We present a novel mapping approach for WENO schemes through the use of an approximate constant mapping function which is constructed by employing an approximation of the classic signum function. The new approximate constant mapping function is designed to meet the overall criteria for a proper mapping function required in the design of the WENO-PM6 scheme. The WENO-PM6 scheme was proposed to overcome the potential loss of accuracy of the WENO-M scheme which was developed to recover the optimal convergence order of the WENO-JS scheme at critical points. Our new mapped WENO scheme, denoted as WENO-ACM, maintains almost all advantages of the WENO-PM6 scheme, including low dissipation and high resolution, while decreases the number of mathematical operations remarkably in every mapping process leading to a significant improvement of efficiency. The convergence rates of the WENO-ACM scheme have been shown through one-dimensional linear advection equation with various initial conditions. Numerical results of one-dimensional Euler equations for the Riemann problems, the Mach 3 shock-density wave interaction and the Woodward-Colella interacting blast waves are improved in comparison with the results obtained by the WENO-JS, WENO-M and WENO-PM6 schemes. Numerical experiments with two-dimensional problems as the 2D Riemann problem, the shock-vortex interaction, the 2D explosion problem, the double Mach reflection and the forward-facing step problem modeled via the two dimensional Euler equations have been conducted to demonstrate the high resolution and the effectiveness of the WENO-ACM scheme. The WENO-ACM scheme provides significantly better resolution than the WENO-M scheme and slightly better resolution than the WENO-PM6 scheme, and compared to the WENO-M and WENO-PM6 schemes, the extra computational cost is reduced by more than 83% and 93%, respectively.

At present, deep learning based methods are being employed to resolve the computational challenges of high-dimensional partial differential equations (PDEs). But the computation of the high order derivatives of neural networks is costly, and high order derivatives lack robustness for training purposes. We propose a novel approach to solving PDEs with high order derivatives by simultaneously approximating the function value and derivatives. We introduce intermediate variables to rewrite the PDEs into a system of low order differential equations as what is done in the local discontinuous Galerkin method. The intermediate variables and the solutions to the PDEs are simultaneously approximated by a multi-output deep neural network. By taking the residual of the system as a loss function, we can optimize the network parameters to approximate the solution. The whole process relies on low order derivatives. Numerous numerical examples are carried out to demonstrate that our local deep learning is efficient, robust, flexible, and is particularly well-suited for high-dimensional PDEs with high order derivatives.

In this paper, a robust modified weak Galerkin (MWG) finite element method for reaction-diffusion equations is proposed and investigated. An advantage of this method is that it can deal with the singularly perturbed reaction-diffusion equations. Another advantage of this method is that it produces fewer degrees of freedom than the traditional WG method by eliminating the element boundaries freedom. It is worth pointing out that, in our method, the test functions space is the same as the finite element space, which is helpful for the error analysis. Optimal-order error estimates are established for the corresponding numerical approximation in various norms. Some numerical results are reported to confirm the theory.

Collocation and Galerkin methods in the discontinuous and globally continuous piecewise polynomial spaces, in short, denoted as DC, CC, DG and CG methods respectively, are employed to solve second-kind Volterra integral equations (VIEs). It is proved that the quadrature DG and CG (QDG and QCG) methods obtained from the DG and CG methods by approximating the inner products by suitable numerical quadrature formulas, are equivalent to the DC and CC methods, respectively. In addition, the fully discretised DG and CG (FDG and FCG) methods are equivalent to the corresponding fully discretised DC and CC (FDC and FCC) methods. The convergence theories are established for DG and CG methods, and their semi-discretised (QDG and QCG) and fully discretized (FDG and FCG) versions. In particular, it is proved that the CG method for second-kind VIEs possesses a similar convergence to the DG method for first-kind VIEs. Numerical examples illustrate the theoretical results.

In this paper, an efficient variational model for multiplicative noise removal is proposed. By using a MAP estimator, Aubert and Aujol [SIAM J. Appl. Math., 68(2008), pp. 925-946] derived a nonconvex cost functional. With logarithmic transformation, we transform the image into a logarithmic domain which makes the fidelity convex in the transform domain. Considering the TV regularization term in logarithmic domain may cause oversmoothness numerically, we propose the TV regularization directly in the original image domain, which preserves more details of images. An alternative minimization algorithm is applied to solve the optimization problem. The $z$-subproblem can be solved by a closed formula, which makes the method very efficient. The convergence of the algorithm is discussed. The numerical experiments show the efficiency of the proposed model and algorithm.

In this work, in order to capture discontinuities correctly in linear elastic solid, augmented internal energy is defined according to the first law of thermodynamics and Hooke’s law. The non-conservative linear elastic system is then rewritten into a conservative form with the help of an augmented total energy equation. We find that the non-physical oscillations occur to the popular HLL and HLLC approximate Riemann solvers when directly applied to simulate the augmented linear elastic solid. We analyze the intrinsic reason by defining a discrepancy factor which can be used to estimate the difference of the total stress across a contact discontinuity, where it is physically required to be continuous. We discover that non-physical oscillations inevitably appear in the vicinity of the contact discontinuity if this factor is away from zero for an approximate Riemann problem solver. In order to overcome this difficulty, we propose an approximate Riemann solver based on the linearized double-shock technique. Theoretical analysis and numerical results show that in comparison to the HLL and HLLC approximate Riemann solvers, the present linearized double-shock Riemann solver can eliminate the non-physical oscillations effectively.

In this paper, we consider parabolic distributed control problems with cost functional of pointwise observation type either in space or in time. First, we show the well-posedness of the optimization problems and derive the first order optimality systems, where the adjoint state can be expressed as the linear combination of solutions to two backward parabolic equations that involve the Dirac delta distribution as source either in space or in time. Second, we use a space-time finite element method to discretize the control problems, where the state variable is approximated by piecewise constant functions in time and continuous piecewise linear polynomials in space, and the control variable is discretized by following the variational discretization concept. We obtain a priori error estimates for the control and state variables with order $\mathcal{O}(k^{\frac{1}{2}}+h)$ up to a logarithmic factor under the $L^2$-norm. Finally, we perform several numerical experiments to support our theoretical results.

In this paper, we propose a fast second-order approximation to the variable-order (VO) Caputo fractional derivative, which is developed based on $L2$-$1_σ$ formula and the exponential-sum-approximation technique. The fast evaluation method can achieve the second-order accuracy and further reduce the computational cost and the acting memory for the VO Caputo fractional derivative. This fast algorithm is applied to construct a relevant fast temporal second-order and spatial fourth-order scheme ($FL2$-$1_σ$ scheme) for the multi-dimensional VO time-fractional sub-diffusion equations. Theoretically, $FL2$-$1_σ$ scheme is proved to fulfill the similar properties of the coefficients as those of the well-studied $L2$-$1_σ$ scheme. Therefore, $FL2$-$1_σ$ scheme is strictly proved to be unconditionally stable and convergent. A sharp decrease in the computational cost and the acting memory is shown in the numerical examples to demonstrate the efficiency of the proposed method.

A family of conforming mixed finite elements with mass lumping on triangular grids are presented for linear elasticity. The stress field is approximated by symmetric $H($div) − $P_k (k ≥ 3)$ polynomial tensors enriched with higher order bubbles so as to allow mass lumping, and the displacement field is approximated by $C^{−1}− P_{k−1}$ polynomial vectors enriched with higher order terms. For both the proposed mixed elements and their mass lumping schemes, optimal error estimates are derived for the stress and displacement in $H$(div) norm and $L^2$ norm, respectively. Numerical results confirm the theoretical analysis.

In this paper, we propose hierarchical absorbing interface conditions to solve the problem of wave propagation in domains with a non-uniform space discretization or grid size inhomogeneity using Padé Via Lanczos (PVL) method. The proposed interface conditions add an auxiliary variable in the wave system to eliminate the spurious reflection at the interface between regions with different mesh sizes. The auxiliary variable with proper boundary condition can suppress the spurious reflection by cancelling the boundary source term produced by the space inhomogeneity in variational perspective. The new hierarchical interface conditions with the help of PVL implementation can effectively reduce the degree of freedom in solving the wave propagation problem.