In typical EGS fracturing projects, the fracture network is denser in the region close to the fracturing wells. The distribution of fractures has a significant impact on the heat extraction performance of enhanced geothermal system (EGS). In order to avoid premature thermal breakthrough and improve the thermal extraction efficiency, this paper studies the heat extraction performance of EGS in heterogeneous fractured reservoirs under different injection-production methods. First, a heterogeneous discrete fracture network generation method based on relative pressure distribution is developed. A thermo-hydraulic-mechanical (THM) coupling model is established to simulate the fluid flow and heat transfer in reservoirs. The heat extraction performance under different production and injection modes is compared with different fracture densities. The results show that injection in the dense side of the anti-parallel well configuration will be more efficient for heat extraction. Under the condition of same injection mass flow rate, the heat extraction performance is positively correlated with the fracture density. Although sparse-side injection is not conducive to fluid flow, under the condition of same injection mass flow rate with anti-parallel well configuration, it will improve the thermal extraction performance.

The WENO-Z+ scheme [F. Acker, R. B. de R. Borges, and B. Costa, An improved WENO-Z scheme, J. Comput. Phys., 313 (2016), pp. 726–753] with two different versions further raised the nonlinear weights with respect to the nonsmooth or less-smooth substencils, by introducing an additional term into the weights formula of the well-validated WENO-Z scheme. These two WENO-Z+ schemes both produce less dissipative solutions than WENO-JS and WENO-Z. However, the recommended one which achieves superior resolutions in the high-frequency-wave regions fails to recover the designed order of accuracy where there exists a critical point, while the other one which obtains the designed order of accuracy at or near critical points is unstable near discontinuities. In the present study, we find that the WENO-Z+ schemes over-amplify the contributions from less-smooth substencils through their additional terms, and hence their improvements of both stability and resolution have been greatly hindered. Then, we develop improved WENO-Z+ schemes by making a set of modifications to the additional terms to avoid the over-amplification of the contributions from less-smooth substencils. The proposed schemes, denoted as WENO-IZ+, maintain the same convergence properties as the corresponding WENO-Z+ schemes. Numerical examples confirm that the new schemes are much more stable near discontinuities and far less dissipative in the region with high-frequency waves than the WENO-Z+ schemes. In addition, improved results have been obtained for one-dimensional linear advection problems, especially over long output times. The excellent performance of the new schemes is also demonstrated in the simulations of 1D and 2D Euler equation test cases.

In this paper, we investigate ultraspherical spectral method for the Ohta-Kawasaki (OK) and Nakazawa-Ohta (NO) models in the disk domain, representing diblock and triblock copolymer systems, respectively. We employ ultraspherical spectral discretization for spatial variables in the disk domain and apply the second-order backward differentiation formula (BDF) method for temporal discretization. To our best knowledge, this is the first study to develop a numerical method for diblock and triblock copolymer systems with long-range interactions in disk domains. We show the energy stability of the numerical method in both semi-discrete and fully-discrete discretizations. In our numerical experiments, we verify the second-order temporal convergence rate and the energy stability of the proposed methods. Our numerical results show that the coarsening dynamics in diblock copolymers lead to bubble assemblies both inside and on the boundary of the disk. Additionally, in the triblock copolymer system, we observe several novel pattern formations, including single and double bubble assemblies in the unit disk. These findings are detailed through extensive numerical experiments.

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.

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 $Log$ marking strategy is proposed. The $Log$ 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.

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-Łojasiewicz 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.

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 $r≥1,$ a space of piecewise polynomials of degree $≤ r−1$ 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.

This paper is devoted to adaptive finite element method for the nonlinear Schrödinger 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 Schrödinger equation. Numerical experiments are presented to illustrate the reliability and efficiency of the proposed error estimators and the corresponding adaptive algorithm.

In this paper, an exponential transformation based lattice Boltzmann (LB) model for solving the $n$-dimensional ($n{\rm D}$) convection-diffusion equation(CDE) is developed. Firstly, a class of exponential transformation is proposed to convert the $n{\rm D}$ 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.

In this paper, the boundary mapped collocation (BMC) approach is presented for the analysis of heat conduction problems involving heat generation and non-homogeneous thermal conductivity. The proposed methodology is introduced to produce the numerical solutions of the temperature field within the framework of the BMC method, a novel boundary meshless method, without resorting to requiring any integral calculation, neither in the domain nor at the boundary. In particular, the arrangement of discrete nodes is restricted to the axis, which brings the spatial dimension down by one. The technique also reduced the traditional complex shape functions to succinct one-dimensional boundary shape functions by using one-dimensional basis functions and weight functions for two- and three-dimensional approximation implementation. In addition, four numerical applications and comparisons with the outcomes of the finite element approach and another meshfree method are used to demonstrate the correctness, convergence, and stability of the BMC method.

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.

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 $p{\rm th}$ 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.

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.

An innovative meshless method is proposed in this paper for the bending problem of arbitrary Kirchhoff plates subjected to external force with various shapes and different boundary conditions. Without using a numerical integral, the deflection of the thin plate is approximated by using the boundary mapped collocation approach. Moreover, the computational domain discretization is just dependent on discretized nodes on the axis, while tensor product nodes have been mapped in the domain automatically. Hence, in the boundary mapped collocation implementation, the approximation functions are derived by employing the one-dimensional moving least squares technique for two-dimensional and higher-dimensional problems. Further, the virtual boundary technique is introduced to enforce the boundary conditions in the proposed method. Additionally, four numerical experiments are presented to illustrate the excellent convergence and high precision of the proposed approach.