• Abstract

Although the genuinely two-dimensional HLL-CPS solver holds the inherent multidimensionality property and capability of resolving contact discontinuities, the conventional low-order (second-order and below) reconstruction methods still limits its application in the two-dimensional complex flows involving shock waves and shear layers. A fifth-order reconstruction method is proposed for the genuinely two-dimensional HLL-CPS solver. The conserved variable vectors at the midpoints of interfaces are approximated by the fifth-order 1D WENO reconstruction. Meanwhile, variables at the corners are evaluated by a dimension-by-dimension reconstruction method consisting of a number of 1D fifth-order WENO sweeps. To avoid introducing spurious oscillations, each reconstruction is carried out in the corresponding local characteristic fields. Numerical results of several benchmark tests indicate  the higher-order accuracy and the multidimensionality property of the proposed scheme. Compared with the 1D HLLE, HLLC and HLL-CPS schemes, the proposed high-order genuinely two-dimensional HLL-CPS solver provides higher resolution for contact discontinuities and presents better robustness against the shock anomalies.

• General Summary

• Abstract

In this paper, we  develop  a new reduced basis (RB) method, named as Single Eigenvalue Acceleration Method (SEAM), for second order parabolic equations with homogeneous Dirichlet boundary conditions. The high-fidelity   numerical method  adopts the backward Euler scheme and  conforming simplicial finite elements  for the temporal and spatial discretizations, respectively.  Under the assumption that the time step size is sufficiently small and time steps   are not very large,  we show that the singular value distribution of the high-fidelity solution matrix $U$ is close to that of a rank one matrix.  We select the eigenfunction associated to the principal eigenvalue of the matrix $U^\top U$   as the basis of    Proper Orthogonal Decomposition (POD) method so as to obtain SEAM and a parallel SEAM. Numerical experiments confirm the efficiency of the new method.

• General Summary

• Abstract

We present a novel two-grid compact finite difference scheme for the viscous Burgers' equation in this paper, where the second-order Crank-Nicolson method is used to deal with the time marching, the compact finite difference formula is used to approximate the spatial second-order term, and the nonlinear convection term is discretized using the developed nonlinear fourth-order operator, providing the scheme with both high fourth-order spatial convergence and a low computational cost. The scheme is then established in three steps, with the first step being the construction of a nonlinear coarse-grid compact finite difference scheme that is solved iteratively using a fixed point iterative method, the second step being the application of the Lagrange interpolation formula to obtain a rough solution on the fine grid, and the third step being the development of the linearized fine-grid compact finite difference scheme. We also perform a convergence and stability analysis on the developed scheme, and the results show that the scheme can achieve spatial fourth-order and temporal second-order convergence. Finally, a number of numerical examples are provided to validate the theoretical predictions.

• General Summary

• Abstract

This paper deals with an efficient two-step time split explicit/implicit scheme applied to a two-dimensional nonlinear unsteady convection-diffusion-reaction equation. The computational cost of the new algorithm at each time level is equivalent to solving a pentadiagonal matrix equation with strictly dominant diagonal elements. Such a bandwidth matrix can be easily inverted using the Gaussian Decomposition and the corresponding linear system should be solved by the back substitution method. The proposed approach is unconditionally stable, temporal second-order accuracy and fourth-order convergence in space. These results suggest that the developed technique is faster and more efficient than a large class of numerical methods studied in the literature for the considered initial-boundary value problem. Numerical experiments are carried out to confirm the theoretical analysis and to demonstrate the performance of the constructed numerical scheme.

• General Summary

• Abstract

Although there are many improvements to WENO3-Z with the target to achieve the optimal order in the occurrence of the first-order critical point ($CP_1$), they mainly address resolution performance, while the robustness of schemes is of less concern and lacks understanding accordingly. In light of our analysis considering the occurrence of critical points within grid intervals, we theoretically prove that it is impossible for a scale-independent scheme that has the stencil of WENO3-Z to fulfill the above order achievement, and current scale-dependent improvements barely fulfill the job when $CP_1$ occurs at the middle of the grid cell. In order to achieve scale-independent improvements, we devise new smoothness indicators that increase the error order from 2 to 4 when $CP_1$ occurs and perform more stably. Meanwhile, we construct a new global smoothness indicator that increases the error order from 4 to 5 similarly, through which new nonlinear weights with regard to WENO3-Z are derived and new scale-independent improvements, namely WENO-Z$_\text{{ES2}}$ and -Z$_\text{{ES3}}$, are acquired. Through 1D scalar and Euler tests, as well as 2D computations, in comparison with typical scale-dependent improvements, the following performances of the proposed schemes are demonstrated: the schemes can achieve third-order accuracy at $CP_1$ no matter its location in the stencil, indicate high resolution in resolving flow subtleties, and manifest strong robustness in hypersonic simulations (e.g., the accomplishment of computations on hypersonic half-cylinder flow with Mach numbers reaching 16 and 19, respectively, as well as essentially non-oscillatory solutions of inviscid sharp double cone flow at $M=9.59$), which contrasts the comparative WENO3-Z improvements.

• General Summary

• Abstract

In this paper, we consider an implicit method for solving the nonlinear time-space fractional Ginzburg-Landau equation. The scheme is based on the $L2$-$1_\sigma$ formula to approximate the Caputo fractional derivative and the weighted and shifted Gr{\"u}nwald difference method to approximate the Riesz space fractional derivative. In order to overcome the non-local property of Riesz space fractional derivatives and the historical dependence brought by Caputo time fractional derivatives, this paper introduces the fractional Sobolev norm and the fractional Sobolev inequality. It is proved in detail that the difference scheme is stable and uniquely solvable by the discrete energy method. In particular, the difference scheme is unconditionally stable when $\gamma\leq 0$, where $\gamma$ is a coefficient of the equation.  Moreover, the scheme is shown to be convergent in $l_h^2$ norm at the optimal order of $\mathcal{O}(\tau^2+h^2)$ with time step $\tau$ and mesh size $h$. Finally, we provide a linearized iterative algorithm, and the numerical results are present to verify the accuracy and efficiency of the proposed scheme.

• General Summary

• Abstract

A quadratic spline collocation method combined with the Crank-Nicolson time discretization of the space fractional diffusion equations gives discrete linear systems, whose coefficient matrix is the sum of a tridiagonal matrix and two diagonal-multiply-Toeplitz-like matrices. By exploiting the Toeplitz-like structure, we split the Toeplitz-like matrix as the sum of a Toeplitz matrix and a rank-2 matrix and Strang's circulant preconditioner is constructed to accelerate the convergence of Krylov subspace method like generalized minimal residual method for solving the discrete linear systems. In theory, both the invertibility of the proposed preconditioner and the clustering spectrum of the corresponding preconditioned matrix are discussed in detail. Finally, numerical results are given to demonstrate that the performance of the proposed preconditioner is better than that of the generalized T. Chan's circulant preconditioner proposed recently by Liu et al. (J. Comput. Appl. Math., 360 (2019),  pp. 138--156) for solving the discrete linear systems of one-dimensional and two-dimensional space fractional diffusion equations.

• General Summary

• Abstract

In this paper, the superconvergence error estimate of a low-order conforming mixed finite element scheme, which is called bilinear-constant scheme, for the Stokes equations with damping is established. In terms of the integral identity technique and dealing with the damping term carefully, the superclose estimates between the interpolation of the exact solution and the finite element solution for the velocity in $H^1$-norm and the pressure in $L^2$-norm are first derived. Then, the global superconvergence results for the velocity in $H^1$-norm and the pressure in $L^2$-norm are derived by a simple postprocessing technique with an economical workload. Finally, some numerical results are presented to demonstrate the correctness of the theoretical analysis.

• General Summary

• Abstract

This paper investigates the reduction of backscatter radar cross

section (RCS) for cavities embedded in the ground plane. It is established by placing a multilayered radar absorbing material (RAM) at the bottom of the cavities. The minimization of the backscatter RCS is formulated as a bound constrained optimization problem. A new vertical mode expansion method is developed to solve the scattering problem, and it is integrated into the gradient projection method to obtain the optimal composite coating. Numerical examples show that the method we developed is efficient and the backscatter RCS can be significantly reduced.

• General Summary

• Abstract

In this paper, we develop an adaptive high-order surface finite element method (FEM) incorporating spectral deferred correction method for chain contour discretization to solve polymeric self-consistent field equations on general curved surfaces. The high-order surface FEM is obtained by the high-order surface geometrical approximation and the high-order function space approximation. Numerical results demonstrate that the precision order of these methods is consistent with theoretical prediction. In order to describe the sharp interface in the strongly segregated system more accurately, an adaptive FEM equipped with a new \textit{Log} marking strategy is proposed. The \textit{Log} marking strategy can not only label the elements that need to be refined or coarsened, but also give the refined or coarsened times, which can make full use of the information of a posterior error estimator and improve the efficiency of the adaptive algorithm. To demonstrate the power of our approach, we investigate the self-assembled patterns of diblock copolymers on several distinct curved surfaces.  Numerical results illustrate the efficiency of the proposed method, especially for strongly segregated systems with economical discretization nodes.

• General Summary

• Abstract

In this paper, we propose a nonconvex and nonsmooth model for image restoration with Cauchy noise removal. The regularization term is composed of a nonconvex Lipschitz potential function with first-order derivatives of an image, while the data fidelity term is introduced based on the maximum a posteriori estimator to Cauchy distribution. To handle the nonconvexity of regularizers, we adopt the proximal linearization technique to convert the original nonconvex model to a series of convex models, which can be easily implemented by alternating direction method with multipliers. Based on the Kurdyka-{\L}ojasiewicz property, we can verify the global convergence of the proposed algorithm by using an abstract convergence framework.

Numerical experiments and comparisons indicate that our method obtains good restorations and is effective for better preserving edges.

• General Summary

• Abstract

Consider a Urysohn integral equation $x - \mathcal{K} (x) = f$, where $f$ and the integral operator $\mathcal{K}$ with kernel of the type of Green's function are given. In the computation of approximate solutions of the given integral equation by Galerkin method, all the integrals are needed to be evaluated by some numerical integration formula. This gives rise to the discrete version of the Galerkin method. For $r \geq 1$, a space of piecewise polynomials of degree $\leq r-1$ with respect to a uniform partition is chosen to be the approximating space. For the appropriate choice of a numerical integration formula, an asymptotic series expansion of the discrete iterated Galerkin solution is obtained at the above partition points. Richardson extrapolation is used to improve the order of convergence. Using this method we can restore the rate of convergence when the error is measured in the continuous case. Numerical examples are given to illustrate this theory.

• General Summary

• Abstract

This paper is devoted to adaptive finite element method for the nonlinear Schr\"{o}dinger equation. The adaptive method is based on the extrapolation technology and a second order accurate, linear and mass preserving finite element scheme. For error control, we take the difference between the numerical gradient and the recovered gradient obtained by the superconvergent cluster recovery method as the spatial discretization error estimator and the difference of numerical approximations between two consecutive time steps as the temporal discretization error estimator. A time-space adaptive algorithm is developed for numerical approximation of the nonlinear Schr\"{o}dinger equation. Numerical experiments are presented to illustrate the reliability and efficiency of the proposed error estimators and the corresponding adaptive algorithm.

• General Summary

• Abstract

In this paper, an exponential transformation based lattice Boltzmann (LB) model for solving the $n$-dimensional ($n$D) convection-diffusion equation(CDE) is developed. Firstly, a class of exponential transformation is proposed to convert the $n$D CDE into a diffusion equation. Then, the converted diffusion equation is solved by the LB model. So, compared to the available LB models for CDE, the present LB model can eliminate the difficulty in treating the convection term. With the direct Taylor expansion method, it is shown that the CDE can be exactly derived from the exponential transformation based LB model. Finally, a variety of numerical tests have been conducted to validate the present LB model. It can be found that the numerical results agree well with the analytical solutions. Moreover, we also find that the present LB model has second-order convergence rate in space, and it is more effective and more stable than the previous LB model for the CDE.

• General Summary

• Abstract

Model Order Reduction is an approximation of the main system so that the simplified system retains important features of the main system. Deep learning technology is a recent breakthrough in artificial neural networks that can find more hidden information from the data. In this paper, a non-intrusive reduced order model (NIROM) based on combining deep neural networks (DNNs) and POD abilities, namely FAE-CAE-LSTM is presented. This method combines the obtained features based on Fully connected autoencoders (FAE), Convolutional autoencoders (CAE), and POD and then, a deep Long short-term memory network is trained by obtained features to predict the pressure and velocity fields at future time instances. We investigate the performance of the proposed methodology by solving two well-known canonical cases: a strong shear flow exhibiting the Kelvin–Helmholtz instability, and flow past a cylinder. The performance of the proposed FAE-CAE-LSTM method in future state prediction of the flow is compared with other NIROM methods such as CAE-LSTM, autoencoder-LSTM, autoencoder-DMD and POD-RNN based models. Results show that the FAE-CAE-LSTM method is considerably capable of predicting fluid flow evolution and obtains the best results in the prediction of the pressure and velocity fields in future time instances.

• General Summary

• Abstract

The incompressible magnetohydrodynamics system with variable density is coupled by the incompressible Navier-Stokes equations with variable density and the Maxwell equations. In this paper, we study a new first-order Euler semi-discrete scheme for solving this system. The proposed numerical scheme is unconditionally stable for any time step size $\tau>0$. Furthermore, a rigorous error analysis is presented and the first-order temporal convergence rate $\mathcal O(\tau)$ is derived by using the method of mathematical induction and the discrete maximal $L^p$-regularity of the Stokes problem. Finally, numerical results are given to support the theoretical analysis.

• General Summary

• Abstract

The stochastic Allen-Cahn equations, as a typical example of nonlinear stochastic partial differential equations, play an important role in phase theory. In this paper, we investigate the rate of convergence in the $p$th mean for a truncated-type explicit Euler time-stepping method applied to the stochastic Allen-Cahn equations as well as using the spectral Galerkin approximation in spatial discretization. Finally, a numerical example is given to confirm the strong convergence order.

• General Summary

• Abstract

A well-conditioned spectral integration (SI) method is introduced, developed and applied to $n$th-order differential equations with variable coefficients and general boundary conditions. The approach is based on integral reformulation techniques which lead to almost banded linear matrices, and the main system to be solved is further banded by utilizing a Schur complement approach. Numerical experiments indicate the spectral integration method can solve high order equations efficiently, oscillatory problems accurately and is adaptable to large systems. Applications in Korteweg-de Vries (KdV) type and Kawahara equations are carried out to illustrate the proposed method is effective to complicated mathematical models.

• General Summary

• Abstract

In this article, we have developed a new time-marching method to simulate sound and soliton propagation in homogeneous and heterogeneous media. The proposed time integration scheme can numerically preserve physical dispersion over a wide wavenumber range and conserve energy while solving wave propagation problems. Here, the Fourier stability analysis

has been used to assess the numerical properties of the developed method. The proposed numerical method's dispersion and dissipation properties have also been compared with the classical fourth-order Runge-Kutta (RK4) method. Stability property contours for the newly proposed method display that the maximum allowable time step is at least five times higher than the RK4 method. The Fourier stability analysis also explains the dispersion error associated with the used spatio-temporal discretization schemes. It is observed that the dispersion error is significantly small for the proposed time integration schemes compared with the RK4 method. The proposed

methods simulate sound propagation problems with fewer computational resources that otherwise demand high computational costs. The efficacy of the proposed time integration methods has been demonstrated by solving

benchmark sound wave propagation problems. Moreover, to test the developed method's efficiency and robustness, we have performed simulations of the sound wave propagation in a layered media, corner-edge model, and damped sine-Gordon equation.

• General Summary

• Abstract

In this paper, the first-order and second-order semi-discrete predictor-multicorrector (PMC) algorithms to solve the 2D/3D unsteady incompressible micropolar fluid equations (IMNSE) are proposed.  In the algorithms, the first-order and second-order BDF formulas are adopted to approximate the time derivative terms. At each time step,  two elliptical sub-problems with Dirichlet conditions are solved at the prediction step, the strategy of projection about linear momentum equation with additional viscosity term and the elliptical sub-problems about angular momentum are solved at the multicorrection step. Furthermore, the unconditional stability and error estimates of the first-order scheme  are proved theoretically. Numerical experiments are carried out to show the effectiveness of the algorithms.

• General Summary