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This paper proposes a linear, unconditionally energy-stable scheme for the Poisson–Nernst–Planck (PNP) equations. Based on a gradient-flow formulation of the PNP equations, the energy factorization approach is applied to linearize the logarithm function at the previous time step, resulting in a linear semi-implicit scheme. Numerical analysis is conducted to illustrate that the proposed fully discrete scheme has desired properties at a discrete level, such as unconditional unique solvability, mass conservation, and energy dissipation. Numerical simulations verify that the proposed scheme, as expected, is first-order accurate in time and second-order accurate in space. Further numerical tests confirm that the proposed scheme can indeed preserve the desired properties. Applications of our numerical scheme to the simulations of electrolyte solutions demonstrate that, as a linear energy stable scheme of efficiency, it will be promising in simulating complicated transport phenomena of charged systems. [ABSTRACT FROM AUTHOR]

In this paper, we have considered the problem of recovering the time dependent source term for the fourth order time fractional pseudoparabolic equation theoretically and numerically, for the first time, by considering an additional measurement at the left boundary of the space interval. This is very challenging and interesting inverse problem with many important applications in various fields of physics and mechanics. The existence of unique solution to the problem has been discussed by means of the contraction principle and Banach Fixed-point theorem on a small time interval and the unique solvability theorem is proved. The stability results for the inverse problem have also been presented. However, since the governing equation is yet ill-posed (very slight errors in the additional input may cause relatively significant errors in the output force), the regularization of the solution is needed. Therefore, to get a stable solution, a regularized objective function is to be minimized for retrieval of the unknown coefficient of the forcing term. The proposed problem is discretized using the quintic B-spline (QnB-spline) collocation technique and has been reshaped as a non-linear least-squares optimization of the Tikhonov regularization function. The stability analysis of the direct numerical scheme has also been presented. The MATLAB subroutine l s q n o n l i n tool has been used to expedite the numerical computations. Both analytical and perturbed data are inverted and the numerical outcomes for two benchmark test examples are reported and discussed. [ABSTRACT FROM AUTHOR]

Multi-physics are widely studied and used in numerical analysis and simulation. Physics-based splitting methods are introduced, in which different methods are applied to solve each individual aspect of a physical system in multiple steps. The implicit methods are typically used for temporal discretization and are unconditionally stable such that the time step size can be large during the computation. However, the implicit methods are more complicated to solve, especially for nonlinear problems. In contrast, the explicit methods are relatively easier to compute, while they may require a much smaller time step size. In this paper, we propose a partially explicit scheme with physics-based splitting. We take the convection diffusion equation as an example. In this scheme, the convection equation is solved using the exact solution, and then we solve the diffusion equation via finite element methods. Some degrees of freedom in the diffusion equation are treated implicitly while others are treated explicitly. We can change the large-scale implicit system into one smaller-scale implicit system and one smaller-scale explicit system, which provides saving in computation. We analyze the stability and the order of accuracy. We also present some numerical results to show the performance of our proposed scheme. [ABSTRACT FROM AUTHOR]

Partial differential equations (PDEs) arise naturally in a wide variety of scientific areas and applications, and their numerical solutions are highly indispensable in many cases. Typical spectral/pseudospectral (PS) methods for solving PDEs work well only for regular domains such as rectangles or disks; however, the application of these methods to irregular domains is not straightforward and difficult enough to consider them less appealing as numerical tools. This research paper endeavors to take advantage of these methods in complex domains by introducing a novel, high-order numerical method that brings into play domain embedding into a regular, square computational domain in combination with integral reformulation, fully exponentially convergent Fourier PS collocation, and the Fourier-Continuation (FC)-Gram method integrated with a novel predictor–corrector-continuation algorithm to improve the accuracy of the extrapolated data. We developed some new formulas to construct the first- and second-order Fourier integration matrices (FIMs) based on equally-spaced nodes within the interval of integration. Modified FIMs were also developed to compute integral approximations of smooth, periodic functions when the upper limits of integration are any random points in the interval of integration. An algorithm for the fast and economic construction of the first-order FIM was also derived. A rounding error analysis based on numerical simulations demonstrates that the rounding errors in the calculation of the elements of the developed FIMs of size N are roughly of order less than or equal to O (N u R) , where u R ≈ 2. 22 × 1 0 − 16 is the unit round-off of the double-precision floating-point system. The powerful features of the proposed method are illustrated through the study of the numerical solution of two-dimensional linear PDEs of Poisson type with constant coefficients and two different sets of boundary conditions. Two test problems are presented to demonstrate the overall capabilities of the proposed method. The current research paper investigations can be very useful when dealing with more complicated problems such as nonlinear PDEs in complex domains, or optimal control problems governed by linear/nonlinear PDEs in complex domains. [ABSTRACT FROM AUTHOR]

Generalized/extended finite element methods (GFEM/XFEM) for an interface problem are generally enriched by a distance function to the interface to handle discontinuities across the interface. Evaluations of the distance function and its gradients are needed for assembling stiffness matrices, which requires certain computational geometry algorithms. The computational complexity grows if the number of integration points is increased. Besides, the mathematical analysis on the effect of numerical integration has not been established in the field of GFEM/XFEM yet. This study designs an improved enriched function for a stable GFEM (SGFEM, a stable version of GFEM) based on the distance function, which only requires evaluating the distance function at nodes of elements that interact the interface (the gradients of distance function are not needed). The optimal convergence order (in the energy norm) of the SGFEM with such an improved enriched function is proven. In turn, we propose a perturbed variational formula, which can be integrated exactly by simple Gaussian rules designed in this paper. We prove that the SGFEM based on the improved enriched function and the perturbed variational formula (exactly integrated by the numerical integration) achieves the optimal convergence order for the interface problem. Namely, we develop a numerical integration rule for the SGFEM such that the optimal convergence order of SGFEM is maintained for the interface problem. The theoretical achievements are verified by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we show how to construct a positive semi-discrete Lagrangian–Eulerian scheme on triangular grids that asymptotically satisfies multidimensional scalar hyperbolic problems and first-order nonlinear systems of conservation laws. The construction of such numerical scheme is based on the improved concept of space–time no-flow curves. For scalar equations, through a sequence of computational grid functions that tend to satisfy the underlying hyperbolic conservation law(s) in a weak sense in the space variable and in a strong sense in the time variable, we prove that the proposed scheme converges to the unique weak solution satisfying a type of Kruzhkov entropy condition. For nonlinear systems of first-order hyperbolic conservation laws, we show that the proposed scheme also satisfies a (weak) positivity principle. This provides an effective numerical analysis tool (based on weak asymptotic methods) to construct a family of continuous functions that asymptotically satisfies scalar conservation laws with discontinuous nonlinearity and systems that have irregular solutions. In order to assess the robustness of the proposed approach, we solve a series of numerical tests on several hyperbolic problems (scalar and systems) using the positive semi-discrete Lagrangian–Eulerian scheme on triangular grids. The results confirm that the proposed method is efficient (since it is Riemann-solver-free) and provides good resolution for scalar models and for the solution of systems of conservation laws. [ABSTRACT FROM AUTHOR]

Abstract This paper proposes a simplified reproducing kernel method to solve the linear Volterra integral equations with variable coefficients. The main idea of the method is to establish a reproducing kernel direct space that can be used in Volterra integral equations. And in the first time, this paper analyzes the convergence order and stability of the approximate solution. Then the uniform convergence of the numerical solution is proved, and the time consuming Schmidt orthogonalization process is avoided. The proposed method is proved to be stable and is not less than the second order convergence. The algorithm is proved to be feasible and stable through some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, the time discrete higher order projection method is proposed and analyzed for the time-dependent viscoelastic flow problem. Our numerical method is based on the time iterative discrete schemes. By the projection method, the considered problem is decoupled into two linear subproblems: One is for the velocity and the other is for the pressure. Unconditional stability of the numerical schemes is established. Convergence results for the velocity and pressure are also derived. Our main results of this paper are that the convergence analysis for the velocity is weakly second order and for the pressure is weakly first order. Finally, some numerical examples are provided to confirm the performances of the developed numerical algorithms. [ABSTRACT FROM AUTHOR]

The variogram is an important element in geostatistical studies that has been used as a tool to investigate the variability and continuity of the study phenomenon. The classical Matheron estimator for calculating the experimental variogram is used in this paper. A new variogram model is proposed, the power-law exponential model and a comparative study of three-time series datasets with the characteristic of being periodic is performed: rainfalls, particulate matter PM10, and Ozone series. Three other models were considered for comparison: spherical, exponential, and quadratic–exponential. The numerical fit and analysis are carried out using the least-squares method provided by the R statistics software, and testing the solutions by the coefficient of determination R 2 and the R M S E measure. Our results provided a good adaptation of the proposed model to the different datasets, satisfying the equality of variances, and also showing a robust fit when adding noise. Moreover, our model presented several advantages over the others, regarding the mentioned characteristics. This suggests that the power-law exponential model is a variant that adds new attributes to the set of existing variogram models. • A new variogram model was proposed to characterize the variations of the spatio-temporal data. • The results of the proposed model were analyzed numerically and studied statistically. • A comparative study is carried out, based on variograms, of three different variables: rainfalls, PM10, and Ozone. • The proposed model has a higher precision, which produces a new guide for further studies. [ABSTRACT FROM AUTHOR]

In this paper, a multi-component series system is considered. The system is monitored periodically. The exact cause of failure is assumed to be masked or missed. In the masked setup, the exact cause of failure is unknown but the set to which it belongs, called the masked set, is known. While in the missing setup, there is no information about the exact cause of failure. A time-dependent covariate-based maintenance policy is proposed such that the maintenance action and cost of the failed components at inspection times depend on several factors and can vary. The component lifetime distributions are considered unknown. The proposed maintenance policy is optimized using some task-driven decision-making statistical learning methods. Finally, the applicability of the proposed theory is analyzed through some numerical analysis. The results are compared to the case where lifetime distributions are known as a benchmark. [ABSTRACT FROM AUTHOR]

In this paper, a modified weak Galerkin method is proposed for the Stokes problem. The numerical scheme is based on a novel variational form of the Stokes problem. The degree of freedoms in the modified weak Galerkin method is less than that in the original weak Galerkin method, while the accuracy stays the same. In this paper, the optimal convergence orders are given and some numerical experiments are presented to verify the theory. [ABSTRACT FROM AUTHOR]

The interior penalty methods using C 0 Lagrange elements ( C 0 IPG ) developed in the last decade for the fourth order problems are an interesting topic in academia at present. In this paper, we apply the methods to the Helmholtz transmission eigenvalue problem which is quadratic and non self-adjoint and has important applications in the inverse scattering theory. We propose a discrete scheme, present a complete error analysis and first prove the C 0 IPG method can capture smooth eigenpairs efficiently. Using the argument in this paper we can prove that the C 0 IPG methods for the fourth order eigenvalue problems in the existing literature can also capture smooth eigenpairs efficiently. Numerical examples confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

Space-fractional partial differential equations are used for simulations of, for example, diffusion of radioactive materials, and financial and other models, which are characterized by heavy-tailed distributions. A number of first order accuracy finite difference methods have been proposed. In the present paper, we introduce second order accuracy finite difference methods with Dirichlet boundary conditions. These methods have a parameter in these schemes, and the parameter stabilizes the schemes. This means that there exist various schemes with second order accuracy, but the stability of each scheme is different. In the present paper, we introduce the most stable scheme for any fractional calculus order by choosing the optimal parameter. In addition, we describe a phenomenon whereby the expected accuracy cannot be obtained if the analytical solution can be expanded to a series having less than second order around boundaries. This also happens in both existing methods and the proposed methods. In the present paper, we develop the stability conditions for the proposed schemes, and numerical examples of second order accuracy and accuracy decay are shown. [ABSTRACT FROM AUTHOR]

In this paper we discuss the multiscale approach for the scattering problem of Maxwell’s equations in a heterogeneous material with a periodic microstructure. The new contributions in this paper are the determination of the multiscale correctors and the strong convergence in the norm of the space H ( curl ; Ω ) with an explicit rate for the approximate solutions (see Theorem 2.4 ). Consequently, we present a multiscale hybrid finite element method–boundary element method (FE–BE). The numerical examples are carried out to validate the theoretical results of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we study variational problems where the cost functional involves the tempered Caputo fractional derivative. Several important optimization conditions are derived to find the optimal solution. Sufficient and necessary conditions are presented for different variational problems. For example, the cases of integral (isoperimetric problem) and holonomic constraints are considered, as well as problems with high order derivatives. A numerical scheme is proposed to determine approximations of the solution and it is illustrated through some examples. [ABSTRACT FROM AUTHOR]

Recently, various evolutionary partial differential equations (PDEs) with a mixed derivative have been emerged and drawn much attention. Nonetheless, their PDE-theoretical and numerical studies are still in their early stage. In this paper, we aim at the unified framework of numerical methods for such PDEs. However, due to the presence of the mixed derivative, we cannot discuss numerical methods without some appropriate reformulation, which is mathematically challenging itself. Therefore, we first propose a novel procedure for the reformulation of target PDEs into a standard form of evolutionary equations. This contribution may become an important basis not only of numerical analysis, but also of PDE-theory. In order to illustrate this point, we establish the global well-posedness of the sine-Gordon equation. After that, we classify and discuss the spatial discretizations based on the proposed reformulation technique. As a result, we show the average-difference method is suitable for the discretization of the mixed derivative. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical solution of an evolutionary variational–hemivariational inequality arising in a dynamic contact problem. The material is assumed to be viscoelastic with short memory. The contact is featured by a normal damped response in the normal direction and by the Tresca friction law in the tangential direction. The linear finite elements are used to discretize the spatial variable. Optimal order error estimates are derived for the discrete velocity and discrete displacement under suitable solution regularity assumptions. [ABSTRACT FROM AUTHOR]

Abstract An open problem in the numerical analysis of spectral methods for fractional differential equations is how to maintain the high-order accuracy for non-smooth solutions. The limited regularity of the exact solution to these equations causes a deterioration in the orders of convergence of standard schemes. In this paper, we derive and analyze an exponentially accurate Jacobi spectral-collocation method for the numerical solution of nonlinear terminal value problems involving the Caputo fractional derivative of rational-order θ ∈ (0 , 1). The main ingredient of the proposed approach is to regularize the solution by a suitable smoothing transformation, which allows us to adjust a parameter in the solution according to different given data to maximize the convergence rate. We systematically describe the necessary steps in the implementation process. Additionally, a comprehensive numerical analysis including error estimates under the L ∞ - and weighted L 2 -norms is derived. The extensive numerical examples that accompany our analysis confirm our theoretical estimates, as well as give additional insights into the convergence behavior of our method for problems with smooth and non-smooth solutions. Highlights • The nonlinear terminal value problem is recast into an equivalent integral equation. • A variable transformation is used to obtain a new equation with a smoother solution. • The Jacobi collocation method is applied to the transformed equation. • A priori error analysis for the method under the L ω θ − 1 , 0 2 - and L ∞ -norms is derived. • The method possesses spectral convergence with efficient computational time. [ABSTRACT FROM AUTHOR]

Abstract In this paper the authors discuss a numerical simulation problem of three-dimensional contamination treatment from nuclear waste. The mathematical model is defined by a nonlinear system of an initial–boundary problem of four partial differential equations: an elliptic equation for the fluid pressure, three convection–diffusion equations for the brine, radionuclides and heat. The pressure appears within the concentration equations and heat conduction equation, and the Darcy velocity controls the concentrations and the temperature. The pressure is solved by the conservative mixed volume element method, and the order of accuracy is improved by the Darcy velocity. The concentrations and temperature are treated by a combination of a multistep method, the upwind approximation and a mixed volume element. The partial derivatives to time variable are approximated by a multistep method. A mixed volume element and an upwind scheme are used to discretize the diffusion and the convection, respectively. The composite method can solve the convection-dominated diffusion problems well because it eliminates numerical dispersion and nonphysical oscillation and has high order computational accuracy. The mixed volume element has the local conservation of mass and energy, and it can obtain the concentrations and temperature and their adjoint vector functions simultaneously. The conservation nature plays an important role in numerical simulation of underground fluid. By the technique of a priori estimates of differential equations, we derive an optimal second order result in L 2 norm. Numerical examples are given to show the effectiveness and practicability of our approach and the composite method is testified as a powerful tool for solving the challenging benchmark problem. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the initial value problem for a quasilinear singularly perturbed delay Volterra integro-differential equation was considered. By the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with the weight and remainder term in integral form, a fitted difference scheme is constructed and analysed. It is shown that the method displays first order uniform convergence in perturbation parameter. Some numerical results are given to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider how to accurately solve the linear system whose coefficient matrix is a generalized sign regular (GSR) matrix with signature (1 , ... , 1 , − 1). A new algorithm with O (n 2) complexity is presented to solve the GSR linear system, provided that parameterization matrices of coefficient matrices are available. We illustrate that no subtraction-cancellation occurs in the computations of the algorithm, which guarantees that all the solution components are computed with a desirable accuracy. An error analysis and numerical experiments are presented to confirm the high accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we deal with epistemic uncertainty in the framework of Dempster–Shafer theory, where basic belief assignments are used to characterize the uncertain parameters. To propagate the mixed epistemic and aleatory uncertainties, we introduce the relevant theoretical basics of DS theory, present the numerical approach based on DS theory combined with generalized polynomial chaos expansion, and conduct the error analysis for the numerical approach. Specifically, to analyze the convergence rate of the numerical solution represented with basic belief assignments, we define a measure based on the Hausdorff distance to quantify the difference between two basic belief assignments. The convergence of the numerical approximation is demonstrated with a few simple examples under different scenarios, and the presented numerical approach is applied to quantify the mixed types of uncertainty in quasi-one-dimensional flow. [ABSTRACT FROM AUTHOR]

In this paper, we introduced the notion of an R ν -generalized solution of the Oseen problem in rotation form in two-dimensional polygonal domain with a reentrant corner at the boundary. Its existence and uniqueness in the nonsymmetric variational formulation of the problem is established. An estimate related to the conservation of the energy balance of the approximation velocity field for the Navier–Stokes problem in the rotation form is obtained. A weighted finite element method is constructed. A series of numerical experiments of test examples based on the proposed method is carried out. A comparative analysis of the approach with the classical FEM is performed. The results of numerical experiments showed a significant advantage of the proposed approach. • Existence and uniqueness of an R ν -generalized solution of Oseen problem is proved. • Crank–Nicolson scheme for solving Navier–Stokes problem is constructed. • Estimate related an energy balance of an approximation velocity field is obtained. • Weighted finite element method is constructed. • Series of numerical experiments of test examples is carried out. [ABSTRACT FROM AUTHOR]

The focus of this paper is the development and analysis of a time filtering process for a linear hyperbolic equation motivated by the modeling of the transcription of ribosomal RNA in bacteria Davis et al. (2021). We demonstrate that a time filter technique can be combined with the classical upwind to produce a new explicit scheme with virtually no dissipation introduced by the method, and the filter can be implemented with minimal computational cost. The analysis shows that the filtered scheme gives the practitioner the ability to adjust the filtering so the dissipation can be made arbitrarily small over a range of time step choices. The analysis also indicates that the filtered scheme has a smaller local truncation error when compared to that of the original upwind method. A CFL condition for the new algorithm is derived, and it is shown to depend explicitly on the filter parameter. Numerical computations illustrate stability and convergence as well as dissipation and dispersion assessments of the filtered upwind scheme. [ABSTRACT FROM AUTHOR]

This paper proposes a new approach to stabilize switched linear systems. In this method, the projections are employed to establish the stabilization of a switched systems class. Hence, we suppose that the subsystems given by a switched linear system can be projected to the same subspace. Under these conditions, we prove that this switched linear system is stabilizable if and only if a low-order switched linear system is also stabilizable. In order to complete this study, we present a counter example that proves that it is not always possible to use the projections. Moreover, the main result in the paper is applied to solve the stabilization of a third-order switched systems class and the static feedback stabilization of a switched linear systems class. Finally, a numerical example is included in order to illustrate the new method obtained to stabilize a third-order switched systems class. [ABSTRACT FROM AUTHOR]

This paper presents a method of numerical verification for the existence of a global-in-time solution to a class of semilinear parabolic equations. Such a method is based on two main theorems in this paper. One theorem gives a sufficient condition for proving the existence of a solution to the semilinear parabolic equations with the initial point t = t ′ ≥ 0 . If the sufficient condition does not hold, the other theorem is used for enclosing the solution for time t ∈ ( 0 , τ ] , τ > 0 in a neighborhood of a numerical solution. Numerical results of obtaining a global-in-time solution for a certain semilinear parabolic equation are also given. [ABSTRACT FROM AUTHOR]

The main contribution of this paper is to propose a new hybrid optimization method for the multiplicative noise and blur removal problem. A degraded image can often be recovered efficiently by minimizing an objective function which consists of a data-fidelity term and a regularization term. In the paper, we apply the quadratic penalty function method combined with the alternating direction method to minimize the corresponding objective function. Numerical experiments are presented to demonstrate the effectiveness of the proposed method. Experimental results illustrate the state-of-the-art performance of the proposed method to handle multiplicative noise and blur removal problem. [ABSTRACT FROM AUTHOR]

Abstract Field experiments of solute transport through heterogeneous porous and fractured media show that the growth of contaminant plumes may convert between diffusive states. In this paper, we propose a multi-term time–space variable-order fractional advection–diffusion model (MTT-SVO-FADM) to describe the underlying transport dynamics. We consider a numerical approach based on the implicit numerical method for numerical solution of this model. A fully-discrete numerical scheme is developed by using the classical finite difference method. The unconditional stability and convergence of the scheme are discussed and theoretically proved. We use a modified grid approximation method (MGAM) to estimate the model's parameters. The MTT-SVO-FADM is then applied to describe transient dispersion observed at a field tracer test and four numerical experiments. The results show that this model can simulate the experimental data more accurately and can efficiently quantify these transitions. Highlights • A multi-term time–space variable-order fractional advection–diffusion model is proposed. • A fully-discrete numerical scheme is developed. • The unconditional stability and convergence of the scheme are discussed and proved. • A modified grid approximation method is used to estimate the model's parameters. • This model is applied to describe transient dispersion observed at a field tracer test and four numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider a coupled system of m (≥ 2) singularly perturbed semilinear reaction–diffusion equations with a discontinuous source term having a discontinuity at a point in the interior of the domain. The diffusion term of each equation is multiplied by small singular perturbation parameter, but these parameters are assumed to be different in magnitude. A numerical method is constructed on a variant of Shishkin mesh. The approximations generated by this method are shown to be almost second order uniformly convergent with respect to all perturbation parameters. Numerical results are in support of the theoretical results. Highlights • A coupled system of m (≥ 2) singularly perturbed semilinear reaction-diffusion equations is considered. • The source term is having a discontinuity at a point in the interior of the domain. • The diffusion term of each equation is multiplied by a small perturbation parameter. • All the perturbation parameters are different in magnitude. • The constructed numerical method has almost second parameter uniform convergence. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a block product (BP) preconditioner is established for saddle point problems. Spectral properties of the BP preconditioned matrix are investigated. A strategy for practical choice of quasi-optimal parameter is given. Numerical results on saddle point linear systems arising from Stokes problems and weighted least square problems show that the proposed BP preconditioner is more economic to implement within Krylov subspace acceleration than some extensively studied preconditioners. [ABSTRACT FROM AUTHOR]

Abstract In this paper, an efficient numerical method is proposed for the 1D elliptic type interface problems. We first construct a broken reproducing kernel space and then apply the simplified reproducing kernel method to the interface problems. The convergence of the proposed method is discussed. Finally, some representative numerical examples are presented to illustrate the accuracy and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract This paper is devoted to numerical analysis of history-dependent variational– hemivariational inequalities arising in contact problems for viscoelastic material. We introduce both temporally semi-discrete approximation and fully discrete approximation for the problem, where the temporal integration is approximated by a trapezoidal rule and the spatial variable is approximated by the finite element method. We analyze the discrete schemes and derive error bounds. The results are applied for the numerical solution of a quasistatic contact problem. For the linear finite element method, we prove that the error estimation for the numerical solution is of optimal order under appropriate solution regularity assumptions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we apply local discontinuous Galerkin methods to the compressible wormhole propagation. Optimal error estimates for the pressure, velocity, porosity and concentration in different norms are established on non-uniform grids. Numerical experiments are presented to verify the theoretical analysis and show the good performance of the LDG scheme for compressible wormhole propagation. [ABSTRACT FROM AUTHOR]

Abstract The spectral conjugate gradient method is an effective method for large-scale unconstrained optimization problems. In this paper, based on Quasi-Newton direction and Quasi-Newton equation, a new spectral conjugate gradient method is proposed. This method is motivated by the three-term modified Polak-Ribiyre-Polyak (PRP) method and spectral parameters. The global convergence of algorithm is proved for general functions under a strong Wolfe line search. Numerical results show that the new algorithm is superior to the three-term modified PRP method. [ABSTRACT FROM AUTHOR]

Abstract In this paper, our main aim is to investigate the stability and strong convergence of an implicit numerical approximations for neutral-type stochastic differential equations with superlinearly growing coefficients. After providing moment boundedness and exponential stability for the exact solutions, we show that the backward Euler–Maruyama numerical method preserves stability and boundedness of moments, and the numerical approximations converge strongly to the true solutions for sufficiently small step size. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present a neural network model for solving semidefinite programming. We show that the equilibrium point of the proposed neural network and the KKT point of the semidefinite programming are equal. By employing Lyapunov function approach, it is also shown that the suggested neural network model is stable in the sense of Lyapunov and globally convergent to an exact optimal solution of the original problem. Some applications of the proposed scheme are stated by modeling of some important problems as semidefinite programming problems. Numerical simulations verify the obtained theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we construct wavelet tight frames with n vanishing moments for Dubuc–Deslauriers 2 n -point semi-regular interpolatory subdivision schemes. Our motivation for this construction is its practical use for further regularity analysis of wide classes of semi-regular subdivision. Our constructive tools are local eigenvalue convergence analysis for semi-regular Dubuc–Deslauriers subdivision, the Unitary Extension Principle and the generalization of the Oblique Extension Principle to the irregular setting by Chui, He and Stöckler. This group of authors derives suitable approximation of the inverse Gramian for irregular B-spline subdivision. Our main contribution is the derivation of the appropriate approximation of the inverse Gramian for the semi-regular Dubuc–Deslauriers scaling functions ensuring n vanishing moments of the corresponding framelets. [ABSTRACT FROM AUTHOR]

Abstract Although the numerical methods to stochastic differential equations with the coefficients of locally Lipschitz and polynomial growth have been discussed commonly by some authors, there are few works on the high strong order numerical methods. In this paper, the mean-square convergence of the general projected explicit Itô–Taylor methods is considered under the assumption that drift and diffusion coefficient functions of stochastic differential equations satisfy the global monotonicity condition, suitable local Lipschitz conditions and polynomial growth conditions. Our analysis follows the idea of stochastic C-stability and stochastic B-consistency. After giving the selection strategy of optimal parameters, we prove that the projected explicit Itô–Taylor methods, owning optimal parameters, share the same mean-square convergence orders with non-projected ones. Finally, two numerical experiments are presented to show the effectiveness of theoretical results. Highlights • The general projected explicit Itô–Taylor methods for SDEs are constructed. • A simple selection strategy for proper projection parameters is introduced. • The methods run well for SDEs under locally Lipschitz, polynomial growth condition. • The methods hold the same mean-square convergence order as non-projection ones. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study the interaction of pedestrian flow and car traffic. A microscopic model for car traffic on road networks is coupled to microscopic pedestrian dynamics by introducing mutual interaction forces. Based on this description coupling conditions for corresponding macroscopic models as well as for mixed microscopic–macroscopic systems can be motivated. In several numerical examples we study the dynamics of the respective descriptions. [ABSTRACT FROM AUTHOR]

Abstract In this paper we provide an expanded mixed finite element method for a class of two-dimensional Sobolev equations. The optimal error estimates for a semi-discrete scheme and a fully discrete scheme are obtained. Also numerical examples are stated to verify our theoretical results. [ABSTRACT FROM AUTHOR]

Abstract Recently, the l p -norm regularization minimization problem (P p λ) has attracted great attention in compressed sensing. However, the l p -norm ‖x‖ p p in problem (P p λ) is nonconvex and non-Lipschitz for all p ∈ (0 , 1) , and there are not many optimization theories and methods proposed to solve this problem. In fact, it is NP-hard for all p ∈ (0 , 1) and λ > 0. In this paper, we study one modified l p -norm regularization minimization problem to approximate the NP-hard problem (P p λ). Inspired by the good performance of Half algorithm in some sparse signal recovery problems, an iterative thresholding algorithm is proposed to solve our modified l p -norm regularization minimization problem (P p , 1 ∕ 2 , ϵ λ). Numerical results on some sparse signal recovery problems show that our algorithm performs effectively in finding the sparse signals compared with some state-of-art methods. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a construction of Marsden's identity for UAH B-splines (i.e. Uniform Algebraic Hyperbolic B-splines) is developed and a clear proof is given. With the help of this identity, quasi-interpolant schemes that produce the space of algebraic hyperbolic functions are derived. Efficient quadrature rules, based on integrating some of these quasi-interpolant schemes, are constructed and studied. Numerical results that illustrate the effectiveness of these rules are presented. [ABSTRACT FROM AUTHOR]

Abstract Time-varying nonlinear optimization (TVNO) problems are considered as important issues in various scientific disciplines and industrial applications. In this paper, the continuous-time derivative dynamics (CTDD) model is developed for obtaining the real-time solutions of TVNO problems. Furthermore, aiming to remedy the weaknesses of CTDD model, a continuous-time zeroing dynamics (CTZD) model is presented and investigated. For potential digital hardware realization, by using bilinear transform, a general four-step Zhang et al discretization (ZeaD) formula is proposed and applied to the discretization of both CTDD and CTZD models. A general four-step discrete-time derivative dynamics (general four-step DTDD) model and a general four-step discrete-time zeroing dynamics (general four-step DTZD) model are proposed on the basis of this general four-step ZeaD formula. Further theoretical analyses indicate that the general four-step DTZD model is zero-stable, consistent and convergent with the truncation error of O (g 4) , which denotes a vector with every entries being O (g 4) with g denoting the sampling period. Theoretical analyses also indicate that the maximal steady-state residual error (MSSRE) has an O (g 4) pattern confirmedly. The efficacy and accuracy of the general four-step DTDD and DTZD models are further illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

Abstract A novel method for synchronization of fractional order complex dynamical networks using a new fractional Proportional–Integral (PI) pinning control scheme is presented in this paper. Stability of this network is proved via the Lyapunov stability theorem, and the synchronizability criterion is obtained. Although one can also design a proportional pinning control scheme to synchronize the whole network, our proposed fractional PI control is more efficient. Numerical simulations on Lorenz oscillator's interaction over a sample network have shown that the proposed fractional PI controller can provide a faster synchronization with less overshoot to the reference state compared to conventional proportional controller with the same power consumption. [ABSTRACT FROM AUTHOR]

Abstract The regularization method could deal with the swamp effect of alternating least squares (ALS) algorithms for tensor decomposition. Usually, the regularization term is a norm of the difference between the solution and the current iterate. In this paper, we show that the norm could be weakened to a seminorm, so the selection of the regularization term could be more flexible. To overcome the swamp effect and avoid the drawback that the Hessian of the subproblem may get close to singular in the iterative process, we propose a seminorm regularized ALS algorithm for solving the canonical tensor decomposition. Moreover, in the new algorithm, we introduce a novel extrapolation in the update of each mode factor which makes an immediate impression on the update of subsequent ones. By assuming the boundness of the infinite sequence of iterates generated by the new algorithm, we establish the global convergence and the (weakly) linear convergence rate of the sequence of iterates Numerical experiments on synthetic and real-world problems illustrate that the new method is efficient and promising. [ABSTRACT FROM AUTHOR]

Abstract In this paper a multivariate negative binomial–Lindley distribution is introduced. The univariate version of this distribution comprises an alternative of the model proposed by Zamani and Ismail (2010). With respect to this latter model, this representation presents some advantages since it provides a tractable model with attractive properties that makes it suitable for application in any field where overdispersion is observed. Some properties of the model are studied including a recurrence for the probabilities of the multivariate and univariate distribution. An EM algorithm is derived to estimate the parameters of this multivariate distribution. Finally, some numerical applications for the univariate and bivariate cases are given to illustrate the performance of this model. [ABSTRACT FROM AUTHOR]

Abstract This paper considers signal recovery in the framework of cumulative coherence. First, we show that the Lasso estimator and the Dantzig selector exhibit similar behavior under the cumulative coherence. Then we estimate the approximation equivalence between the Lasso and the Dantzig selector by calculating prediction loss difference under the condition of cumulative coherence. And we also prove that the cumulative coherence implies the restricted eigenvalue condition. Last, we illustrate the advantages of cumulative coherence condition for three class matrices, in terms of the recovery performance of sparse signals via extensive numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract This paper considers how to solve a class of tensor equations arising from the unified definition of tensor–vector products. Of special interest is the order-3 tensor equation whose solutions are the intersection of a group of quadrics from a geometric point of view. Inspired by the method of alternating projections for set intersection problems, we develop a hybrid alternating projection algorithm for solving order-3 tensor equations. The local linear convergence of the alternating projection method is established under suitable conditions. Some numerical experiments are conducted to evaluate the effect of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Abstract We consider a stochastic differential equations with time-dependent delay in this paper. We first obtain the existence, uniqueness and polynomial stability of exact solution to this equation under suitable conditions. Then for the numerical method of the corresponding SDDE, we present a so called modified truncated Euler–Maruyama(MTEM) method and consider the almost sure and mean square polynomial stability of this numerical method. By using the well known discrete semimartingale convergence theorem, sufficient conditions are obtained for both bounded and unbounded delay δ to ensure the polynomial stability of the corresponding numerical approximation. Results suggest that the MTEM method replicates the polynomial stability of given SDDE under suitable conditions. Examples are presented to illustrate the conclusion. [ABSTRACT FROM AUTHOR]

In this paper a numerical analysis of in situ-biorestoration is presented. Improved stability and error estimates are derived for a P 1 finite-element method applied to coupled system of non linear partial differential equations modeling flow in porous media. These results improve upon previously derived stability and error estimates in two respects: first, a stability of the approximate solution is demonstrated in the case of a non linear diffusive flux λ ( u ) ∇ u with weaker norm assumptions than before and depends only on the initial conditions and time interval, and second, error estimates are optimal as in linear case. Extensions include the finite element approximation of flow field, described by Darcy’s law under the stream function–vorticity formulation because of its multiple advantages. Finally, numerical simulations for a correlation exponential plume are presented. [ABSTRACT FROM AUTHOR]