and implemented by the Runge-Kutta (RK) DG method in this paper. The solver of node velocity in the scheme has good adaptability for many problems. Without considering the material derivatives of basis functions and the Jacobian matrix associated with the map between Lagrangian space and Eulerian space, our scheme is relatively succinct. A HWENO reconstruction is used to eliminate the false oscillations whose stencils involve only the von Neumann neighborhood so that the scheme keeps compact within the DG method. Finally, some numerical examples are presented to illustrate the accuracy, resolution, and robustness of our scheme.

In this article, we propose a novel stabilized physics informed neural networks method (SPINNs) for solving wave equations. In general, this method not only demonstrates theoretical convergence but also exhibits higher efficiency compared to the original PINNs. By replacing the $L^2$ norm with $H^1$ norm in the learning of initial condition and boundary condition, we theoretically proved that the error of solution can be upper bounded by the risk in SPINNs. Based on this, we decompose the error of SPINNs into approximation error, statistical error and optimization error. Furthermore, by applying the approximating theory of $ReLU^3$ networks and the learning theory on Rademacher complexity, covering number and pseudo-dimension of neural networks, we present a systematical non-asymptotic convergence analysis on our method, which shows that the error of SPINNs can be well controlled if the number of training samples, depth and width of the deep neural networks have been appropriately chosen. Two illustrative numerical examples on 1-dimensional and 2-dimensional wave equations demonstrate that SPINNs can achieve a faster and better convergence than classical PINNs method.