We propose an efficient iterative convolution thresholding method for the formulation of flow networks where the fluid is modeled by the Darcy–Stokes flow with the presence of volume sources. The method is based on the minimization of the dissipation power in the fluid region with a Darcy term. The flow network is represented by its characteristic function and the energy is approximated under this representation. The minimization problem can then be approximately solved by alternating: 1) solving a Brinkman equation to model the Darcy–Stokes flow and 2) updating the characteristic function by a simple convolution and thresholding step. The proposed method is simple and easy to implement. We prove mathematically that the iterative method has the total energy decaying property. Numerical experiments demonstrate the performance and robustness of the proposed method and interesting structures are observed.

The quantum lattice Boltzmann (qlB) algorithm solves the 1D Dirac equations and has been used to solve approximately the classical (i.e., non-relativistic) Schrödinger equation. We point out that the qlB method actually approximates the hyperbolic version of the non-relativistic Schrödinger equation, whose solution is thus obtained at the price of an additional small error. Such an error is of order of $(ω_c\tau)^{−1},$ where $ω_c:=\frac{mc^2}{h}$ is the Compton frequency, $ħ$ being the reduced Planck constant, $m$ the rest mass of the electrons, $c$ the speed of light, and $\tau$ a chosen reference time (i.e., 1 s), and hence it vanishes in the non-relativistic limit $c → +∞.$ This asymptotic result comes from a singular perturbation process which does not require any boundary layer and, consequently, the approximation holds uniformly, which fact is relevant in view of numerical approximations. We also discuss this occurrence more generally, for some classes of linear singularly perturbed partial differential equations.

A new type of finite volume WENO schemes for hyperbolic problems was devised in [33] by introducing the order-preserving (OP) criterion. In this continuing work, we extend the OP criterion to the WENO-Z-type schemes. We firstly rewrite the formulas of the Z-type weights in a uniform form from a mapping perspective inspired by extensive numerical observations. Accordingly, we build the concept of the locally order-preserving (LOP) mapping which is an extension of the order-preserving (OP) mapping and the resultant improved WENO-Z-type schemes are denoted as LOP-GMWENO-X. There are four major advantages of the LOP-GMWENO-X schemes superior to the existing WENO-Z-type schemes. Firstly, the new schemes can amend the serious drawback of the existing WENO-Z-type schemes that most of them suffer from either producing severe spurious oscillations or failing to obtain high resolutions in long calculations of hyperbolic problems with discontinuities. Secondly, they can maintain considerably high resolutions on solving problems with high-order critical points at long output times. Thirdly, they can obtain evidently higher resolution in the region with high-frequency but smooth waves. Finally, they can significantly decrease the post-shock oscillations for simulations of some 2D problems with strong shock waves. Extensive benchmark examples are conducted to illustrate these advantages.

We develop a second-order continuous finite element method for solving the static Eikonal equation. It is based on the vanishing viscosity approach with a homotopy method for solving the discretized nonlinear system. More specifically, the homotopy method is utilized to decrease the viscosity coefficient gradually, while Newton’s method is applied to compute the solution for each viscosity coefficient. Newton’s method alone converges for just big enough viscosity coefficients on very coarse grids and for simple 1D examples, but the proposed method is much more robust and guarantees the convergence of the nonlinear solver for all viscosity coefficients and for all examples over all grids. Numerical experiments from 1D to 3D are presented to confirm the second-order convergence and the effectiveness of the proposed method on both structured or unstructured meshes.

In this paper, a new mixed finite element scheme using element-wise stabilization is introduced for the biharmonic equation with variable coefficient on Lipschitz polyhedral domains. The proposed scheme doesn’t involve any integration along mesh interfaces. The gradient of the solution is approximated by $H$(div)-conforming $BDM_{k+1}$ element or vector valued Lagrange element with order $k+1,$ while the solution is approximated by Lagrange element with order $k+2$ for any $k≥0.$ This scheme can be easily implemented and produces symmetric and positive definite linear system. We provide a new discrete $H^2$-norm stability, which is useful not only in analysis of this scheme but also in ${\rm C}^0$ interior penalty methods and DG methods. Optimal convergences in both discrete $H^2$-norm and $L^2$-norm are derived. This scheme with its analysis is further generalized to the von Kármán equations. Finally, numerical results verifying the theoretical estimates of the proposed algorithms are also presented.

In this paper, effective Eulerian algorithms are introduced for the computation of the forward finite time Lyapunov exponent (FTLE) of smooth flow fields. The advantages of the proposed algorithms mainly manifest in two aspects. First, previous Eulerian approaches for computing the FTLE field are improved so that the Jacobian of the flow map can be obtained by directly solving a corresponding system of partial differential equations, rather than by implementing certain finite difference upon the flow map, which can significantly improve the accuracy of the numerical solution especially near the FTLE ridges. Second, in the proposed algorithms, all computations are done on the fly, that is, all required partial differential equations are solved forward in time, which is practically more natural. The new algorithms still maintain the optimal computational complexity as well as the second order accuracy. Numerical examples demonstrate the effectiveness of the proposed algorithms.

We apply the monotonicity correction to the finite element method for the anisotropic diffusion problems, including linear and quadratic finite elements on triangular meshes. When formulating the finite element schemes, we need to calculate the integrals on every triangular element, whose results are the linear combination of the two-point pairs. Then we decompose the integral results into the main and remaining parts according to coefficient signs of two-point pairs. We apply the nonlinear correction to the positive remaining parts and move the negative remaining parts to the right side of the finite element equations. Finally, the original stiffness matrix can be transformed into a nonlinear M-matrix, and the corrected schemes have the positivity-preserving property. We also give the monotonicity correction to the time derivative term for the time-dependent problems. Numerical experiments show that the corrected finite element method has monotonicity and maintains the convergence order of the original schemes in $H^1$-norm and $L^2$-norm, respectively.

Robust quantum control with uncertainty plays a crucial role in practical quantum technologies. This paper presents a method for solving a quantum control problem by combining neural network and symplectic finite difference methods. The neural network approach provides a framework that is easy to establish and train. At the same time, the symplectic methods possess the norm-preserving property for the quantum system to produce a realistic solution in physics. We construct a general high dimensional quantum optimal control problem to evaluate the proposed method and an approach that combines a neural network with forward Euler’s method. Our analysis and numerical experiments confirm that the neural network-based symplectic method achieves significantly better accuracy and robustness against noises.

We propose a numerical method for a non-selfadjoint Steklov eigenvalue problem of the Helmholtz equation. The problem is formulated using boundary integrals. The Nyström method is employed to discretize the integral operators, which leads to a non-Hermitian generalized matrix eigenvalue problems. The spectral indicator method (SIM) is then applied to calculate the (complex) eigenvalues. The convergence is proved using the spectral approximation theory for (non-selfadjoint) compact operators. Numerical examples are presented for validation.

Extrapolation cascadic multigrid (EXCMG) method with conjugate gradient smoother is very efficient for solving the elliptic boundary value problems with linear finite element discretization. However, it is not trivial to generalize the vertex-centred EXCMG method to cell-centered finite volume (FV) methods for diffusion equations with strongly discontinuous and anisotropic coefficients, since a non-nested hierarchy of grid nodes are used in the cell-centered discretization. For cell-centered FV schemes, the vertex values (auxiliary unknowns) need to be approximated by cell-centered ones (primary unknowns). One of the novelties is to propose a new gradient transfer (GT) method of interpolating vertex unknowns with cell-centered ones, which is easy to implement and applicable to general diffusion tensors. The main novelty of this paper is to design a multigrid prolongation operator based on the GT method and splitting extrapolation method, and then propose a cell-centered EXCMG method with BiCGStab smoother for solving the large linear system resulting from linear FV discretization of diffusion equations with strongly discontinuous and anisotropic coefficients. Numerical experiments are presented to demonstrate the high efficiency of the proposed method.

This study aimed to specialise a directional $\mathcal{H}^2 (\mathcal{D}\mathcal{H}^2)$ compression to matrices arising from the discontinuous Galerkin (DG) discretisation of the hypersingular equation in acoustics. The significant finding is an algorithm that takes a DG stiffness matrix and finds a near-optimal $\mathcal{D}\mathcal{H}^2$ approximation for low and high-frequency problems. We introduced the necessary special optimisations to make this algorithm more efficient in the case of a DG stiffness matrix. Moreover, an automatic parameter tuning strategy makes it easy to use and versatile. Numerical comparisons with a classical Boundary Element Method (BEM) show that a DG scheme combined with a $\mathcal{D}\mathcal{H}^2$ gives better computational efficiency than a classical BEM in the case of high-order finite elements and $hp$ heterogeneous meshes. The results indicate that DG is suitable for an auto-adaptive context in integral equations.