• Abstract

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.

• General Summary

• Abstract

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.

• 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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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The effects of non-physical mixing on interface development are still not reasonably evaluated when diffuse interface methods (DIMs) are employed to treat the two-medium flows with immiscible interfaces, especially for compressible multi-medium flows with geometrical source terms. In this work, we simulate radially symmetric multi-medium flows employing the sharp interface methods (SIMs) and DIMs to evaluate their performance such as pressure dislocations, mass conservation, and convergence. The $\gamma$-based model and the five-equation transport model are two common DIMs, which are extended to equations with geometrical source terms combined with discontinuous Galerkin (DG) methods. For the SIMs, we employ the classical modified ghost fluid method (MGFM) and its second-order extension (2nd-MGFM) developed recently. Numerical results exhibit that the 2nd-MGFM is more effective in maintaining the interfacial pressure equilibrium relative to the MGFM. The DIMs can always maintain the pressure continuity naturally and total mass conservation, which is not available for SIMs. Further, under the premise of immiscible interfaces defined artificially, the DIMs cannot provide satisfactory single medium mass conservation, while the SIMs have a smaller mass error. In addition, compared to the other three methods, the 2nd-MGFM has higher confidence for radially symmetric flows, matching the exact solution in sparser grids.

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In this work, a new lattice Boltzmann model for a class of viscous wave equation is proposed through the variable transformation, which eliminates the mixed third order partial derivative term of time and space. Some numerical tests are performed to validate the present model, and the results show that the present model has a second-order convergence rate in space.

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In this paper, we propose two efficient block preconditioners to solve the

mass-conserved Ohta-Kawasaki equation with finite element discretization.

We also study the spectral distribution of these two preconditioners, i.e.,  Schur complement preconditioner and the modified Hermitian and

skew-Hermitian splitting (MHSS in short) preconditioner. Besides, Newton method and Picard method are used to address the implicitly nonlinear term. We rigorously analyze the convergence of Newton method.  Finally, we offer numerical examples to support the theoretical analysis and indicate the efficiency of the proposed preconditioners for the mass-conserved Ohta-Kawasaki equation.

• General Summary