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This paper employs a numerical approach to investigate the impact of liquidity shocks on European options in modeling markets. To accurately capture the behavior of European options under liquidity shocks, a coupled system of differential equations is employed, consisting of a degenerate parabolic equation and a diffusion-free equation. The primary focus is on developing and analyzing implicit-explicit methods for numerically simulating European option pricing, specifically considering the presence of liquidity shocks while ensuring the positivity of the solution. The paper also includes convergence analysis and establishes the discrete comparison principle for the developed methods. Numerical experiments are conducted using both uniform and nonuniform meshes to validate the theoretical findings, demonstrating the efficiency and accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

The Diffusion Peterlin Viscoelastic Fluid (DPVF) model describes the movement of specific incompressible polymeric fluids. In this paper, we introduce and evaluate a new low-complexity linear-time filter finite element (FE) method for the DPVF model. In order to avoid the value at time t = − Δ t , the proposed time filter method consists of three steps, including a post-processing step. Firstly, a first-order Euler backward nonlinear fully discrete mixed FE scheme is employed to compute the numerical solutions at time t 1 = Δ t. For n ≥ 1 , we obtain the intermediate values ( u ˜ h n + 1 , p ˜ h n + 1 , d ˜ h n + 1) in Step II using a fully implicit backward Euler scheme. At the same time level, we proceed with these intermediate values ( u ˜ h n + 1 , p ˜ h n + 1 , d ˜ h n + 1) using the linear time filters. The linear time filters step does not significantly increase computational complexity. However, it can enhance temporal convergence accuracy from first order to second order for backward Euler time filter (BE time filter), and from second order to three order for BDF2 time filter. We demonstrate the almost unconditional stability of the scheme. Error estimates for the time filter method are derived and presented. Several numerical experiments are conducted to validate the theoretical findings and showcase the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, an improved projection scheme has been constructed for solving the nonstationary magnetohydrodynamic equations, based on the modular grad-div stabilization. Owing to the utilization of the projection scheme and some delicate implicit-explicit approach to the nonlinear coupled terms, the proposed scheme is linear, decoupled and unconditionally energy stable. In addition, the modular grad-div stabilization technique is introduced to improve the conservation of Gauss's law and mass, and reduce the splitting error of the projection method, which is simple to implement with a small intrusion step into existing code. In this way, the developed scheme can prevent the solver from breakdown and enhance computational efficiency with the increasing of grad-div parameters. What's more, we present a rigorous convergence analysis. Finally, several numerical experiments are performed to illustrate the theoretical consequences and their effectiveness. [ABSTRACT FROM AUTHOR]

Efficient finite difference/spectral approximation is proposed for the time-fractional diffusion equation with an inverse square potential on the unit ball in this paper. The efficiency of the method lies in the use of ball functions to mimic the singular behavior of solutions in space and L1 scheme on graded mesh to compensate for the initial weak singularity of solutions in time. The stability and convergence of the fully discrete scheme in both L 2 -norm and H 1 -norm are analyzed. Numerical examples are presented to illustrate the accuracy and efficiency of our proposed discrete scheme. [ABSTRACT FROM AUTHOR]

This paper is devoted to a study of a history-dependent mixed hemivariational-variational inequality arising in contact mechanics. The contact problem concerns the deformation of a viscoelastic body with long memory, subject to a general friction law on one part of the boundary and a frictionless Signorini condition on another part of the boundary. The solution existence and uniqueness of the history-dependent mixed hemivariational-variational inequality based on a Lagrange multiplier approach are proved. Then, a fully discrete scheme is introduced and studied. The trapezoidal rule is used to approximate the integral in the history-dependent operator. For the spatial discretization, the linear finite elements are used to discretize the displacement field, and dual basis functions are used in the approximation of the Lagrange multiplier. Optimal order error estimates are derived for the displacement and the Lagrange multiplier under appropriate solution regularity assumptions. Numerical results are presented to illustrate the theoretical prediction of the convergence orders. [ABSTRACT FROM AUTHOR]

This paper presents globally numerical properties of a new numerical scheme for a reaction-diffusion advection susceptible-infected-susceptible (SIS) model. A new numerical treatment technique is introduced in spatial discretization of advection-diffusion equation, which enables the numerical solutions to preserve the stability and positivity with less stepsize restrictions. The convergence, biological significance and globally stability of numerical solutions is explored in the paper. A threshold value, named by numerical basic reproduction number and denoted by R 0 Δ x , is introduced in the numerical stability analysis of the model. It is proved the numerical disease free equilibrium (DFE) is globally asymptotically stable if R 0 Δ x < 1 and unstable if R 0 Δ x > 1. It is shown the numerical basic number R 0 Δ x replicates the asymptotic behaviors of the basic reproduction number R 0 for the model. Some numerical experiments are given in the end to confirm the conclusions. [ABSTRACT FROM AUTHOR]

The paper focuses on proposing and analyzing a nonlinear interior penalty discontinuous Galerkin finite element (IPDGFE) method for the Allen–Cahn equation, which is a reaction–diffusion model with a nonlinear singular perturbation arising from the phase separation process. We firstly present a fully discrete IPDGFE formulation based on the modified Crank–Nicolson scheme and a mid-point approximation of the potential term f (u). We then derive the energy-stability and the second-order-in-time error estimates for the proposed IPDGFE method under some regularity assumptions on the initial function u 0. There are two key works in our paper. One is to establish a second-order-in-time and energy-stable IPDGFE scheme. The other is to use a discrete spectrum estimate to handle the midpoint of the discrete solutions u m and u m + 1 in the nonlinear term, instead of using the standard Gronwall inequality technique, so we obtain that all our error bounds depend on the reciprocal of the perturbation parameter ϵ only in some lower polynomial order, instead of exponential order. As a nontrivial byproduct of our paper, we also analyze the convergence of the zero-level sets of fully discrete IPDGFE solutions to the curvature-driven geometric flow. Finally, numerical experiments are provided to demonstrate the good performance of our presented IPDGFE method, including the time and space error estimates of the discrete solutions, discrete energy-stability, and the convergence of numerical interfaces governed by the curvature-driven geometric flow in the classical motion and generalized motion. [ABSTRACT FROM AUTHOR]

An efficient solution of the three-dimensional Helmholtz equation is known to be crucial in many applications, especially geophysics. In this paper, we present and test two preconditioning approaches for the discrete problem resulting from the second order finite-difference discretization of this equation. The first approach combines shifted-Laplacian preconditioner with inversion of a separable matrix, corresponding to the horizontally-layered velocity model, using fast Fourier based transforms. The second approach is novel and involves a special transformation resulting in a preconditioner with a contraction operator (CO preconditioner). The two approaches have near the same arithmetical complexity; however, the second approach, developed in this paper, provides a faster convergence of an iterative solver as illustrated by numerical experiments and analysis of the spectral properties of the preconditioned matrices. Our numerical experiments involve parallel modeling of highly heterogeneous lossy and lossless media at different frequencies. We show that the CO-based solver can tackle problems with hundreds of millions of unknowns on a conventional cluster node. The CO preconditioned solver demonstrates a very moderate increase of iteration count with the frequency. We have conducted a comparison of the performance of the developed method versus open-source parallel sweeping preconditioner. The results indicate that, the CO solver is several times faster with respect to the wall-clock time and consumes substantially less memory than the code based on the sweeping preconditioner at least in the example we tested. [ABSTRACT FROM AUTHOR]

In this paper we propose a solution strategy for the Cahn-Larché equations, which is a model for linearized elasticity in a medium with two elastic phases that evolve subject to a Ginzburg-Landau type energy functional. The system can be seen as a combination of the Cahn-Hilliard regularized interface equation and linearized elasticity, and is non-linearly coupled, has a fourth order term that comes from the Cahn-Hilliard subsystem, and is non-convex and nonlinear in both the phase-field and displacement variables. We propose a novel semi-implicit discretization in time that uses a standard convex-concave splitting method of the nonlinear double-well potential, as well as special treatment to the elastic energy. We show that the resulting discrete system is equivalent to a convex minimization problem, and propose and prove the convergence of alternating minimization applied to it. Finally, we present numerical experiments that show the robustness and effectiveness of both alternating minimization and the monolithic Newton method applied to the newly proposed discrete system of equations. We compare it to a system of equations that has been discretized with a standard convex-concave splitting of the double-well potential, and implicit evaluations of the elasticity contributions and show that the newly proposed discrete system is better conditioned for linearization techniques. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes a stabilized embedded discontinuous Galerkin (EDG) method for linear elasticity problems. A stabilized term is added in the scheme to ensure locking-free approximations. With a special designed interpolation, we prove the Stokes stable elements in space [ H 0 1 (Ω) ] d × L 0 2 (Ω) are also stable in [ H D 1 (Ω) ] d × L 2 (Ω) on general simplicial mesh, and therefore, we prove the underlying EDG method is robust in the sense that the derived a priori error estimates are optimal and uniform with respect to the Lamé constant λ. Numerical experiments are provided to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is devoted to the numerical analysis of a control constrained distributed optimal control problem subject to a time fractional diffusion equation with non-smooth initial data. The solutions of state and co-state are decomposed into singular and regular parts, and some growth estimates are obtained for the singular parts. By following the variational discretization concept, a full discretization is applied to the corresponding state and co-state equations by using linear conforming finite element method in space and piecewise constant discontinuous Galerkin method in time. Error estimates are derived by employing the growth estimates. In particular, graded temporal grids are adopted to obtain the first-order temporal accuracy. Finally, numerical experiments are performed to verify the theoretical results. [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]

Early estimation of biological growth of organisms is an indispensable task in ecology and related research areas. The biological growth is always time-continuous, while our observations of the phenomenon are time-discrete in practice. The formalism of the discrete costly observation (DCO) enables us to mathematically bridge the two qualitatively different processes. This formalism is still germinating, and its practical applications have not been carried out. This paper presents a first application of the DCO formalism to a cost-effective early estimation problem of the biological growth, and its mathematical and numerical analysis. Growth dynamics of organisms, which are fishery resources in this paper, is governed by a stochastic differential equation whose solution is observed discretely. The optimality equation to be solved for finding the most cost-effective observation policy is derived as a fixed point problem based on degenerate parabolic partial differential equations. The fixed point problem turns out to be uniquely solvable. A recursive approximation of the fixed point problem is presented and its solvability in a viscosity sense is discussed. A finite different scheme is then employed to fully-discretize the recursive equations. The present model is finally applied to a problem of Japanese smelt Plecoglossus altivelis altivelis (Ayu): an important inland fishery resource in Japan. [ABSTRACT FROM AUTHOR]

The time-fractional Burgers-Huxley (TFBH) equation is a typical fractional parabolic equation, it is an evolutionary model describing the propagation of neural pulses. The high-accuracy numerical method for studying TFBH equation has important scientific significance and engineering application value. In this paper, a high-order compact difference scheme is constructed for inhomogeneous TFBH equation. The time-fractional derivative is discretized by L 1 formula, and the spatial derivative is approximated by fourth-order precision compact approximation. We analyzed the existence and uniqueness of difference scheme solution, and proved the stability and convergence of the fourth-order compact scheme using the energy method. Numerical experiments show that the scheme converges to O (τ 2 − α + h 4) under the strong regularity assumption. Under the condition of weak regularity, the scheme converges to O (τ α + h 4). It shows that the scheme has good robustness and high accuracy for solving inhomogeneous TFBH equation. [ABSTRACT FROM AUTHOR]

This paper presents a quasistatic problem of 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. We recall existence and uniqueness results obtained in Sofonea et al. (2017). The main aim of this paper is to present a fully discrete scheme for numerical approximation together with an error estimation of a solution to this problem. Finally, computational simulations are performed to illustrate the mathematical model. [ABSTRACT FROM AUTHOR]

This paper is devoted to numerical analysis of a new class of elliptic variational–hemivariational inequalities in the study of a family of contact problems for elastic ideally locking materials. The contact is described by the Signorini unilateral contact condition and the friction is modeled by a nonmonotone multivalued subdifferential relation allowing slip dependence. The problem involves a nonlinear elasticity operator, the subdifferential of the indicator function of a convex set for the locking constraints and a nonconvex locally Lipschitz friction potential. Solution existence and uniqueness result on the inequality can be found in Migórski and Ogorzaly (2017). In this paper, we introduce and analyze a finite element method to solve the variational–hemivariational inequality. We derive a Céa type inequality that serves as a starting point of error estimation. Numerical results are reported, showing the performance of the numerical method. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a novel element-free smoothed radial point interpolation method (EFS-RPIM) for solving 2D and 3D solid mechanics problems. The idea of the present technique is that field nodes and smoothing cells (SCs) used for smoothing operations are created independently and without using a background grid, which saves tedious mesh generation efforts and makes the pre-process more flexible. In the formulation, we use the generalized smoothed Galerkin (GS-Galerkin) weak-form that requires only discrete values of shape functions that can be created using the RPIM. By varying the amount of nodes and SCs as well as their ratio, the accuracy can be improved and upper bound or lower bound solutions can be obtained by design. The SCs can be of regular or irregular polygons. In this work we tested triangular, quadrangle, n -sided polygon and tetrahedron as examples. Stability condition is examined and some criteria are found to avoid the presence of spurious zero-energy modes. This paper is the first time to create GS-Galerkin weak-form models without using a background mesh that tied with nodes, and hence the EFS-RPIM is a true meshfree approach. The proposed EFS-RPIM is so far the only technique that can offer both upper and lower bound solutions. Numerical results show that the EFS-RPIM gives accurate results and desirable convergence rate when comparing with the standard finite element method (FEM) and the cell-based smoothed FEM (CS-FEM). [ABSTRACT FROM AUTHOR]

Abstract In this paper the authors deal with a class of parabolic type Kirchhoff equations, which were considered in Han and Li (2018), where global existence and finite time blow-up of solutions were studied when the initial energy was subcritical, critical and supercritical. Their results are complemented in this paper in the sense that a new blow-up criterion will be given for nonnegative initial energy and upper and lower bounds for blow-up time will be derived. [ABSTRACT FROM AUTHOR]

In this paper, for a two-grid decoupling finite element scheme for the mixed Navier-Stokes/Darcy model with Beavers-Joseph-Saffman's interface condition, we establish the optimal error estimate for the approximate solution. Our analysis shows that the fine grid decoupled problems, that is the Navier-Stokes equations and the Darcy equation, can be solved simultaneously and achieve the optimal convergence order. [ABSTRACT FROM AUTHOR]

In this paper, the Black–Scholes PDE is solved numerically by using the high order numerical method. Fourth-order central scheme and fourth-order compact scheme in space are performed, respectively. The comparison of these two kinds of difference schemes shows that under the same computational accuracy, the compact scheme has simpler stencil, less computation and higher efficiency. The fourth-order backward differentiation formula (BDF4) in time is then applied. However, the overall convergence order of the scheme is less than O ( h 4 + δ 4 ) . The reason is, in option pricing, terminal conditions (also called pay-off function) is not able to be differentiated at the strike price and this problem will spread to the initial time, causing a second-order convergence solution. To tackle this problem, in this paper, the grid refinement method is performed, as a result, the overall rate of convergence could revert to fourth-order. The numerical experiments show that the method in this paper has high precision and high efficiency, thus it can be used as a practical guide for option pricing in financial markets. [ABSTRACT FROM AUTHOR]

The diffusive-viscous wave equation arises in a variety of applications in geophysics, and it plays an important role in seismic exploration. In this paper, semi-discrete and fully discrete numerical methods are introduced to solve a general initial-boundary value problem of the diffusive-viscous wave equation. The spatial discretization is carried out through the finite element method, whereas the time derivatives are approximated by finite differences. Optimal order error estimates are derived for the numerical methods. Numerical results on a test problem are reported to illustrate the numerical convergence orders. [ABSTRACT FROM AUTHOR]

This paper presents the numerical analysis for a variant of the nonlinear multiscale Dynamic Diffusion (DD) method for the advection-diffusion-reaction equation initially proposed by Arruda et al. [1] and recently studied by Valli et al. [2]. The new DD method, based on a two-scale approach, introduces locally and dynamically an extra stability through a nonlinear operator acting in all scales of the discretization. We prove existence of discrete solutions, stability, and a priori error estimates. We theoretically show that the new DD method has convergence rate of O (h 1 / 2) in the energy norm, and numerical experiments have led to optimal convergence rates in the L 2 (Ω) , H 1 (Ω) , and energy norms. [ABSTRACT FROM AUTHOR]

The goal of this paper is to introduce and study a new class of fractional differential hemivariational inequality formulated by an evolutionary hemivariational inequality and a fractional differential equation in Banach spaces. By employing the Rothe method and the surjectivity result, we derive the existence of unique solution for such a problem under some mild conditions. Moreover, we use the fully discrete scheme to approximate the fractional differential hemivariational inequality and provide an error estimate for the approximation. Finally, the main results are applied to obtain the unique solvability as well as the numerical analysis for a viscoelastic frictional contact problem with adhesion. [ABSTRACT FROM AUTHOR]

In this paper, we obtain very general decay rate estimates associated to a wave–wave transmission problem subject to a nonlinear damping locally distributed employing arguments firstly introduced in Lasiecka and Tataru (1993) and we shall present explicit decay rate estimates as considered in Alabau-Boussouira (2005) and Cavalcanti et al. (2007). In addition, we implement a precise and efficient code to study the behavior of the transmission problem when k 1 ≠ k 2 and when one has a nonlinear frictional dissipation g ( u t ) . More precisely, we aim to numerically check the general decay rate estimates of the energy associated to the problem established in first part of the paper. [ABSTRACT FROM AUTHOR]

The implementation of higher-order finite element schemes that can handle multi-level hanging nodes is known to be a difficult task. In fact, most of the available literature on hanging nodes in finite element schemes restricts to one-level hanging nodes resulting from symmetric bisections. The intent of this paper is to provide all data structures and algorithms that are necessary for an implementation of a H 1 -conforming higher-order finite element method. The corresponding finite element spaces are defined via tensor products of hierarchic as well as nodal polynomials for quadrilateral and hexahedron based meshes with unsymmetric, multi-level hanging nodes and arbitrary anisotropic polynomial degree distributions, where special care is given to possible orientation problems. The meshes may even result from non-matching refinements. Given these data structures and algorithms, an extension for Serendipity spaces is described in detail along with some other techniques to improve computational efficiency. Numerical results from an implementation based on these data structures and algorithms serve as a validation and show the broad possibilities that highly flexible higher-order finite element schemes have to offer. To the best of our knowledge, this paper offers the most comprehensive numerical results so far for various three-dimensional benchmark problems using in particular finite element spaces defined for hierarchical as well as nodal polynomials on meshes with unsymmetric refinement ratios as well as (multi-level) hanging nodes. Most notably, the numerical results indicate that unsymmetric refinements are indeed favorable over symmetric refinements with respect to convergence rates. However, the actual optimal refinement ratio for a given problem seems to depend on the type and magnitude of singularities to be resolved as well as on the chosen (full) tensor product or Serendipity finite element spaces. In addition to these numerical results, we find that systems of equations defined via finite element spaces using the nodal Lagrange polynomials with Gauss–Lobatto quadrature points as support points yield drastically improved condition numbers. [ABSTRACT FROM AUTHOR]

The main aim of the paper is the application of a global way of defining the domain and global integration over the domain to the process of numerical solving of elastoplastic problems by the parametric integral equations system (PIES). The paper presents mathematical formalism of PIES for elastoplastic boundary value problems and its numerical implementation. Two crucial elements of the numerical algorithm are: parametric surfaces used to define yield regions and 2D series applied to approximate solutions. Characteristics of PIES are presented in comparison with FEM and BEM. The paper consists of practical examples solved by PIES, whose results are confronted with other numerical, experimental and analytical solutions. [ABSTRACT FROM AUTHOR]

In this paper, a reduced-order finite element (FE) method preserving the unconditional energy-stability is proposed to simulate the Allen-Cahn phase field model, based on the proper orthogonal decomposition (POD) method with the snapshot technique. We first derive the full order FE formulation of the Allen-Cahn model and compute its FE full solutions, from which we choose a few spatio-temporal solutions as snapshots. Based on the POD technique, we then build a set of optimal POD bases maximizing the energy content in the original ensemble data, and in the new low-dimensional space spanned by the POD bases, we establish a low-order numerical model of stable reduced-order FE (SROFE) formulation for the Allen-Cahn phase field model. We also prove error estimates of the SROFE solutions of the Allen-Cahn phase field model. Finally, some numerical results are provided to test the validity of the SROFE formulation. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the time-domain Maxwell’s equations coupled to another partial differential equation, which arises from modeling of light and structure interaction at the nanoscale. One major contribution of this paper is that the well-posedness is rigorously justified for the first time. Then we propose a fully-discrete finite element method to solve this model. It is interesting to note that we need use curl conforming, divergence conforming, and L 2 finite elements for this model. Numerical stability and optimal error estimate of the scheme are proved. Numerical results justifying our theoretical convergence rate are presented. [ABSTRACT FROM AUTHOR]

Particle-based methods based on material models using spheres can provide good approximations for many physical phenomena at both the micro and macroscales. The point of departure for the simulations, in general, is a dense arrangement of spherical particles (sphere pack) inside a given container. For generic domains, the generation of a sphere pack may be complex and time-consuming, especially if the pack must comply with a prescribed sphere size distribution and the stability requirements of the simulation. The primary goal of this paper is to present an efficient algorithm that is capable of producing packs with millions of spheres following a statistical sphere size distribution inside complex arbitrary domains. This algorithm uses a new strategy to ensure that the sphere size distribution is preserved even when large particles are rejected in the growing process. The paper also presents numerical results that enable an evaluation of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The paper presents a numerical procedure for solving a class of optimal control problems for heterogeneous systems. The latter are described by parameterized systems of ordinary differential equations, coupled by integrals along the parameter space. Such problems arise in economics, demography, epidemiology, management of biological resources, etc. The numerical procedure includes discretization and a gradient projection method for solving the resulting discrete problem. A main point of the paper is the performed error analysis, which is based on the property of metric regularity of the system of necessary optimality conditions associated with the considered problem. [ABSTRACT FROM AUTHOR]

In this paper, by constructing an objective function and using the gradient search, full-rank and reduced-rank gradient-based algorithms are suggested for solving generalized coupled Sylvester matrix equations. It is proved that the reduced-rank iterative algorithm is convergent for proper initial iterative values. By analyzing the spectral radius of the related matrices, the convergence properties are studied and the optimal convergence factor of the reduced-rank algorithm is determined. The relationship between the reduced-rank algorithm and the full-rank algorithm is discussed. Consequently, the computation load can be reduced greatly for solving a class of matrix equation. A numerical example is provided to illustrate the effectiveness of the proposed algorithms and testify the conclusions suggested in this paper. [ABSTRACT FROM AUTHOR]

In this paper, the discretization of a non-symmetric elliptic obstacle problem with h p -adaptive H 1 ( Ω ) -conforming finite elements is discussed. For this purpose, a higher-order mixed finite element discretization is introduced where the dual space is discretized via biorthogonal basis functions. The h p -adaptivity is realized via automatic adaptive mesh refinement based on a residual a posteriori error estimation which is also derived in this paper. The use of biorthogonal basis functions leads to unilateral box constraints and componentwise complementarity conditions enabling the highly efficient application of a quadratically converging semi-smooth Newton scheme, which can be modified to ensure global convergence. h p -adaptivity usually implies meshes with hanging nodes and varying polynomial degrees which have to be handled appropriately within the H 1 ( Ω ) -conforming finite element discretization. This is typically done by using so-called connectivity matrices. In this paper, a procedure is proposed which efficiently computes these matrices for biorthogonal basis functions. Finally, the applicability of the theoretical findings is demonstrated with several numerical experiments. [ABSTRACT FROM AUTHOR]

The PLEIADES software environment is devoted to the thermomechanical simulation of nuclear fuel elements behaviour under irradiation. This platform is co-developed in the framework of a research cooperative program between Électricité de France ( EDF ), AREVA and the French Atomic Energy Commission ( CEA ). As many thermomechanical solvers are used within the platform, one of the PLEAIADES’s main challenge is to propose a unified software environment for capitalisation of material knowledge coming from research and development programs on various nuclear systems. This paper introduces a tool called mfront which is basically a code generator based on C++ (Stroustrup and Eberhardt, 2004). Domain specific languages are provided which were designed to simplify the implementations of new material properties, mechanical behaviours and simple material models. mfront was recently released under the GPL open-source licence and is available on its web site: http://tfel.sourceforge.net/ . The authors hope that it will prove useful for researchers and engineers, in particular in the field of solid mechanics. mfront interfaces generate code specific to each solver and language considered. In this paper, after a general overview of mfront functionalities, a particular focus is made on mechanical behaviours which are by essence more complex and may have significant impact on the numerical performances of mechanical simulations. mfront users can describe all kinds of mechanical phenomena, such as viscoplasticity, plasticity and damage, for various types of mechanical behaviour (small strain or finite strain behaviour, cohesive zone models). Performance benchmarks, performed using the Code_Aster finite element solver, show that the code generated using mfront is in most cases on par or better than the behaviour implementations written in fortran natively available in this solver. The material knowledge management strategy that was set up within the PLEIADES platform is briefly discussed. A material database named sirius proposes a rigorous material verification workflow. We illustrate the use of mfront through two case of studies: a simple FFC single crystal viscoplastic behaviour and the implementation of a recent behaviour for the fuel material which describes various phenomena: fuel cracking, plasticity and viscoplasticity. [ABSTRACT FROM AUTHOR]

In this paper, a numerical analysis of two velocity–two pressure models for flows of solid particles and fluids is presented. First, a formal exploitation of the weak formulation of such models asserts that they are amenable to integration via projection methods. The challenging issues in the algorithm development for these models are then documented and suitable numerical methodologies for their remedy are devised. Subsequently, an algorithm for the integration of the models of interest is proposed. This is a two-phase projection method on collocated grids that utilizes a fractional-step time-marching scheme. It is further endowed with an interface detection-and-treatment methodology to properly account for the stiffness induced by the presence of moving and deforming material interfaces. The efficiency and robustness of the proposed numerical method are assessed in a series of numerical experiments that are delineated in the last part of this paper. [ABSTRACT FROM AUTHOR]

In this paper, a system of partial differential equations, that can be used to describe the drug release from a biodegradable viscoelastic polymeric platform, is studied from an analytical and numerical point of view. The system is defined in a moving boundary domain and its stability is analysed. From numerical point of view, a numerical method is proposed and its convergence properties are established. In the context of the drug release from biodegradable polymeric platforms, the qualitative behaviour of the differential system is numerically illustrated. [ABSTRACT FROM AUTHOR]

This paper presents a new finite element solver for linear elasticity on quadrilateral and hexahedral meshes based on enrichment of the classical bilinear or trilinear Lagrangian elements. It solves the primal variable displacement in the strain–div formulation and can handle both displacement and traction boundary conditions. It is a locking-free solver based on conforming finite elements. The solver has second order accuracy in displacement and first order accuracy in stress and dilation (divergence of displacement), as validated by theoretical analysis and illustrated by numerical experiments on benchmarks. deal.II implementation is also discussed. [ABSTRACT FROM AUTHOR]

In this paper, based on interior penalty discontinuous Galerkin method and mixed finite element method we consider a strongly conservative discretization for the rearranged Stokes–Biot model. Theoretical analysis and numerical modeling are carried out. From the view of mathematics, we firstly present the existence and uniqueness of solution of the numerical scheme. Then, the analysis of stability and priori error estimates are derived. From the point of numerical simulation, we report the numerical examples under uniform meshes, which well validate the analysis of convergence and the strong mass conservation. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical computation and analysis for a class of two-dimensional time–space fractional convection–diffusion equations. An implicit difference scheme is derived for solving this class of equations. It is proved under some suitable conditions that the derived difference scheme is stable and convergent. Moreover, the convergence orders of the scheme in time and space are also given. In order to accelerate the convergence rate, by combining the Kronecker product splitting (KPS) preconditioner with the generalized minimal residual (GMRES) method, a preconditioning strategy for implementing the difference scheme is introduced. Finally, several numerical examples are presented to illustrate the computational accuracy and efficiency of the methods. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose and analyze a numerical method to solve transient eddy current problems formulated in terms of the magnetic field intensity. Space discretization is based on Nédélec edge elements, while a backward Euler scheme is used for time discretization; the curl-free constraint in the dielectric domain is imposed by means of a penalty strategy. Convergence of the penalized problem as the penalty parameter goes to zero is proved for the continuous and the discrete problems, for the latter uniformly in the discretization parameters. Optimal order error estimates for the convergence of the discrete penalized problem with respect to the penalty and the discretization parameters are also proved. Finally, some numerical tests are reported to assess the performance of this approach. [ABSTRACT FROM AUTHOR]

The paper is devoted to the analysis of passive scalar transport models, used in the lattice Boltzmann method. The case of the pure advection process, without physical diffusion, is considered. The models, proposed by other authors, the modifications of these models and new high-order finite-difference schemes, based on the extended Runge–Kutta-like formulae, are analyzed. The attention is focused on the stability analysis, analysis of the fictitious numerical effects and on the investigation of the sensitivity of accuracy order to the parameter values. The stability analysis is based on the von Neumann method. The influence of the parameter values on the stability is demonstrated. Stability conditions are obtained. Numerical dispersion and diffusion are analyzed. Test problems with discontinuous and smooth initial conditions are considered. The sensitivity of the accuracy order on the parameter values is analyzed. [ABSTRACT FROM AUTHOR]

In the paper we consider a treatment of Bernoulli type shape optimization problems in three dimensions by the combination of the boundary element method and the hierarchical algorithm based on the subdivision surfaces. After proving the existence of the solution on the continuous level we discretize the free part of the surface by a hierarchy of control meshes allowing to separate the mesh necessary for the numerical analysis and the choice of design parameters. During the optimization procedure the mesh is updated starting from its coarse representation and refined by adding design variables on finer levels. This approach serves as a globalization strategy and prevents geometry oscillations without any need for remeshing. We present numerical experiments demonstrating the capabilities of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, 17-point and 25-point finite difference (FD) schemes for the Helmholtz equation with perfectly matched layer (PML) in the two-dimensional domain are presented. It is shown that the 17-point FD scheme is inconsistent in the presence of PML; however, the 25-point FD scheme is pointwise consistent. An error analysis for the numerical approximation of the exact wavenumber is also presented. We present the global and refined 25-point finite difference schemes based on minimizing the numerical dispersion. Numerical experiments are given to illustrate the improvement of the accuracy and the reduction of the numerical dispersion. [ABSTRACT FROM AUTHOR]

In this paper we provide a derivation of a 1D N-phase flow model in a porous medium under the condition of vertical equilibrium which generalizes the three-phase flow model developed in Guerrero et al. (2013). We identify, from a mathematical point of view, a set of conditions on the capillary pressures that ensure that the linearized, purely parabolic, initial value problem is well-posed. For the numerical simulation of the model, we advocate the use of Implicit-Explicit (IMEX)-Runge-Kutta (RK) schemes for time evolution with a Weighted-Essentially Non-Oscillatory (WENO) spatial discretization of the convective terms. In previous papers (Donat et al., 2013), (Guerrero et al., 2013), we showed that IMEX-RK methods can be useful in the numerical simulation of 1D two-phase and three-phase flows, since the stability restrictions for the time-step of these schemes are less severe than those of fully explicit schemes. On the other hand, their implementation requires, in general, a fairly intensive use of a nonlinear system solver, so that the efficiency and robustness of IMEX schemes for multi-phase flow is directly related to this nonlinear technique. In this paper we describe an efficient nonlinear system solver, based on an appropriate fixed-point iteration technique, in order to find the solution of the nonlinear systems that result from the implicit discretization of the nonlinear diffusive terms in the model. In addition, we implement zero-flux boundary conditions fully consistent with the vertical equilibrium assumptions. A set of numerical examples confirm the efficiency, robustness and reliability of the proposed numerical technique. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the one-dimensional time-fractional diffusion–wave equation with the fractional derivative of order , is revisited. This equation interpolates between the diffusion and the wave equations that behave quite differently regarding their response to a localized disturbance: whereas the diffusion equation describes a process, where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the wave equation. For the time-fractional diffusion–wave equation, the propagation speed of a disturbance is infinite, but its fundamental solution possesses a maximum that disperses with a finite speed. In this paper, the fundamental solution of the Cauchy problem for the time-fractional diffusion–wave equation, its maximum location, maximum value, and other important characteristics are investigated in detail. To illustrate analytical formulas, results of numerical calculations and plots are presented. Numerical algorithms and programs used to produce plots are discussed. [Copyright &y& Elsevier]

Abstract: The aim of the paper is the description of fractional-order differential equations in terms of exact mechanical models. This result will be archived, in the paper, for the case of linear multiphase fractional hereditariness involving linear combinations of power-laws in relaxation/creep functions. The mechanical model corresponding to fractional-order differential equations is the extension of a recently introduced exact mechanical representation (Di Paola and Zingales (2012) [33] and Di Paola et al. (2012) [34]) of fractional-order integrals and derivatives. Some numerical applications have been reported in the paper to assess the capabilities of the model in terms of a peculiar arrangement of linear springs and dashpots. [Copyright &y& Elsevier]

Abstract: This paper seeks to make a systematic study over a series of lid-driven flow in various deep cavities using the filter-matrix lattice Boltzmann (FMLB) model. A concise description of the FMLB model is presented in this paper, and important numerical considerations for effective use of the FMLB model are also clearly elucidated. In particular, the selection of a free parameter employed to appropriately control the weight of the third-order terms in the FMLB solution vector is carefully examined, resulting in some general suggestions that may render the FMLB stability consistently secured for simulations of different cavity flow scenarios. Employing the FMLB and the lattice Bhatnagar–Gross–Krook (LBGK) methods for comparison purpose, the first series of test cases correspond to the lid-driven cavity flows with a low Reynolds number ( ) at a variety of aspect ratios; the simulation results demonstrate that the FMLB model is superior to the LBGK method in terms of numerical stability and, particularly, the FMLB result can reach quite good agreement with the benchmark solution even if the aspect ratio goes up to 15. Then, the FMLB model is used to compute the steady flows for deep cavities with aspect ratios ranging from 1.5 to 7 and elevated Reynolds numbers ranging from 100 to 5000; a number of features of steady flows, such as the locations, strengths, and sizes of the vortices, as well as the effects of Reynolds number and aspect ratio on the vortex structure, are all predicted by the FMLB model with an obviously improved accuracy when compared to some other available numerical results. [Copyright &y& Elsevier]

Abstract: Understanding the structure of diffusion flames is often complicated by the dependence of flame structure on the boundary conditions, e.g. composition, temperature and flow field (e.g., strain rate in a counterflow flame.) In this paper, the local carbon-to-oxygen atom ratio (C/O ratio) is applied as a variable to interpret the flame and soot zone structures of counterflow diffusion flames from numerical results with detailed chemical kinetics and transport. Radical pool and soot precursor zones are shown to be clearly delineated in C/O ratio space. The boundary of these two zones, as well as the flame location, are shown to be independent of both stoichiometric mixture fraction ( ) and strain rate when interpreted in C/O space. The kinetic ratio is used to study the characteristics of key chemical reactions and to identify regions of equilibrium for these reactions. The results of this paper indicate that the C/O ratio is a valuable variable for interpreting flame structure and soot precursor chemistry for diffusion flames. [Copyright &y& Elsevier]

Abstract: Similarity coefficients (also known as coefficients of association) are important measurement techniques used to quantify the extent to which objects resemble one another. Due to privacy concerns, the data owner might not want to participate in any similarity measurement if the original dataset will be revealed or could be derived from the final output. There are many different measurements used for numerical, structural and binary data. In this paper, we particularly consider the computation of similarity coefficients for binary data. A large number of studies related to similarity coefficients have been performed. Our objective in this paper is not to design a specific similarity coefficient. Rather, we are demonstrating how to compute similarity coefficients in a secure and privacy preserved environment. In our protocol, a client and a server jointly participate in the computation. At the end of the protocol, the client will obtain all summation variables needed for the computation while the server learns nothing. We incorporate cryptographic methods in our protocol to protect the original dataset and all other intermediate results. Note that our protocol also supports dissimilarity coefficients. [Copyright &y& Elsevier]

Abstract: This paper discusses a numerical subgrid resolution approach for solving the Stokes–Brinkman system of equations, which is describing coupled flow in plain and in highly porous media. Various scientific and industrial problems are described by this system, and often the geometry and/or the permeability vary on several scales. A particular target is the process of oil filtration. In many complicated filters, the filter medium or the filter element geometry are too fine to be resolved by a feasible computational grid. The subgrid approach presented in this paper is aimed at describing how these fine details are accounted for by solving auxiliary problems in appropriately chosen grid cells on a relatively coarse computational grid. This is done via a systematic and careful procedure of modifying and updating the coefficients of the Stokes–Brinkman system in chosen cells. This numerical subgrid approach is motivated from one side from homogenization theory, from which we borrow the formulations for the so-called cell problem, and from the other side from the numerical upscaling approaches, such as Multiscale Finite Volume, Multiscale Finite Element, etc. Results on the algorithm’s efficiency, both in terms of computational time and memory usage, are presented. Comparison of the full fine grid solution (when possible) of the Stokes–Brinkman system with the subgrid solution of the upscaled Stokes–Brinkman system (including effective permeabilities for the quasi-porous cells), are presented in order to evaluate the accuracy and the efficiency. Advantages and limitations of the considered subgrid approach are discussed. [Copyright &y& Elsevier]