• Abstract

We propose a numerical method to solve the Monge-Ampere equation which admits a classical convex solution. The Monge-Ampère equation is reformulated into an equivalent first-order system. We adopt a novel reconstructed discontinuous approximation space which consists of piecewise irrotational polynomials. This space allows us to solve the first-order system in two sequential steps. In the first step, we solve a nonlinear system to obtain the approximation to the gradient. A Newton iteration is adopted to handle the nonlinearity of the system. The approximation to the primitive variable is obtained from the approximate gradient by a trivial least squares finite element method in the second step. Numerical examples in both two and three dimensions are presented to show an optimal convergence rate in accuracy. It is interesting to observe that the approximation solution is piecewise convex. Particularly, with the reconstructed approximation space, the proposed method numerically demonstrates a remarkable robustness. The convergence of the Newton iteration does not rely on the initial values. The dependence of the convergence on the penalty parameter in the discretization is also negligible, in comparison to the classical discontinuous approximation space.

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In this paper, we propose a variational multiscale method (VMM) for the stationary incompressible magnetohydrodynamics equations. This method is defined by large-scale spaces for the velocity field and the magnetic field, which aims to solve flows at high Reynolds numbers. We provide a new VMM formulation and prove its stability and convergence. Finally, some numerical experiments are presented to indicate the optimal convergence of our method.

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The Richtmyer–Meshkov instability of interfaces separating elastic-plastic materials from vacuum is investigated by numerical simulation using a multi-material solid mechanics algorithm based on an Eulerian framework. The research efforts are directed to reveal the influence of the initial perturbation and material strength on the deformation of the perturbed interface impacted by an initial shock. By varying the initial amplitude $(k\xi_0)$ of the perturbed interface and the yield stress $(σ_{\Upsilon}),$ three typical modes of interface deformation have been identified as the broken mode, the stable mode and the oscillating mode. For the broken mode, the interface width (i.e., the bubble position with respect to that of the spike) increases continuously resulting in a final separation of the spike from the perturbed interface. For the stable mode, the interface width grows to saturation and then maintains a nearly constant value in the long term. For the oscillating mode, the wavy-like interface moving forward obtains an aperiodic oscillation of small amplitude, namely, the interface width varies in time slightly around zero. The intriguing difference of the typical modes is interpreted qualitatively by comparing the early-stage wave motion and the commensurate pressure and effective stress. Further, the subsequent interface deformation is illustrated quantitatively via the time series of the interface positions and velocities of these three typical modes.

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The background numerical noise $ε_0$ is determined by the maximum of truncation error and round-off error. For a chaotic system, the numerical error $ε(t)$ grows exponentially, say, $ε(t) = ε_0 {\rm exp}(κt),$ where $κ > 0$ is the so-called noise-growing exponent. This is the reason why one can not gain a convergent simulation of chaotic systems in a long enough interval of time by means of traditional algorithms in double precision, since the background numerical noise $ε_0$ might stop decreasing because of the use of double precision. This restriction can be overcome by means of the clean numerical simulation (CNS), which can decrease the background numerical noise $ε_0$ to any required tiny level. A lot of successful applications show the novelty and validity of the CNS. In this paper, we further propose some strategies to greatly increase the computational efficiency of the CNS algorithms for chaotic dynamical systems. It is highly suggested to keep a balance between truncation error and round-off error and besides to progressively enlarge the background numerical noise $ε_0,$ since the exponentially increasing numerical noise $ε(t)$ is much larger than it. Some examples are given to illustrate the validity of our strategies for the CNS.

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This investigation focused on the influence of the radial inertia effect on the propagation behavior of stress waves in thin-walled tubes subjected to combined longitudinal and torsional impact loads. Generalized characteristics theory was used to analyze the main features of the characteristic wave speeds and simple wave solutions in thin-walled tubes. The incremental elastic-plastic constitutive relations described by the rate-independent plasticity were adopted, and the finite difference method was used to investigate the evolution and propagation behaviors of combined elasticplastic stress waves in thin-walled tubes when the radial inertial effect was considered. The numerical results were compared with those obtained when the radial inertia effect was not considered. The results showed that the speed of the coupled stress wave increased when the radial inertia effect was considered. The hardening modulus of the material in the plastic stage had a greater impact on the coupled slow waves than on the coupled fast waves.

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In this paper, we present a quadratic auxiliary variable (QAV) technique to develop a novel class of arbitrarily high-order energy-preserving algorithms for the Camassa-Holm equation. The QAV approach is first utilized to transform the original equation into a reformulated QAV system with a consistent initial condition. Then the reformulated QAV system is discretized by applying the Fourier pseudo-spectral method in space and the symplectic Runge-Kutta methods in time, which arrives at a class of fully discrete schemes. Under the consistent initial condition, they can be rewritten as a new fully discrete system by eliminating the introduced auxiliary variable, which is rigorously proved to be energy-preserving and symmetric. Ample numerical experiments are conducted to confirm the expected order of accuracy, conservative property and efficiency of the proposed methods. The presented numerical strategy makes it possible to directly apply a special class of Runge-Kutta methods to develop energy-preserving algorithms for a general conservative system with any polynomial energy.

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In this article, a robust, effective, and scale-invariant weighted compact nonlinear scheme (WCNS) is proposed by introducing descaling techniques to the nonlinear weights of the WCNS-Z/D schemes. The new scheme achieves an essentially non-oscillatory approximation of a discontinuous function (ENO-property), a scale-invariant property with an arbitrary scale of a function (Si-property), and an optimal order of accuracy with smooth function regardless of the critical point (Cp-property). The classical WCNS-Z/D schemes do not satisfy Si-property intrinsically, which is caused by a loss of sub-stencils’ adaptivity in the nonlinear interpolation of a discontinuous function when scaled by a small scale factor. A new nonlinear weight is devised by using an average of the function values and the descaling function, providing the new WCNS schemes (WCNS-Zm/Dm) with many attractive properties. The ENO-property, Si-property and Cp-property of the new WCNS schemes are validated numerically. Results show that the WCNS-Zm/Dm schemes satisfy the ENO-property and Si-property, while only the WCNS-Dm scheme satisfies the Cp-property. In addition, the Gaussian wave problem is solved by using successively refined grids to verify that the optimal order of accuracy of the new schemes can be achieved. Several one-dimensional shock tube problems, and two-dimensional double Mach reflection (DMR) problem and the Riemann IVP problem are simulated to illustrate the ENO-property and Si-property of the scale-invariant WCNS-Zm/Dm schemes.

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In this paper, we construct and analyze a Crank-Nicolson fitted finite volume scheme for pricing European options under regime-switching Kou’s jump-diffusion model which is governed by a system of partial integro-differential equations (PIDEs). We show that this scheme is consistent, stable and monotone as the mesh sizes in space and time approach zero, hence it ensures the convergence to the solution of continuous problem. Finally, numerical experiments are performed to demonstrate the efficiency, accuracy and robustness of the proposed method.

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In the last two decades, many edge detection methods have been developed and widely used in image processing for edge detection and the hybrid compact-WENO finite difference (hybrid) schemes for solving the system of hyperbolic conservation laws with solutions containing both discontinuous and complex fine-scale structures. However, many edge detection methods include the problem-dependent parameters such as the high order multi-resolution (MR) analysis (Harten, JCP, 49 (1983)). Therefore, we combined the Tukey’s boxplot method with MR analysis (Gao et al., JSC, 73 (2017)) to overcome this problem in a sense. But the Tukey’s boxplot method needs to sort the data at the beginning of Runge-Kutta time integration method, which is relatively time-consuming and inefficient. In this study, we employ the PauTa criterion and remove the problem-dependent parameters in the MR analysis. Furthermore, two new edge detection approaches, which are based on second-order central difference scheme and Ren’s idea (Ren et al., JCP, 192 (2003)), are also proposed. The accuracy, efficiency and robustness of the hybrid scheme with the new edge detectors are verified by numerous classical one- and two-dimensional examples in the image processing and compressible Euler equations with discontinuous solutions.

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In this paper, a quadratic finite volume method (FVM) for parabolic problems is studied. We first discretize the spatial variables using a quadratic FVM to obtain a semi-discrete scheme. We then employ the backward Euler method and the Crank-Nicolson method respectively to further disctetize the time vatiable so as to derive two full-discrete schemes. The existence and uniqueness of the semi-discrete and full-discrete FVM solutions are established and their optimal error estimates are derived. Finally, we give numerical examples to illustrate the theoretical results.

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The study of impacts of down-up hill road segment on the density threshold of traffic shock formation in ring road vehicular flow is helpful to the deep understanding of sags' bottleneck effect. Sags are freeway segments along which the gradient increases gradually in the traffic direction. The main aim of this paper is to seek the density threshold of shock formation of vehicular flow in ring road with down-up hill segment, because down-up hill roadway segment is a source to cause capacity reduction that is an attractive topic in vehicular traffic science. To seek the density threshold numerically, a viscoelastic continuum model [1] is extended and used. To solve the model equations, a fifth-order weighted essentially non-oscillatory scheme for spatial discretization, and a 3rd order Runge-Kutta scheme for time partial derivative term are used. Validation by existing observation data and the Navier-Stokes like model [2] extended as EZM is done before conducting extensive numerical simulations. For ring road vehicular flow with three separated down-up hill segments, it is found that the density threshold of shock formation decreases monotonically with the relative difference of free flow speed, this variation can be simply fitted by a third order polynomial.

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This paper aims to present a new well-balanced, accurate and fast finite volume scheme on unstructured grids to solve hyperbolic conservation laws. It is a scheme that combines both finite volume approach and characteristic method. In this study, we consider a shallow water system with Coriolis effect and bottom friction stresses where this new Finite Volume Characteristics (FVC) scheme has been applied. The physical and mathematical properties of the system, including the C-property, have been well preserved.
First, we developed this approach by preserving the advantages of the finite volume discretization such as conservation property and the method of characteristics, in order to avoid Riemann solvers and to enhance the accuracy without any complexity of the MUSCL reconstruction. Afterward, a discretization was applied to the bottom source term that leads to a well-balanced scheme satisfying the steady-state condition of still water. A semi-implicit treatment will also be presented in this study to avoid stability problems due to source terms. Finally, the proposed finite volume method is verified on several benchmark tests and shows good agreement with analytical solutions and experimental results; moreover, it gives a noteworthy accuracy and rapidity improvement compared to the original approaches.

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This article aims to identify the partial topological structures of delayed complex network. Based on the drive-response concept, a more universal model, which includes nonlinear couplings, stochastic perturbations and multi-weights, is considered into drive-response networks. Different from previous methods, we obtain identification criteria by combining graph-theoretic method and adaptive synchronization. After that, the partial topological structures of stochastic multi-weighted complex networks with or without time delays can be identified successfully. Moreover, response network can reach synchronization with drive network. Ultimately, the effectiveness of the proposed theoretical results is validated through numerical simulations.

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Gas distribution function plays a crucial role in the description of gas flows at the mesoscopic scale. In the presence of non-equilibrium flow, the distribution function loses its rotational symmetricity, making the mathematical derivation difficult. From both the Chapman-Enskog expansion and the Hermite polynomial expansion (Grad’s method), we observe that the non-equilibrium effect is closely related to the peculiar velocity space $(C).$ Based on this recognition, we propose a new methodology to construct the non-equilibrium distribution function from the perspective of polynomial expansion in the peculiar velocity space of molecules. The coefficients involved in the non-equilibrium distribution function can be exactly determined by the compatibility conditions and the moment relationships. This new framework allows constructing non-equilibrium distribution functions at any order of truncation, and the ones at the third and the fourth order have been presented in this paper for illustration purposes. Numerical validations demonstrate that the new method is more accurate than the Grad’s method at the same truncation error for describing non-equilibrium effects. Two-dimensional benchmark tests are performed to shed light on the applicability of the new method to practical engineering problems.

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A generalized homotopy-based Coiflet-type wavelet method for solving strongly nonlinear PDEs with nonhomogeneous edges is proposed. Based on the improvement of boundary difference order by Taylor expansion, the accuracy in wavelet approximation is largely improved and the accumulated error on boundary is successfully suppressed in application. A unified high-precision wavelet approximation scheme is formulated for inhomogeneous boundaries involved in generalized Neumann, Robin and Cauchy types, which overcomes the shortcomings of accuracy loss in homogenizing process by variable substitution. Large deflection bending analysis of orthotropic plate with forced boundary moments and rotations on nonlinear foundation is used as an example to illustrate the wavelet approach, while the obtained solutions for lateral deflection at both smally and largely deformed stage have been validated compared to the published results in good accuracy. Compared to the other homotopy-based approach, the wavelet scheme possesses good efficiency in transforming the differential operations into algebraic ones by converting the differential operators into iterative matrices, while nonhomogeneous boundary is directly approached dispensing with homogenization. The auxiliary linear operator determined by linear component of original governing equation demonstrates excellent approaching precision and the convergence can be ensured by iterative approach.

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In this article, we develop a new well-balanced finite volume central weighted essentially non-oscillatory (CWENO) scheme for one- and two-dimensional shallow water equations over uneven bottom. The well-balanced property is of paramount importance in practical applications, where many studied phenomena can be regarded as small perturbations to the steady state. To achieve the well-balanced property, we construct numerical fluxes by means of a decomposition algorithm based on a novel equilibrium preserving reconstruction procedure and we avoid applying the traditional hydrostatic reconstruction technique accordingly. This decomposition algorithm also helps us realize a simple source term discretization. Both rigorous theoretical analysis and extensive numerical examples all verify that the proposed scheme maintains the well-balanced property exactly. Furthermore, extensive numerical results strongly suggest that the resulting scheme can accurately capture small perturbations to the steady state and keep the genuine high-order accuracy for smooth solutions at the same time.

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Based on a subspace method and a linear approximation method, a convex algorithm is designed to solve a kind of non-convex PDE constrained fractional optimization problem in this paper. This PDE constrained problem is an infinite-dimensional Hermitian eigenvalue optimization problem with non-convex and low regularity. Usually, such a continuous optimization problem can be transformed into a large-scale discrete optimization problem by using the finite element methods. We use a subspace technique to reduce the scale of discrete problem, which is really effective to deal with the large-scale problem. To overcome the difficulties caused by the low regularity and non-convexity, we creatively introduce several new artificial variables to transform the non-convex problem into a convex linear semidefinite programming. By introducing linear approximation vectors, this linear semidefinite programming can be approximated by a very simple linear relaxation problem. Moreover, we theoretically prove this approximation. Our proposed algorithm is used to optimize the photonic band gaps of two-dimensional Gallium Arsenide-based photonic crystals as an application. The results of numerical examples show the effectiveness of our proposed algorithm, while they also provide several optimized photonic crystal structures with a desired wide-band-gap. In addition, our proposed algorithm provides a technical way for solving a kind of PDE constrained fractional optimization problems with a generalized eigenvalue constraint.

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In this paper, a node-based smoothed finite element method (NS-FEM) with linear gradient fields (NS-FEM-L) is presented to solve elastic wave scattering by a rigid obstacle. By using Helmholtz decomposition, the problem is transformed into a boundary value problem with coupled boundary conditions. In numerical analysis, the perfectly matched layer (PML) and transparent boundary condition (TBC) are introduced to truncate the unbounded domain. Then, a linear gradient is constructed in a node-based smoothing domain (N-SD) by using a complete order of polynomial. The unknown coefficients of the smoothed linear gradient function can be solved by three linearly independent weight functions. Further, based on the weakened weak formulation, a system of linear equation with the smoothed gradient is established for NS-FEM-L with PML or TBC. Some numerical examples also demonstrate that the presented method possesses more stability and high accuracy. It turns out that the modified gradient makes the NS-FEM-L-PML and NS-FEM-L-TBC possess an ideal stiffness matrix, which effectively overcomes the instability of original NS-FEM. Moreover, the convergence rates of $L^2$ and $H^1$ semi-norm errors for the two NS-FEM-L models are also higher.

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This paper proposes a high order deep domain decomposition method (HOrderDeepDDM) for solving high-frequency interface problems, which combines high order deep neural network (HOrderDNN) with domain decomposition method (DDM). The main idea of HOrderDeepDDM is to divide the computational domain into some sub-domains by DDM, and apply HOrderDNNs to solve the high-frequency problem on each sub-domain. Besides, we consider an adaptive learning rate annealing method to balance the errors inside the sub-domains, on the interface and the boundary during the optimization process. The performance of HOrderDeepDDM is evaluated on high-frequency elliptic and Helmholtz interface problems. The results indicate that: HOrderDeepDDM inherits the ability of DeepDDM to handle discontinuous interface problems and the power of HOrderDNN to approximate high-frequency problems. In detail, HOrderDeepDDMs $(p>1)$ could capture the high-frequency information very well. When compared to the deep domain decomposition method (DeepDDM), HOrderDeepDDMs $(p >1)$ converge faster and achieve much smaller relative errors with the same number of trainable parameters. For example, when solving the high-frequency interface elliptic problems in Section 3.3.1, the minimum relative errors obtained by HOrderDeepDDMs $(p =9)$ are one order of magnitude smaller than that obtained by DeepDDMs when the number of the parameters keeps the same, as shown in Fig. 4.

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In this paper spectral Galerkin approximation of optimal control problem governed by fractional elliptic equation is investigated. To deal with the nonlocality of fractional Laplacian operator the Caffarelli-Silvestre extension is utilized. The first order optimality condition of the extended optimal control problem is derived. A spectral Galerkin discrete scheme for the extended problem based on weighted Laguerre polynomials is developed. A priori error estimates for the spectral Galerkin discrete scheme is proved. Numerical experiments are presented to show the effectiveness of our methods and to verify the theoretical findings.

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