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.

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 $U^TU$ 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.

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.

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.

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.

In this paper, we consider an implicit method for solving the nonlinear time-space fractional Ginzburg-Landau equation. The scheme is based on the $L2-1_σ$ formula to approximate the Caputo fractional derivative and the weighted and shifted Grünwald 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 $\gamma≤0,$ where $\gamma$ is a coefficient of the equation. Moreover, the scheme is shown to be convergent in $l^2_h$ 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 presented to verify the accuracy and efficiency of the proposed scheme.

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.

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.

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.

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.