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This paper focuses on the problem of image restoration under Cauchy noise. The variational method, which constructs the data fidelity term involving the Cauchy distribution by MAP estimator, has been proven to be a successful approach. In this paper, a nonlinear diffusion equation is proposed to deal with it. The main ingredients of the proposed equation are a gray level based diffusivity that estimates the amplitude of the noise and a classical gradient based diffusivity that controls the anisotropic diffusion according to the image's local structure. The proposed equation has the nondivergence form, and its properties, including the existence, uniqueness, and stability of solutions, are established by the notion of viscosity solution. Experimental results show the superiority of the proposed equation over variational methods in restoring small details of images. [ABSTRACT FROM AUTHOR]

A nonoscillation criterion for a second-order linear difference equation is established correcting a result in [1]. [Copyright &y& Elsevier]

In this paper, several domain decomposition algorithms are considered to solve parallelly non-steady diffusion equations. The contributions of this work are threefold. The first one is that we present effective parallel domain decomposition schemes underlying the classical and monotone finite volume element discretization. These schemes adopt a three-stage technique, which is also used in the difference method and the finite volume method. Positivity of the parallel scheme based on monotone finite volume element method is proved. The second one is that we propose a new domain decomposition computational strategy to improve the correction step in prediction-correction frame such that the whole parallel process is simplified. The third one is that we consider a parallel algorithm build on quadratic finite volume element method, which preserves local conservation and has higher convergence rates. These parallel schemes are of intrinsic parallelism since we apply the prediction-correction technique to the interface of subdomains coming from an artificial division of computational domain or multi-medium physical domain. The proposed parallel schemes need only local communication among neighboring processors. Numerical results are presented to illuminate the accuracy, stability and parallelism of the parallel schemes. [ABSTRACT FROM AUTHOR]

In this paper, we propose a dual-mixed formulation for stationary viscoplastic flows with yield, such as the Bingham or the Herschel-Bulkley flow. The approach is based on a Huber regularization of the viscosity term and a two-fold saddle point nonlinear operator equation for the resulting weak formulation. We provide the uniqueness of solutions for the continuous formulation and propose a discrete scheme based on Arnold-Falk-Winther finite elements. The discretization scheme yields a system of slantly differentiable nonlinear equations, for which a semismooth Newton algorithm is proposed and implemented. Local superlinear convergence of the method is also proved. Finally, we perform several numerical experiments in two and three dimensions to investigate the behavior and efficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, we propose, analyze, and test an efficient algorithm for computing ensemble average of incompressible magnetohydrodynamic (MHD) flows, where instances/members correspond to varying kinematic viscosity, magnetic diffusivity, body forces, and initial conditions. The algorithm is decoupled in Elsässer variables and permits a shared coefficient matrix for all members at each time-step. Thus, the algorithm is much more computationally efficient than separately computing simulations for each member using usual MHD algorithms. We prove the proposed algorithm is unconditionally stable and convergent. Several numerical tests are given to support the predicted convergence rates. Finally, we test the proposed scheme and observe how the physical behavior changes as the coupling number increases in a lid-driven cavity problem with mean Reynolds number R e ≈ 15000 , and as the deviation of uncertainties in the initial and boundary conditions increases in a channel flow past a step problem. [ABSTRACT FROM AUTHOR]

In this paper, we point out the main disadvantage of the existing methods for solving an inverse source problems: The smoothness of the solutions is limited by the boundary condition. A Fourier extension method with a modified Tikhonov regularization is presented to overcome the limitation. Compared with previous studies, the convergence result of the new method is established under a weaker a priori assumption. Numerical experiments further verify the advantage of the method. [ABSTRACT FROM AUTHOR]

• Alternating method of Schwarz is developed for dispersed composites. • A symbolic-numerical algorithm is analyzed. • New analytical formulas for the effective properties are derived. The generalized alternating method of Schwarz can be presented as an infinite sequence of all the mutual interactions between inclusions in the boundary value problem stated for a composite. Schwarz's method for overlapping domains has been assigned to decomposition methods widely used in pure numerical computations. In the same time, the generalized alternating method for non-overlapping inclusions is convenient in symbolic-numerical computations described in this paper. Implementation of the explicit and implicit schemes yields new approximate analytical formulas for the effective properties of dispersed composites. Moreover, the exactly derived precision of their validity in concentration and in contrast parameter is established. The first order iteration corresponds to the famous Maxwell's self-consistent approach. [ABSTRACT FROM AUTHOR]

In this paper, we propose a plane wave activation based neural network (PWNN) to solve the Helmholtz equation with constant coefficients and relatively large wave number k efficiently. Since the complex activation function e i x → is introduced to be the activation function of the neural network, PWNN significantly improves the computational speed and accuracy as compared to traditional activation based neural networks (TANN) and the finite difference method (FDM), for relatively large wave number problems. We establish a new upper bound for error estimates for homogeneous Helmholtz solutions in two dimensions by plane waves, which is different from the previous estimates by generalized harmonic polynomials. Based on the new error estimates, the theoretical guidance is given for choosing the number of neural network's neurons and the initial value to accelerate network training. The analyses of computational complexity related to the wave number k are given for PWNN with two layers, TANN, the plane wave partition of unity method (PWPUM) and FDM. Numerical experiments in 2D and 3D are performed to demonstrate the efficiency and accuracy of PWNN. Especially for large wave number problems, like k = 500 , PWNN can get the solution with relative error less than 10 − 4 in less than 20 seconds, which is more efficient than other methods. [ABSTRACT FROM AUTHOR]

This paper aims to study the Galerkin finite element method (FEM) of both the semi-discrete and the fully-discrete schemes with linear triangular element for the time-dependent Joule heating problem. Through some new estimating approaches such as the different mathematical induction assumption, the mean-value technique and the special characters of this element, the superclose and superconvergence estimates for the related variables in the H 1 -norm are derived under lower regularities of the solutions of the considered problem rigorously, which simply the proof process and improve the corresponding results in the existing literature. Finally, numerical results are provided to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we develop and analyze an efficient spectral method for the transmission eigenvalue problem. By rewriting the original problem into its equivalent fourth order coupled linear eigenvalue problem, a new variational formulation is proposed based on a mixed scheme. With the help of the spectral theory of the compact operator and the approximation property of the orthogonal projection operators in non-uniform weighted Soblev spaces, error estimates for both the eigenvalues and eigenfunctions approximations are established in the d (d = 2 , 3) dimensional setting. Numerical examples are presented to show that the method converges exponentially, and is effective and efficient for computing the eigenvalues. [ABSTRACT FROM AUTHOR]

The particular and homogeneous solutions of two-dimension (2D) thermal-elastic problems are expressed by using the virtual boundary element method (VBEM) and the superposition principle. Accordingly, two matrix equations are needed and formed for particular and homogeneous solutions by the novel nonsingular boundary element method with the radial basis function interpolation (RBFI), namely virtual boundary meshfree Galerkin method (VBMGM). Considering the boundary conditions of heat conduction and combining the Galerkin method, virtual source functions of particular solutions can be solved by first using VBMGM. Applying boundary conditions of transformation, virtual source functions of homogeneous solutions can be obtained by second employing VBMGM. The coefficients of two matrix equations are symmetrical. Formed equations are nonsingular and have merits of virtual boundary element method, meshfree method, and Galerkin method. The detailed expressions, calculation steps, and simplified flow chart are given in detail, that is convenient for other scholars to study other more complex thermo-elastic problems through the method of this paper. Three examples are calculated. Comparing their calculation results with other methods, the stability and the precision of VBMGM for 2D thermal-elastic problems are validated. [ABSTRACT FROM AUTHOR]

The quadrature error associated with a regular quadrature rule for evaluation of a layer potential increases rapidly when the evaluation point approaches the surface and the integral becomes nearly singular. Error estimates are needed to determine when the accuracy is insufficient and a more costly special quadrature method should be utilized. The final result of this paper are such quadrature error estimates for the composite Gauss-Legendre rule and the global trapezoidal rule, when applied to evaluate layer potentials defined over smooth curved surfaces in R 3. The estimates have no unknown coefficients and can be efficiently evaluated given the discretization of the surface, invoking a local one-dimensional root-finding procedure. They are derived starting with integrals over curves, using complex analysis involving contour integrals, residue calculus and branch cuts. By complexifying the parameter plane, the theory can be used to derive estimates also for curves in R 3. These results are then used in the derivation of the estimates for integrals over surfaces. In this procedure, we also obtain error estimates for layer potentials evaluated over curves in R 2. Such estimates combined with a local root-finding procedure for their evaluation were earlier derived for the composite Gauss-Legendre rule for layer potentials written in complex form [4]. This is here extended to provide quadrature error estimates for both complex and real formulations of layer potentials, both for the Gauss-Legendre and the trapezoidal rule. Numerical examples are given to illustrate the performance of the quadrature error estimates. The estimates for integration over curves are in many cases remarkably precise, and the estimates for curved surfaces in R 3 are also sufficiently precise, with sufficiently low computational cost, to be practically useful. [ABSTRACT FROM AUTHOR]

• We propose a novel framework to design decoupled numerical algorithms. • The derived numerical schemes are linear and structure preserving. • Its applications on hydrodynamic models are elaborated to highlight its power. This paper presents a general numerical platform for designing accurate, efficient, and stable numerical algorithms for incompressible hydrodynamic models that obey thermodynamical laws. The obtained numerical schemes are automatically linear in time. It decouples the hydrodynamic variable and other state variables such that only small-size linear problems need to be solved at each time marching step. Furthermore, if the classical velocity projection method is utilized, the velocity field and pressure field can be decoupled. In the end, only a few elliptic-type equations shall be solved in each time step. This strategy is made possible through a sequence of model reformulations by exploring the models' thermodynamic structures. The generalized Onsager principle directly guides these reformulation procedures. In the reformulated but equivalent models, the reversible and irreversible components can be identified, guiding the numerical platform to decouple the reversible and irreversible dynamics. This eventually leads to decoupled numerical algorithms, given that the coupling terms only involve irreversible dynamics. To further demonstrate the numerical platform's power, we apply it to several specific incompressible hydrodynamic models. The energy stability of the proposed numerical schemes is shown in detail. The second-order accuracy in time is verified numerically through time step refinement tests. Several benchmark numerical examples are presented to further illustrate the proposed numerical framework's accuracy, stability, and efficiency. [ABSTRACT FROM AUTHOR]

The solution of d -dimensional fractional partial differential equations (FPDEs) is challenging in numerical computation. This paper proposes a low-rank preconditioned conjugate gradient (PCG) method for discretizations with a low-rank right-hand side stemming from the d -dimensional fractional Riesz equation. At each iteration step, a combination of low-rank in Tensor Train (TT) format technique and Fast Fourier Transform (FFT) is used to accelerate multilevel Toeplitz matrix-vector multiplications. Furthermore, a low-rank circulant preconditioner is constructed based on the spectrum of the coefficient matrix. The reported numerical results demonstrate the competitiveness of the new method compared with a state-of-the-art matrix-free approach based on FFT. [ABSTRACT FROM AUTHOR]

In this paper, we develop an efficient virtual element method to solve the two-dimension time fractional convection diffusion reaction equation involving the Caputo fractional derivative with non-smooth solutions in the time direction. The equation exhibits a weak singularity near the initial time. We first use the exponential transformation to eliminate the convection terms, and obtain the time fractional diffusion reaction equation. By utilizing properties of the energy projection operator and continuous Grönwall inequality, we derive estimates of the approximation error in the L 2 -norm and H 1 -norm under semi-discrete on polygonal meshes. Based on the L 1 scheme on graded mesh in time, the fully-discrete scheme is proved to be unconditionally stable and the optimal convergence results are derived with regards to the L 2 -norm and H 1 -norm. Finally, some numerical experiments are implemented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a finite temperature multiscale method for simulating complex dynamics with different spatial scales accurately and efficiently. We separate the domain into an MD (Molecular Dynamics) subdomain and an MC (MacrosCopic) subdomain, with a handshaking subdomain between two scales. We cast the coarse grid over the whole domain using a matching differential operator approach. We reassign the coarse grid displacement in the MD subdomain with an average of the MD solution. Using a spectral decomposition in a handshaking subdomain, we split the displacement of atoms into its mean and fluctuation parts. The mean part is reassigned by the coarse grid linear interpolation, which is the main technique to realize information exchange from the coarse grid to fine grid consistently. A two-way artificial boundary condition is proposed for eliminating boundary reflections and injecting heat sources. With the two-way boundary condition, we design the heat jet approach and combine with the handshaking subdomain, which is used to implement temperature control in crystalline solids. Numerical tests of linear lattice and non-linear lattice with Morse potential in one space dimension illustrate the accuracy and effectiveness of resolving information exchange between the two scales subject to thermal fluctuations at a given temperature. [ABSTRACT FROM AUTHOR]

The lattice Boltzmann method (LBM) has often been employed for numerical analysis of fluid dynamics problems and is described as an algorithm based on a simple fully explicit method. When LBM can be applied to structural mechanics problems, there is a possibility that topology optimization can be performed efficiently for non-linear elastic fields and the topology optimization of fluid-structure interaction problems can be performed easily with the single calculation method. We propose a topology optimization method for isotropic linear elastic body problems using LBM in this paper, as a first step to realize this. First, the analysis approach of the isotropic linear elastic field using LBM is constructed with incorporating the expansion technique of the governing equations for applying to a topology optimization framework. Next, based on the density approach, the optimization problem is formulated, and by using the adjoint lattice Boltzmann method (ALBM), the design sensitivity is derived. Finally, the validity of the proposed method is demonstrated with several numerical examples. [ABSTRACT FROM AUTHOR]

This paper presents a modified characteristic projection finite element method for the Kelvin-Voigt viscoelastic fluid equations. Then, the unconditional stability and optimal convergence of numerical solutions in L 2 -norms and H 1 -norms were presented. In addition, some numerical results will be given to testify the theoretical analysis. The numerical results show that the convergence orders are optimal, which show the theoretical analysis is right. The numerical results show that our method is robust for different Reynolds numbers. The numerical results imply that the numerical algorithm also converges for different retardation time. It means that our numerical method is robust for different retardation time. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a first-order and a second-order time-stepping schemes for the anisotropic phase-field dendritic crystal growth model. The proposed schemes are based on an auxiliary variable approach for the Allen-Cahn equation and delicate treatment of the terms coupling the Allen-Cahn equation and temperature equation. The idea of the former is to introduce suitable auxiliary variables to facilitate construction of high order stable schemes for a large class of gradient flows. We propose a new technique to treat the coupling terms involved in the crystal growth model, and introduce suitable stabilization terms to result in totally decoupled schemes, which satisfy a discrete energy law without affecting the convergence order. A delicate implementation demonstrates that the proposed schemes can be realized in a very efficient way. That is, it only requires solving four linear elliptic equations and a simple algebraic equation at each time step. A detailed comparison with existing schemes is given, and the advantage of the new schemes is emphasized. As far as we know this is the first second-order scheme that is totally decoupled, linear, unconditionally stable for the dendritic crystal growth model with variable mobility parameter. [ABSTRACT FROM AUTHOR]

In this paper, numerical predictions are performed for a planar prefilming atomizer using the lattice Boltzmann (LB) method. The main target is to inspect the effects of surface wettability on the primary liquid film breakup at the prefilmer edge. To this end, a phase-field-based multiple-relaxation-time LB model for two-phase flow is derived and verified for the present study, as well as the wetting boundary scheme. After that, the effects of surface wettability are assessed by specifying different contact angles during the simulations. Numerical results indicate that the whole primary breakup process of the liquid film can be roughly divided into three interactive stages: (i) liquid accumulation at the trailing edge; (ii) ligament formation and its flapping movements; (iii) ligament breakup and droplet formation. It is visible that the liquid accumulation and the following ligament breakup show similar periodic patterns with quite consistent frequencies. Besides, when the contact angle is increased, the volume of liquid accumulated at the prefilmer edge declines, the maximum flapping region of ligaments shrinks, the ligament breakup length reduces, but the ligament breakup frequency increases. Based on the above-mentioned results, it can be concluded that the prefilmer with hydrophobic surfaces benefits the primary liquid film breakup in the close vicinity of the prefilmer edge. [ABSTRACT FROM AUTHOR]

• A polygonal element 2DRSBd2 without 2D bubble function for 2D Poisson problem. • The element 2DRSBd2 possesses the second order completeness. • An element 3DRSB-2DRSBd2 without 3D bubble function for 3D elastostatic problem. • Two 3D elements 3DSB-2DRSBd2 and 3DRSB-2DRSBd2 possess the second order completeness. By the theoretical analysis on the scaled boundary finite element (SBFEM), some appropriate 2D or 3D bubble functions are necessary in order that the elements can possess the second order completeness. However, the extra 2D bubble function is difficult to be adopted as facet shape function for 3D polyhedral element. In this paper, a revised-scaled-boundary element (denoted by 2DRSBd2) only corresponding to the boundary nodal displacements is constructed by elimination of the 2D bubble function derived from the constant right-hand term in 2D Poisson equation. This polygonal element 2DRSBd2 possesses the second order completeness and the third order convergence in L 2 -norm. Then we construct two polyhedral elements 3DSB-2DRSBd2 and 3DRSB-2DRSBd2 based on the polygonal element 2DRSBd2. The difference between the two 3D elements is: the 3DSB-2DRSBd2 element is constructed with three 3D bubble functions derived from the constant body loads in three dimensional elastostatic problem, while the 3D revised-scaled-boundary element 3DRSB-2DRSBd2 is constructed by elimination of those 3D bubble functions. Both elements 3DSB-2DRSBd2 and 3DRSB-2DRSBd2 possess the second order completeness and the third order convergence in L 2 -norm. Numerical examples verify that they have good accuracy and insensitive to mesh distortion. [ABSTRACT FROM AUTHOR]

In this paper, we consider an orthogonal spline collocation (OSC) method to solve the fourth-order multi-term subdiffusion equation. The L1 method on graded meshes is employed in temporal direction. In the spatial direction, we develop the orthogonal spline collocation method. The stability and convergence of the above mentioned method are proved theoretically, and it's still valid if α 1 → 1 −. A new robust error bound is obtained, which is a more natural and desirable result and does not blow up as α 1 → 1 −. Some numerical results are also provided to verify our theoretical analysis and demonstrate the effectiveness of the OSC method. [ABSTRACT FROM AUTHOR]

In this paper, we construct a backward Euler full-discrete two-grid finite volume element scheme for solving time-dependent Schrödinger equation. Combining the idea of the two-grid discretization, the original coupling system is solved in the coarse grid space, and the decoupling system with two independent Poisson problems is solved in the fine grid space, which ensures the accuracy and improves the computational efficiency. We further prove the optimal error estimate of the scheme rigorously. Finally, the numerical simulation results show that the two-grid method is more effective than the standard finite volume element method in solving coupled partial differential problems. [ABSTRACT FROM AUTHOR]

In this paper, a fast finite volume method is proposed for the initial and boundary value problems of spatial fractional diffusion equations on nonuniform meshes. The discretizations of the Riemann-Liouville fractional derivatives lead to unstructured dense coefficient matrices, differing from the Toeplitz-like structure under the uniform mesh. The fast algorithm is proposed by using the sum-of-exponentials (SOE) technique to the spatial kernel x α − 1 , α ∈ (0 , 1). Then, the matrix-vector multiplications of the resulting coefficient matrices could be implemented in O (m log 2 ⁡ m) operations, where m denotes the size of matrices. Iterative solvers are preferably applied to obtain the numerical solution. The proposed fast scheme is proved to be unconditionally stable for sufficiently accurate SOE approximation. Meanwhile, a banded preconditioner is exploited to accelerate the Krylov subspace method. Numerical experiments are provided to demonstrate the efficiency of the proposed fast algorithm. [ABSTRACT FROM AUTHOR]

Not only coughing and sneezing, but even normal breathing can produce aerosols, because rupture of liquid plugs forms microdroplets during pulmonary airway reopening. Aerosols are important carriers of various viruses, such as influenza, SARS, MERS, and COVID-19. To control airborne disease transmission, it is important to understand aerosol formation, which is related to the pressure drop, liquid plug, and film. In addition, the detrimental pressure and shear stress at the airway wall produced in the process of airway reopening have also attracted a lot of attention. In this paper, we proposed a multiphase lattice Boltzmann method to numerically simulate pulmonary airway reopening, in which the gas-liquid transition is directly driven by the equation of state. After validating the numerical model, two rupture cases with and without aerosol formation were compared and analyzed. We found that injury of the epithelium in the case with aerosol formation was almost the same as that without aerosol formation, even though the pressure drop in the airway increased by about 50%. Further investigation showed that the aerosol size and maximum differences of the wall pressure and shear stress increased with pressure drop in the pulmonary airway. A similar trend was observed when the thickness of the liquid plug became larger, while an opposite trend occurred when the thickness of the liquid film increased. The model can be extended to study generation and transmission of bioaerosols carrying the influenza or coronavirus. [ABSTRACT FROM AUTHOR]

In this paper, we derive the time-fractional Cahn–Hilliard equation from continuum mixture theory with a modification of Fick's law of diffusion. This model describes the process of phase separation with nonlocal memory effects. We analyze the existence, uniqueness, and regularity of weak solutions of the time-fractional Cahn–Hilliard equation. In this regard, we consider degenerating mobility functions and free energies of Landau, Flory–Huggins and double-obstacle type. We apply the Faedo–Galerkin method to the system, derive energy estimates, and use compactness theorems to pass to the limit in the discrete form. In order to compensate for the missing chain rule of fractional derivatives, we prove a fractional chain inequality for semiconvex functions. The work concludes with numerical simulations and a sensitivity analysis showing the influence of the fractional power. Here, we consider a convolution quadrature scheme for the time-fractional component, and use a mixed finite element method for the space discretization. [ABSTRACT FROM AUTHOR]

Numerical modeling of phase change problems with convection is known to be computationally expensive. The main challenge comes from the coupling between Navier–Stokes and heat energy equations. In this paper, we develop a new scheme for phase change problems based on a projection method. The proposed method reduces the size of the system by splitting the temperature, the velocity, and the pressure fields while preserving the accuracy of the simulations. A single-domain approach using a variant of the enthalpy-porosity formulation is employed. Incompressible Navier–Stokes problem with Boussinesq approximation for thermal effects in solid and liquid regions is considered. We regularize the discontinuous variables such as latent heat and material properties by a continuous and differentiable hyperbolic tangent function. The robustness and effectiveness of the proposed scheme are illustrated by comparing the numerical results with numerical and experimental benchmark. [ABSTRACT FROM AUTHOR]

In this paper we propose a modeling setting and a numerical Riemann problem solver at the junction of one dimensional shallow-water channel networks. The junction conditions take into account the angles with which the channels intersect and include the possibility of channels with different sections. The solver is illustrated with several numerical tests which underline the importance of the angle dependence to obtain reliable solutions. [ABSTRACT FROM AUTHOR]

This paper develops a positivity-preserving and free energy dissipative hybrid scheme for the multi-dimensional Poisson-Nernst-Planck equations with multiple ions on unstructured polygonal and polyhedral meshes. It is built on the linear virtual element method for the Poisson-type equation for the electrostatic potential and a cell-centered finite volume scheme for the Nernst-Planck equations for the ion concentrations, and preserves the positivity of the ion concentrations, the mass conservation, the free energy dissipation and steady-state properties. The existence of the numerical solutions is also discussed. Numerical results demonstrate the effectiveness of the proposed two- and three-dimensional schemes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the numerical approximation of mixed formulations for the acoustic eigenvalue problem with axial symmetry. It is well known that, in the full three-dimensional problem, using Raviart–Thomas elements for the fluid displacements yields a discretization free of spurious eigenvalues. However, it has been recently shown that this is not the case in the axisymmetric pure displacement formulation. Two mixed formulations are introduced in order to avoid the spurious modes. A discretization based on Raviart–Thomas mixed method is proposed and analyzed. We provide a convergence result under rather general conditions, as well as absence of spurious modes. Moreover, we prove quasi-optimal order error estimates under additional regularity assumptions. Finally, we report numerical results which allow us to confirm the theoretical estimates. [ABSTRACT FROM AUTHOR]

• The implicit LBFS is proposed aiming to efficiently simulate both steady and unsteady flows. • The proposed solver works in the framework of finite volume method on both structured and unstructured meshes. • With a local compressible lattice Boltzmann model, the implicit LBFS is able to simulate subsonic and supersonic flows. The lattice Boltzmann flux solver (LBFS) is a novel Boltzmann-typed solver under the finite volume framework, which has many advantages over traditional lattice Boltzmann equation (LBE) solvers. However, the existing versions of LBFS are explicit in temporal discretization. Therefore, they are of low computational efficiency and are mostly applied to the simulation of flows around simple geometries. In this paper, an implicit lattice Boltzmann flux solver for compressible flows is proposed, which can efficiently march in time with a large CFL number, and therefore facilitates the simulation of flows around complex geometries in practical engineering applications. Aiming to obtain the time-accurate results of unsteady flows, the implicit LBFS is further combined with the dual time-stepping technique to efficiently simulate unsteady flows, in which a sub-iteration is applied on the pseudo time level. In the sub-iteration, the Jacobian matrix is preconditioned by the lower-upper symmetric Gauss-Seidel (LUSGS) method and solved by the generalized minimal residual method (GMRES). The accuracy and efficiency of the proposed solver are demonstrated by various simulations in a wide Mach number regime. To further demonstrate its performance in practical engineering applications, the proposed solver is also used to simulate the flow over complex aircraft models. [ABSTRACT FROM AUTHOR]

In the diffusion based processes, the gradient operator performs well in the case of Gaussian noise and can distinguish between edges (large gradient) and the noise-contaminated areas, while impulse noise can cause large gradients and consequently be preserved like the edges. Therefore, in this paper, the ENI operator which distinguishes well between impulse noise and signal is analyzed and using it, an efficient diffusion based method for impulse denoising is proposed. The proposed method is carried out in 2 phases using the residual feedback and the localization of the noisy pixels. The experimental results confirm the efficiency of the proposed method according to subjective and objective criteria. [ABSTRACT FROM AUTHOR]

This paper proposes a new model for image shadow removal. The model reformulates a recent osmosis model with nonlocal differential operators. This allows to benefit from distant pixels similarities and thus improves restoration results. Some properties of this model are established and discussed, making it suitable for our application. Experimental results show that the nonlocal model obtained very good qualitative and quantitative results compared with state-of-the-art techniques. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a spectral Petrov-Galerkin method for an optimal control problem governed by a two-sided space-fractional diffusion-advection-reaction equation. Taking into account the effect of singularities near the boundary generated by the weak singular kernel of the fractional operator, we establish the regularity of the problem in weighted Sobolev space. Error estimates are provided for the presented spectral Petrov-Galerkin method and the convergence orders of the state and control variables are determined. Furthermore, a fast projected gradient algorithm with a quasi-linear complexity is presented to solve the resulting discrete system. Numerical experiments show the validity of theoretical findings and efficiency of the proposed fast algorithm. [ABSTRACT FROM AUTHOR]

In this paper, the unconditionally optimal H 1 -norm error estimate for a two-step fast L1 scheme is considered for a nonlinear subdiffusion equation, which involves the considerations of both the initial singularity of the solution and the computational complexity to evaluate the Caputo derivative. For the fully discrete scheme, we use the fast L1 scheme developed in [28] to speed up the evaluation of the Caputo derivative on nonuniform time step, employ the Galerkin finite element (FE) method to discretize the spatial direction, and take the Newton linearized method to approximate the nonlinear term. While the theory of the L 2 -norm error estimate is well studied, there are few works on the unconditional convergence of H 1 -norm error estimate of the linearized Galerkin FE scheme for nonlinear subdiffusion equations. The main reason is that it requires to find a suitable test function in the FE space. To this end, we here respectively take the time-discrete operator and Laplace operator as the suitable test function in the FE space, and combine an improved discrete fractional Grönwall inequality to achieve the unconditional error estimate in H 1 -norm. Numerical results are provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

Ultrasound enhanced drug transport is a multiphysics problem involving acoustic waves propagation, bioheat transfer, and drug transport. In this paper, we study a model for this problem that consists of a wave-type equation for acoustic pressure, a diffusion-reaction equation for bioheat transfer, and a convection-diffusion-reaction equation for drug transport. We focus in particular on the numerical analysis of such a system. We propose and derive convergence estimates for a piecewise linear finite element method (FEM) with quadrature. We prove that the FEM is second order convergent for concentration in a discrete L 2 -norm. Since concentration depends on the gradient of acoustic pressure, this result shows that the FEM is superconvergent. Note that one expects the FEM to be convergent of order one in the L 2 -norm because the optimal convergence rate for the acoustic pressure is two in the H 1 -norm. We include numerical results backing the theoretical findings and an application of the model to a laboratory experiment of ultrasound enhanced transdermal drug delivery. [ABSTRACT FROM AUTHOR]

The paper is devoted to the numerical solution of algebraic systems of the type (A α + q I) u = f , 0 < α < 1 , q > 0 , u , f ∈ R N , where A is a symmetric and positive definite matrix. We assume that A is obtained by finite difference approximation of a second order diffusion problem in Ω ⊂ R d , d = 1 , 2 so that A α + q I approximates the related fractional diffusion–reaction operator or could be a result of a time-stepping procedure in solving time-dependent sub-diffusion problems. We also assume that a method of optimal complexity for solving linear systems with matrices A + c I , c ≥ 0 is available. We analyze and study numerically a class of solution methods based on the best uniform rational approximation (BURA) of a certain scalar function in the unit interval. The first such method, originally proposed in Harizanov et al. (2018) for numerical solution of fractional-in-space diffusion problems, was based on the BURA r α (ξ) of ξ 1 − α in 0 , 1 through scaling of the matrix A by its largest eigenvalue. Then the BURA of t − α in 1 , ∞) is given by t − 1 r α (t) and correspondingly, A − 1 r α (A) is used as an approximation of A − α. Further, this method was improved in Harizanov et al. (2019) using the same concept but by scaling the matrix A by its smallest eigenvalue. In this paper we consider the BURA r α (ξ) of 1 ∕ (ξ − α + q) for ξ ∈ (0 , 1 . Then we define the approximation of (A α + q I) − 1 as r α (A − α). We also propose an alternative method that uses BURA of ξ α to produce certain uniform rational approximation (URA) of 1 ∕ (ξ − α + q). Comprehensive numerical experiments are used to demonstrate the computational efficiency and robustness of the new BURA and URA methods. [ABSTRACT FROM AUTHOR]

The localized method of fundamental solutions (LMFS) is an efficient meshless collocation method that combines the concept of localization and the method of fundamental solutions (MFS). The resultant system of linear algebraic equations in the LMFS is sparse and banded and thus, drastically reduces the storage and computational burden of the MFS. In the LMFS, the moving least square (MLS) approximation, based on fundamental solutions, is used to construct approximate solution at each node. In this paper, this fundamental solutions-based MLS approximation, named as an augmented MLS (AMLS) approximation, is generalized to any point in the computational domain. Computational formulas, theoretical properties and error estimates of the AMLS approximation are derived. Then, taking Laplace equation as an example, this paper sets up a framework for the theoretical error analysis of the LMFS. Finally, numerical results are presented to verify the efficiency and theoretical results of the AMLS approximation and the LMFS. Convergence and comparison researches are conducted to validate the accuracy, convergence and efficiency. [ABSTRACT FROM AUTHOR]

The discontinuous Petrov–Galerkin (DPG) method minimizes a residual in a non-standard norm. This paper shows that the minimization of this residual is equivalent to the minimization of a residual in a L 2 norm. Since such residuals are well known from least squares finite element methods, this novel interpretation allows to extend results for least squares methods to the DPG method and vice versa. This paper exemplifies the benefits of this possibility by the verification of an asymptotic exactness result for a DPG method for the Helmholtz equation, the design of a locking-free DPG method for linear elasticity, and an investigation of the spectral condition number. [ABSTRACT FROM AUTHOR]

This paper represents a sequel to Jureczka and Ochal (2019) where numerical solution of a quasistatic contact problem is considered for an elastic body in frictional contact with a moving foundation. The model takes into account wear of the contact surface of the body caused by the friction. Some preliminary error analysis for a fully discrete approximation of the contact problem was provided in Jureczka and Ochal (2019). In this paper, we consider a more general fully discrete numerical scheme for the contact problem, derive optimal order error bounds and present computer simulation results showing that the numerical convergence orders match the theoretical predictions. [ABSTRACT FROM AUTHOR]

The paper is concerned with the space of C 1 continuous polynomial splines of degree 5 on general triangulations such that the restriction to each triangle corresponds to a Bell element. It is investigated how the standard construction of Bell elements, which is based on prescribing values and derivatives up to order two at the vertices of the triangulation, can be used to define different bases. The most straightforward one is the nodal basis, which lacks stability, but it is shown that this can be improved by scaling the interpolation data. However, the prime focus of the paper is on the construction of B-spline-like bases that depend on the choice of special domain triangles associated with the vertices of the triangulation. A B-spline-like basis consists of functions that are locally supported and form a partition of unity. If the triangulation satisfies certain geometric constraints (e.g. is acute) and the domain triangles are chosen suitably, the basis is also non-negative. The stability analysis of the derived bases is addressed theoretically as well as numerically, the latter with the examples of least square fitting and PDE solving. [ABSTRACT FROM AUTHOR]

This paper is an attempt for solving operator-valued quantum stochastic optimal control problems, in Fock space. For this purpose, the dynamics of the classical system state is described by Hudson–Parthasarathy type Quantum Stochastic Differential Equation (QSDE) in Fock space and then by associating a quadratic performance criterion with the QSDE, a Quantum Stochastic Linear Quadratic Regulator (QS-LQR) optimal control problem is formulated. Also, an algorithm for solving the QS-LQR optimal control problem is designed. For solving the resulting optimal control problem, a new HJB equation is obtained. Thereby, the operator valued control process is obtained. Two theorems are proved to facilitate the algorithm. In this paper, for the first time, the optimal strategy for trading stock is designed via the presented method. For this purpose, Merton portfolio allocation problem is solved. The simulation results show that portfolio optimal performances, minimum risk and maximum return are achieved via presented method. [ABSTRACT FROM AUTHOR]

The paper discusses the efficiency of the classical BiCGStab method and several of its modifications for solving systems with multiple right-hand side vectors. These iterative methods are widely used for solving systems with large sparse matrices. The paper presents execution time analytical model for the time to solve the systems. The BiCGStab method and several modifications including the Reordered BiCGStab and Pipelined BiCGStab methods are analysed and the range of applicability for each method providing the best execution time is highlighted. The results of the analytical model are validated by the numerical experiments and compared with results of other authors. The presented results demonstrate an increasing role of the vector operations when performing simulations with multiple right-hand side vectors. The proposed merging of vector operations allows to reduce the memory traffic and improve performance of the calculations by about 30%. [ABSTRACT FROM AUTHOR]

In this paper, the original finite volume lattice Boltzmann method (FVLBM) on an unstructured grid (Part I of these twin papers) is extended to simulate turbulent flows. To model the turbulent effect, the k − ω SST turbulence model is incorporated into the present FVLBM framework and is also solved by the finite volume method. Based on the eddy viscosity hypothesis, the eddy viscosity is computed from the solution of the k − ω SST model, and the total viscosity is modified by adding this eddy viscosity to the laminar (kinematic) viscosity given in the Bhatnagar–Gross–Krook collision term. Because of solving for the collision term with the explicit method in the original FVLBM scheme, the computational efficiency is much lower for simulating high Reynolds number flow. This is due to the fact that the largest time step decided by the stability condition of the collision term, which is less than twice the relaxation time, is much smaller than that decided by the CFL condition. In order to enhance the computational efficiency, the three-stage second-order implicit–explicit (IMEX) Runge–Kutta method is used for temporal discretization, and the time step can be one or two orders of magnitude larger as compared with the explicit Euler forward scheme. Although the computational cost is increased, the final computational efficiency is enhanced by about one-order of magnitude and good results can also be obtained at a large time step through the test case of a lid-driven cavity flow. Two turbulent flow cases are carried out to validate the present method, including flow over a backward-facing step and flow around a NACA0012 airfoil. The numerical results are found to be in agreement with experimental data and numerical solutions, demonstrating the applicability of the present FVLBM coupled with the k − ω SST model to accurately predict the incompressible turbulent flows. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the numerical application of radial basis functions (RBFs) approximation in the reconstruction process of well known ENO/WENO schemes. The resulted schemes are employed for approximating the viscosity solution of Hamilton–Jacobi (H–J) equations. The accuracy in the smooth area is enhanced by locally optimizing the shape parameter according to the results. It is revealed that the proposed schemes in this research prepare more accurate reconstructions and sharper solution near singularities by comparing the RBFENO/RBFWENO schemes and the classical ENO/WENO schemes for some benchmark examples. Looking at the several numerical examples in 1D, 2D and 3D illustrate that the proposed schemes in this paper perform better than the traditional ENO/WENO schemes for solving H–J equations. [ABSTRACT FROM AUTHOR]

In order to study the flow behavior at high Reynolds numbers, two modified models, known as the multiple-relaxation-time lattice Boltzmann method (MRT-LBM) and large-eddy-simulation lattice Boltzmann method (LES-LBM), have been employed in this paper. The MRT-LBM was designed to improve numerical stability at high Reynolds numbers, by introducing multiple relaxation time terms, which consider the variations of density, energy, momentum, energy flux and viscous stress tensor. As a result, MRT-LBM is capable of dealing with turbulent flows considering energy dispersion and dissipation. In the present paper, this model was employed to simulate the flow at turbulent Reynolds numbers in wall-driven cavities. Two-sided wall driven cavity flow was studied for the first time, based on MRT-LBM, at Reynolds numbers ranging from 2 × 1 0 4 to 1 × 1 0 6 , and employing a very large resolution2048 × 2048. It is found that whenever top and bottom lids are moving in the opposite directions, and the Reynolds number is higher than 2 × 1 0 4 , the flow is chaotic, although some quasi-symmetric properties still remain, fully disappearing at Reynolds numbers between 2 × 1 0 5 and 3 × 1 0 5 . Furthermore, between this Reynolds numbers range, 2 × 1 0 5 < Re < 3 × 1 0 5 , the quasi-symmetric structures turn into a much smaller and fully chaotic eddies. The LES-LBM model implements the large eddy simulation turbulent model into the conventional LBM, allowing to study the flow at turbulent Reynolds numbers. LES-LBM combined with Quadruple-tree Cartesian cutting grid (tree grid) was employed for the first time to characterize the flow dynamics over a cylinder and a hump, at relatively high Reynolds numbers. In order to construct the macroscopic quantities in the virtual boundaries separating two different grid levels, a set of new schemes were designed. The coupling of the LES-LBM and tree grid drastically reduced the computational time required to perform the simulations, thus, allowing to minimize the hardware requirements. LES-LBM model is shown to be much more efficient when combined with the tree grid instead of using the standard Cartesian grid. [ABSTRACT FROM AUTHOR]

In this paper a novel method is presented to solve problems with weak singularities in two-dimensional heterogeneous media using equilibrated singular basis functions. This especially includes crack problems in composites of functionally graded material types. The method is mainly presented in a boundary formulation, although it may be found quite useful in other mesh-based or mesh-less approaches as the eXtended Finite Element Method (XFEM). The present paper considers harmonic and elasticity problems. The most distinguished advantage of the present method is that the solution progress advances without absolutely any knowledge of the analytical singularity order of the problem. To this end the partial differential equation of the problem is approximately satisfied in a weighted residual approach. After developing the formulation in a mapped polar co-ordination, some primary basis functions generated from Chebyshev polynomials and trigonometric functions, along with corresponding weight functions are employed. The numerical examples, either selected from the well-known literature or solved by well-established techniques, will demonstrate the capability of the method in problems related to composite materials. [ABSTRACT FROM AUTHOR]

This paper aims to seek proper stabilized Galerkin methods for solving the two-dimensional coupled Burgers' equations at high Reynolds numbers. The stabilization techniques employed here include the streamline upwind/Petrov–Galerkin (SUPG) method, the spurious oscillations at layers diminishing (SOLD) method and the characteristic Galerkin (CG) method. The first two methods are combined with the Crank–Nicolson scheme for time discretization and the last one is applied in its semi-implicit version. Different from most of the studies on the equations which are usually devoted to improving the accuracy of computed solution in the case of low Reynolds numbers, this paper mainly focuses on keeping the stability of the solution at high Reynolds numbers, which is significant in practical applications and also challenging in numerical computation. We study two problems, equipped with mixed boundary conditions and only Dirichlet boundary conditions, respectively. Numerical experiments reveal that the SUPG method is optimal for the former problem, and the SOLD method is more appropriate for the latter one. In addition, the performances of these methods demonstrate the difference between the two problems, which is seldom mentioned previously and might be helpful to other conventional methods intending to solve the problems at high Reynolds numbers. And last, since SOLD methods have rarely been utilized to solve nonlinear unsteady problems before, this study also indicates the potential of this class of methods to solve nonlinear unsteady convection-dominated problems. [ABSTRACT FROM AUTHOR]