• Abstract

In a previous paper, a technique was suggested to avoid order reduction with any explicit exponential Runge-Kutta method when integrating initial boundary value nonlinear problems with time-dependent boundary conditions. In this paper, we significantly simplify the full discretization formulas to be applied under conditions which are nearly always satisfied in practice. Not only a simpler linear combination of $\varphi_j$-functions is given for both the stages and the solution, but also the information required on the boundary is so much simplified that, in order to get local order three, it is no longer necessary to resort to numerical differentiation in space. In many cases, even to get local order 4. The technique is then shown to be computationally competitive against other widely used methods with high enough stiff order through the standard method of lines.

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In this paper, an effective oscillation-free discontinuous Galerkin (DG) scheme for a nonlinear stochastic convection-dominated problem is formulated and analyzed. The proposed oscillation-free scheme is capable to capture the steep fronts of solution automatically and distinguish the influence of the convection domination and noise perturbation. Under proper regularity assumptions, the optimal convergence rates in space and time are rigorously proved with the techniques of variational solution and conditional expectation. In the numerical simulation, the classical SIPG scheme and the proposed oscillation-free DG scheme are both performed and compared. The numerical convergence rates tests are first carried out to verify the theoretical results. The benchmark tests having the steep behaviors are further provided to illustrate the effectiveness and robustness of our proposed oscillation-free DG scheme.

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In this paper, we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations (PDEs). The main idea is to use a neural network to learn the solution map of the PDEs and to do so in an element-wise fashion. This map takes input of the element geometry and the PDE’s parameters on that element, and gives output of two operators: (1) the in2out operator for inter-element communication, and (2) the in2sol operator (Green’s function) for element-wise solution recovery. A significant advantage of this approach is that, once trained, this network can be used for the numerical solution of the PDE for any domain geometry and any parameter distribution without retraining. Also, the training is significantly simpler since it is done on the element level instead on the entire domain. We call this approach element learning. This method is closely related to hybridizable discontinuous Galerkin (HDG) methods in the sense that the local solvers of HDG are replaced by machine learning approaches. Numerical tests are presented for an example PDE, the radiative transfer or radiation transport equation, in a variety of scenarios with idealized or realistic cloud fields, with smooth or sharp gradient in the cloud boundary transition. Under a fixed accuracy level of $10^{−3}$ in the relative $L^2$ error, and polynomial degree $p = 6$ in each element, we observe an approximately $5$ to $10$ times speed-up by element learning compared to a classical finite element-type method.

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The strong convergence of an explicit full-discrete scheme is investigated for the stochastic Burgers-Huxley equation driven by additive space-time white noise, which possesses both Burgers-type and cubic nonlinearities. To discretize the continuous problem in space, we utilize a spectral Galerkin method. Subsequently, we introduce a nonlinear-tamed exponential integrator scheme, resulting in a fully discrete scheme. Within the framework of semigroup theory, this study provides precise estimations of the Sobolev regularity, $L^∞$ regularity in space, and Hölder continuity in time for the mild solution, as well as for its semi-discrete and full-discrete approximations. Building upon these results, we establish moment boundedness for the numerical solution and obtain strong convergence rates in both spatial and temporal dimensions. A numerical example is presented to validate the theoretical findings.

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We analyze the long-time behavior of numerical schemes for a class of monotone stochastic partial differential equations (SPDEs) driven by multiplicative noise. By deriving several time-independent a priori estimates for the numerical solutions, combined with the ergodic theory of Markov processes, we establish the exponential ergodicity of these schemes with a unique invariant measure, respectively. Applying these results to the stochastic Allen-Cahn equation indicates that these schemes always have at least one invariant measure, respectively, and converge strongly to the exact solution with sharp time-independent rates. We also show that these numerical invariant measures are exponentially ergodic and thus give an affirmative answer to a question proposed in [J. Cui et al., Stochastic Process. Appl., 134 (2021)], provided that the interface thickness is not too small.

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This study proposes a class of augmented subspace schemes for the weak Galerkin (WG) finite element method used to solve eigenvalue problems. The augmented subspace is built with the conforming linear finite element space defined on the coarse mesh and the eigenfunction approximations in the WG finite element space defined on the fine mesh. Based on this augmented subspace, solving the eigenvalue problem in the fine WG finite element space can be reduced to the solution of the linear boundary value problem in the same WG finite element space and a low dimensional eigenvalue problem in the augmented subspace. The proposed augmented subspace techniques have the second order convergence rate with respect to the coarse mesh size, as demonstrated by the accompanying error estimates. Finally, a few numerical examples are provided to validate the proposed numerical techniques.

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In this paper, we develop the stabilization-free virtual element method for the Helmholtz transmission eigenvalue problem on anisotropic media. The eigenvalue problem is a variable-coefficient, non-elliptic, non-selfadjoint and nonlinear model. Separating the cases of the index of refraction $n≠1$ and $n≡1,$ the stabilization-free virtual element schemes are proposed, respectively. Furthermore, we prove the spectral approximation property and error estimates in a unified theoretical framework. Finally, a series of numerical examples are provided to verify the theoretical results, show the benefits of the stabilization-free virtual element method applied to eigenvalue problems, and implement the extensions to high-order and high-dimensional cases.

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In this paper, we propose and analyze two second-order accurate finite difference schemes for the one-dimensional heat equation with concentrated capacity on a computational domain $Ω = [a, b].$ We first transform the target equation into the standard heat equation on the domain excluding the singular point equipped with an inner interface matching (IIM) condition on the singular point $x = ξ ∈ (a, b),$ then adopt Taylor’s expansion to approximate the IIM condition at the singular point and apply second-order finite difference method to approximate the standard heat equation at the nonsingular points. This discrete procedure allows us to choose different grid sizes to partition the two sub-domains $[a, ξ]$ and $[ξ, b],$ which ensures that $x = ξ$ is a grid point, and hence the proposed schemes can be generalized to the heat equation with more than one concentrated capacities. We prove that the two proposed schemes are uniquely solvable. And through in-depth analysis of the local truncation errors, we rigorously prove that the two schemes are second-order accurate both in temporal and spatial directions in the maximum norm without any constraint on the grid ratio. Numerical experiments are carried out to verify our theoretical conclusions.

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The $k$-th ($k = 3, 4, 5$) order backward differential formula (${\rm BDF}k$) is applied to develop the high order energy stable schemes for the molecular beam epitaxial model with slope selection. The numerical schemes are established by combining the convex splitting technique with the $k$-th order accurate Douglas-Dupont stabilization term in the form of $S\tau^{k−1}∆_h(\phi^n−\phi^{n−1}).$ With the help of the new constructed discrete gradient structure of the $k$-th order explicit extrapolation formula, the stabilized ${\rm BDF}k$ scheme is proved to preserve energy dissipation law at the discrete levels and unconditionally stable in the energy norm. By using the discrete orthogonal convolution kernels and the associated convolution embedding inequalities, the $L^2$ norm error estimate is established under a weak constraint of time-step size. Numerical simulations are presented to demonstrate the accuracy and efficiency of the proposed numerical schemes.

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For complex-valued or quaternionic neural networks, scholars and researchers usually decompose them into real-valued systems. The decomposed real-valued systems are equivalent to original systems. Then, the dynamical behaviors of real-valued systems obtained are investigated, including stability, synchronization, and chaos etc. In this paper, a class of quaternionic neural networks with time-varying delays is investigated. First, by designing a suitable PI controller, synchronization of the considered chaotic system is realized. By using a non-decomposition method and structuring a novel Lyapunov functional, sufficient conditions are derived to guarantee synchronization between the drive-response systems. It is worth mentioning that, unlike other methods, our approach does not require breaking down the quaternionic neural networks into four separate real-valued systems. Furthermore, we demonstrate the practical application of these chaotic quaternionic neural networks with time-varying delays in image encryption and decryption. Based on one sequence of chaotic signal from state trajectory of single quaternion-valued neuron and a new encryption algorithm, the application of chaotic system proposed, that is, image encryption, is researched. The process of image decryption is simply the reverse of the encryption process. Finally, numerical simulation examples are provided to validate the effectiveness of the designed PI controller and performance of image encryption and decryption.

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This paper deals with the numerical solutions of two-dimensional (2D) semi-linear reaction-diffusion equations (SLRDEs) with piecewise continuous argument (PCA) in reaction term. A high-order compact difference method called I-type basic scheme is developed for solving the equations and it is proved under the suitable conditions that this method has the computational accuracy $\mathcal{O}(\tau^2+h^4_x+h^4_y),$ where $\tau,$ $h_x$ and $h_y$ are the calculation stepsizes of the method in $t$-, $x$- and $y$-direction, respectively. With the above method and Newton linearized technique, a ${\rm II}$-type basic scheme is also suggested. Based on the both basic schemes, the corresponding ${\rm I}$- and ${\rm II}$-type alternating direction implicit (ADI) schemes are derived. Finally, with a series of numerical experiments, the computational accuracy and efficiency of the four numerical schemes are further illustrated.

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Resonant tunneling diodes (RTDs) exhibit a distinctive characteristic known as negative resistance. Accurately calculating the tunneling bias energy is indispensable for the design of quantum devices. This paper conducts a thorough investigation into the current-voltage (I-V) characteristics of RTDs utilizing various numerical methods. Through a series of numerical experiments, we verified that the transfer matrix method ensures robust convergence in I-V curves and proficiently determines the tunneling bias for energy potential functions with discontinuities. Our numerical analysis underscores the significant impact of variations in effective mass on I-V curves, emphasizing the need to consider this effect. Furthermore, we observe that increasing the doping concentration results in a reduction in tunneling bias and an enhancement in peak current. Leveraging the unique features of the I-V curve, we employ shallow neural networks to accurately fit the I-V curves, yielding satisfactory results with limited data.

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This report presents a series of implicit-explicit (IMEX) variable stepsize algorithms for natural convection equations. The presented method requires a minimally intrusive modification to an existing program, does not add to the computational complexity, and is conceptually simple. Here, IMEX means the nonlinear term is treated fully explicitly, while the remaining terms are treated implicitly. Due to the increasing demand for low memory solvers, the addition of time adaptive can improve the accuracy and efficiency of the algorithms. For the first-order algorithm, we prove the stability of the variable stepsize backward Euler scheme combined with Adams-Bashforth 2 (VSS BE-AB2) and analyze convergence. Then, the stability of Constant Timestep Filtered-BE-AB2 (BE-AB2+F) is proved. Moreover, we construct adaptive algorithms by extending the approach to variable stepsize. Finally, numerical tests confirm the convergence rates of our method and validate the theoretical results.

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Anderson acceleration is a kind of effective method for improving the convergence of the general fixed point iteration. In the linear case, Anderson acceleration can be used to improve the convergence rate of matrix splitting based iterative methods. In this paper, by using Anderson acceleration on general splitting iterative methods for linear systems, three classes of methods are given. The first one is obtained by directly applying Anderson acceleration on splitting iterative methods. For the second class of methods, Anderson acceleration is used periodically in the splitting iteration process. The third one is constructed by combining the Anderson acceleration and split iteration method in each iteration process. The key of this class of method is to determine a combination coefficient for Anderson acceleration and split iteration method. One optimal combination coefficient is given. Some theoretical results about the convergence of the considered three methods are established. Numerical experiments show that the proposed methods are effective.

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In this paper, we propose a pressure-robust weak Galerkin (WG) finite element scheme to solve the Stokes-Darcy problem. To construct the pressure-robust numerical scheme, we use the divergence-free velocity reconstruction operator to modify the test function on the right side of the numerical scheme. This numerical scheme is easy to implement because it only need to modify the right side. We prove the error between the velocity function and its numerical solution is independent of the pressure function and viscosity coefficient. Moreover, the errors of the velocity function reach the optimal convergence orders under the energy norm, as validated by both theoretical analysis and numerical results.

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The main purpose of this paper is to give stability analysis and error estimates of the ultra-weak local discontinuous Galerkin (UWLDG) method coupled with a spectral deferred correction (SDC) temporal discretization method up to fourth order, for solving the fourth-order equation. The UWLDG method introduces fewer auxiliary variables than the local discontinuous Galerkin method and no internal penalty terms are required for stability, which is efficient for high order partial differential equations (PDEs). The SDC method we adopt in this paper is based on second-order time integration methods and the order of accuracy is increased by two for each additional iteration. With the energy techniques, we rigorously prove the fully discrete schemes are unconditionally stable. By the aid of special projections and initial conditions, the optimal error estimates of the fully discrete schemes are obtained. Furthermore, we generalize the analysis to PDEs with higher even-order derivatives. Numerical experiments are displayed to verify the theoretical results.

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An analysis of logistic stochastic differential equations (SDEs) with general power-law and driven by a Wiener process is conducted. We prove existence of unique, strong Markovian, continuous solutions. The solutions live (a.s.) on bounded domains $D = [0, K]$ required by applications to biology, ecology and physics with nonrandom threshold parameter $K>0$ (i.e. the maximum carrying constant). Moreover, we present and justify nonstandard numerical methods constructed by specified balanced implicit methods (BIMs). Their weak and $L^p$-convergence follows from the fact that these methods with local Lipschitz-continuous coefficients of logistic SDEs “produce” positive numerical approximations on bounded domain [0, K] (a.s.). As commonly known, standard numerical methods such as Taylor-type ones for SDEs fail to do that. Finally, asymptotic stability of nontrivial equilibria $x_∗=K$ is proven for both continuous time logistic SDEs and discrete time approximations by BIMs. We exploit the technique of positive, sufficiently smooth and Lyapunov functionals governed by well-known Dynkin’s formula for SDEs.

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The Rytov approximation has been commonly used to obtain reconstructed images for optical tomography. However, the method requires linearization of the nonlinear inverse problem. Here, we demonstrate nonlinear Rytov approximations by developing the inverse Rytov series for the time-dependent diffusion equation. The method is verified by a solid-phantom experiment.

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In this paper, we present a posteriori error estimates of the weak Galerkin finite element method for the steady-state Poisson-Nernst-Planck equations. The a posteriori error estimators for the electrostatic potential and ion concentrations are constructed. The reliability and efficiency of the estimators are verified by the upper and lower bounds of the energy norm of the error. The a posteriori error estimators are applied to the adaptive weak Galerkin algorithm for triangle, quadrilateral and polygonal meshes with hanging nodes. Finally, numerical results demonstrate the effectiveness of the adaptive algorithm guided by our constructed estimators.

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An explicit numerical method is developed for a class of non-autonomous time-changed stochastic differential equations, whose coefficients obey Hölder’s continuity in terms of the time variables and are allowed to grow super-linearly in terms of the state variables. The strong convergence of the method in the finite time interval is proved and the convergence rate is obtained. Numerical simulations are provided.

• General Summary