A class of multidomain hybrid methods of direct discontinuous Galerkin (DDG) methods and central difference (CD) schemes for the viscous terms is proposed in this paper. Both conservative and nonconservative coupling modes are discussed. To treat the shock wave, the nonconservative coupling mode automatically switch to conservative coupling mode to preserve the conservative property when discontinuities pass through the artificial interface. To maintain the accuracy of the hybrid methods, the Lagrange interpolation polynomials and their derivatives are reconstructed to handle the coupling cells in the DDG subdomain, while the values of ghost points for the CD subdomain are calculated by the approximate polynomials from the DDG methods. The linear stabilities of these methods are demonstrated in detail through von-Neumann analysis. The multidomain hybrid DDG and CD methods are then extended to one- and two-dimensional hyperbolic-parabolic equations. Numerical results validate that the multidomain hybrid methods are high-order accurate in the smooth regions, robust for viscous shock simulations and capable to save computational cost.

In this study, based on an iterative method to solve nonlinear equations, a third-order convergent iterative method to compute the Moore-Penrose inverse of a tensor with the Einstein product is presented and analyzed. Numerical comparisons of the proposed method with other methods show that the average number of iterations, number of the Einstein products, and CPU time of our method are considerably less than other methods. In some applications, partial and fractional differential equations that lead to sparse matrices are considered as prototypes. We use the iterates obtained by the method as a preconditioner, based on tensor form to solve the multilinear system $\mathcal{A}∗_N\mathcal{X}=\mathcal{ B}.$ Finally, several practical numerical examples are also given to display the accuracy and efficiency of the new method. The presented results show that the proposed method is very robust for obtaining the Moore-Penrose inverse of tensors.

This study aims to investigate the polar decomposition of tensors with the Einstein product for the first time. The polar decomposition of tensors can be computed using the singular value decomposition of the tensors with the Einstein product. In the following, some iterative methods for finding the polar decomposition of matrices have been developed into iterative methods to compute the polar decomposition of tensors. Then, we propose a novel parametric iterative method to find the polar decomposition of tensors. Under the obtained conditions, we prove that the proposed parametric method has the order of convergence four. In every iteration of the proposed method, only four Einstein products are required, while other iterative methods need to calculate multiple Einstein products and one tensor inversion in each iteration. Thus, the new method is superior in terms of efficiency index. Finally, the numerical comparisons performed among several well-known methods, show that the proposed method is remarkably efficient and accurate.

We apply the local method of fundamental solutions (LMFS) to boundary value problems (BVPs) for the Laplace and homogeneous biharmonic equations in annuli. By appropriately choosing the collocation points, the LMFS discretization yields sparse block circulant system matrices. As a result, matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs) can be used for the solution of the systems resulting in considerable savings in both computational time and storage requirements. The accuracy of the method and its ability to solve large scale problems are demonstrated by applying it to several numerical experiments.

In this paper, we formulate and analyse a kind of parareal-RKN algorithms with energy conservation for Hamiltonian systems. The proposed algorithms are constructed by using the ideas of parareal methods, Runge-Kutta-Nyström (RKN) methods and projection methods. It is shown that the algorithms can exactly preserve the energy of Hamiltonian systems. Moreover, the convergence of the integrators is rigorously analysed. Three numerical experiments are carried out to support the theoretical results presented in this paper and show the numerical behaviour of the derived algorithms.

The simulation of multi-domain, multi-physics mathematical models with uncertain parameters can be quite demanding in terms of algorithm design and computation costs. Our main objective in this paper is to examine a physical interface coupling between two random dissipative systems with uncertain parameters. Due to the complexity and uncertainty inherent in such interface-coupled problems, uncertain diffusion coefficients or friction parameters often arise, leading to considering random systems. We employ Monte Carlo methods to produce independent and identically distributed deterministic heat-heat model samples to address random systems, and adroitly integrate the ensemble idea to facilitate the fast calculation of these samples. To achieve unconditional stability, we introduce the scalar auxiliary variable (SAV) method to overcome the time constraints of the ensemble implicit-explicit algorithm. Furthermore, for a more accurate and stable scheme, the ensemble data-passing algorithm is raised, which is unconditionally stable and convergent without any auxiliary variables. These algorithms employ the same coefficient matrix for multiple linear systems and enable easy parallelization, which can significantly reduce the computational cost. Finally, numerical experiments are conducted to support the theoretical results and showcase the unique features of the proposed algorithms.

In this paper, we propose a method for solving semilinear elliptical equations using a ResNet with ${\rm ReLU}^2$ activations. Firstly, we present a comprehensive formulation based on the penalized variational form of the elliptical equations. We then apply the Deep Ritz Method, which works for a wide range of equations. We obtain an upper bound on the errors between the acquired solutions and the true solutions in terms of the depth $\mathcal{D},$ width $\mathcal{W}$ of the ${\rm ReLU}^2$ ResNet, and the number of training samples $n.$ Our simulation results demonstrate that our method can effectively overcome the curse of dimensionality and validate the theoretical results.

In this paper, we construct, analyze, and numerically validate a family of divergence-free virtual elements for Stokes equations with nonlinear damping on polygonal meshes. The virtual element method is $\mathbf{H}^1$-conforming and exact divergence-free. By virtue of these properties and the topological degree argument, we rigorously prove the well-posedness of the proposed discrete scheme. The convergence analysis is carried out, which imply that the error estimate for the velocity in energy norm does not explicitly depend on the pressure. Numerical experiments on various polygonal meshes validate the accuracy of the theoretical analysis and the asymptotic pressure robustness of the proposed scheme.

In this work, we propose an explicit second order scheme for decoupled mean-field forward backward stochastic differential equations with jumps. The stability and the rigorous error estimates are presented, which show that the proposed scheme yields a second order rate of convergence, when the forward mean-field stochastic differential equation is solved by the weak order 2.0 Itô-Taylor scheme. Numerical experiments are carried out to verify the theoretical results.