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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]

In this research paper, we delve into the analysis of a generalized discrete reaction-diffusion system. Our study begins with the discretization of a generalized reaction-diffusion model, achieved through second-order and L 1-difference approximations. We explore the local stability of its unique solution, both in the absence and presence of the diffusion term. To determine the conditions for global asymptotic stability of the steady-state solution, we employ suitable techniques including the direct Lyapunov method. To illustrate the practical application of this theoretical framework, we provide several numerical simulations that examine both the Lengyel-Epstein reaction-diffusion model and the discrete Degn-Harrison reaction-diffusion model. These simulations serve to validate the predictions of asymptotic stability. [ABSTRACT FROM AUTHOR]

In this paper, we propose a failure-informed multi-stage training algorithm for approximating the double-valley dark soliton solutions of the three-component nonlinear Schrödinger equation (NLSE). Although the physics-informed neural network (PINN) method has achieved remarkable results in many equations, it is ineffective when directly applied to the considered equation due to the non-vanishing background of dark solitons and the complexity of the multi-valley structure. To improve the approximation ability of the PINN, the algorithm we proposed generates effective samples adaptively based on Gaussian mixture distribution by combining the time-adaptive strategy, failure-informed enrichment technique, and pre-fixed technique. Furthermore, the algorithm is extended to learn the solution operator of three-component NLSE within a specialized solution space. The results of numerical simulation demonstrate that the improved algorithm has better approximation ability and faster convergence rate in approximating the solution and solution operator of the considered equation while using a smaller training set. • We explore a deep learning method for solving three-component nonlinear Schrödinger equation. • We proposed an improved PINNS by time-adaptive strategy, failure-informed enrichment technique, and pre-fixed technique. • The data-driven double-valley dark soliton solutions are derived by the proposed method. • The proposed method can be easily extended to the physics-informed DeepONets. [ABSTRACT FROM AUTHOR]

This paper mainly studies the high-order stable numerical solutions of the time-space fractional diffusion equation. The Fourier spectral method is used for discretization in space and the Spectral Deferred Correction (SDC) method is used for numerical solutions in time. Therefore, a high-precision numerical discretization scheme for solving the fractional diffusion equation is obtained, and the convergence and stability of the scheme are proved. Finally, several numerical examples are given to illustrate the effectiveness and feasibility of the numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper, we consider output tracking and disturbance rejection for a 1-D wave equation with in-domain feedback/recirculation of an intermediate point velocity. The performance output is an intermediate point displacement and the control matches the general disturbance. We first use the active disturbance rejection control method to design a disturbance estimator to estimate the disturbance. Then, we propose an observer and show that the obtained observer converges exponentially to the original system. Finally, an output feedback controller is constructed in terms of the designed observer. It is verified that the resulting closed-loop system is well-posed and all internal signals are bounded. Moreover, the output exponentially tracks the reference signal. Some numerical simulations are given to validate the theoretical results. [ABSTRACT FROM AUTHOR]

Biological structures are hierarchical and non-homogeneous objects. In particular, in each type of the biological cell various organelles exist. Also, in the cell many nonlinear processes, like the synaptic transmission, take place. The mesh, created for the purpose of numerical simulation, should reflect accurately their nature that is usually complicated. This, in connection with the fact that the mesh should fit precisely the aforementioned structures, create substantially new problems in comparison with well worked-out engineering applications. In this paper the problem of refining the mesh of a presynaptic bouton is considered. The refinement was tested by performing simulations of nonlinear diffusion-reaction type of propagation of neurotransmitters. The results for the original and for the refined mesh were compared. The latter reflected the nature of the biological object more precisely. The benefits and the disadvantages of the proposed refinement algorithm are discussed in detail. [ABSTRACT FROM AUTHOR]

In this paper, an efficient dimension splitting method is applied to the two-dimensional (2D) and three-dimensional (3D) Cahn-Hilliard type equations with significant applications in physical, biological and computer science. The proposed method can greatly reduce the storage requirements and computational complexity in two and three dimensions, and preserve the high precision and mass conservation. The dimension splitting method is based on auxiliary variables for spatial derivative terms and the operator splitting approach. It converts the 2D or 3D problem into a series of one-dimensional (1D) problems which can be solved by using multi-thread. The stability and accuracy of the proposed method are improved by a local stabilizing approach and extrapolation. The analyses of discrete energy and discrete mass conservation property are shown. Numerical examples confirm the advantages of proposed method. And a variety of simulations, such as the phase separations, curvature driven flows, diblock copolymers, tumor growth and volume restoration, are performed with respect to practical applications. • Dimension splitting method for Cahn-Hilliard type equations. • The global stability is achieved by the local stability of solving each sub-problem. • Fourth-order mass conservative scheme based on the compact differencing and extrapolation. • A large number of numerical examples for practical applications. [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]

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, we report numerical simulations for dynamic fracture and fragmentation problems in brittle materials using a recently developed split strain energy diffusive mass-field damage model in terms of standard finite element method (FEM). The enhanced constitutive laws for mass source and mass flux by means of strain energy decomposition are adopted to overcome certain drawbacks of the original non-split model. We present theoretical formulations in a more general way that is suitable and valid for both small and finite deformation regimes. The coupled system of equations is derived in which the momentum is governed to describe the time-dependent deformation of brittle solids whereas the mass-diffusion balance equations are for the evolution of mass density. We also develop simple techniques for determining crack velocity and estimating amount of mass loss. Numerical examples are considered for both small and finite deformations. The accuracy and performance of the developed theory for dynamic brittle fracture are illustrated through comparison and verification, which are to compare the computed results with respect to experimental data and other numerical methods in terms of crack path trajectories, dynamic response, energy dissipation, and other relevant numerical aspects. • Dynamic description of localized mass-field damage model at finite deformation for brittle fracture is presented. • Localized mass-field damage model enhanced by energy decomposition for dynamic analysis is used. • Dynamic crack propagation for problems with complex geometries and fragmentation simulation are studied. • Prediction of dissipated energy agrees well with the theoretical value. • It provides an alternative approach for modeling dynamic fracture in solids. [ABSTRACT FROM AUTHOR]

Equilibrium position, as well as dynamic patterns of suspended particles, plays a dominant role in particle manipulation of segregation, transporting, or sorting. The current paper aims to investigate the falling manners of non-circular particles in an enclosure while the pulsatile flow is involved as a counter-flow. The direct-forcing immersed boundary method is coupled with the lattice Boltzmann method to simulate this widespread issue numerically. Corresponding boundary conditions are hired to take into account pulsatile flow. Findings were first successfully verified against the existing literature and the accuracy of the results is well-demonstrated. For the first time, a series of proof-of-principal simulations with an increased level of geometric complexity, particle aspect ratio, as well as the number and initial configuration (release positions), is carried out. The force resulting from the particle acceleration is also considered. It is shown that considering such factors, particle settling manners would markedly differ. The collective behavior of particles is also studied, and it is revealed that the presence of neighboring particles makes the influence of the particle shape less clear; however, pulsatile flow presents considerable differences in the settling manners of particles. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a reaction-diffusion model incorporating delay and nonlocal effects. The normal form of double Hopf bifurcation is derived. The diffusive model of pollen tube tip growth is discussed and numerical simulations show that spatially homogeneous and inhomogeneous periodic solutions can be both stable or connected by a heteroclinic orbit under certain conditions. In addition, the diffusive Lotka-Volterra model with delay and nonlocality is considered and spatially inhomogeneous quasi-periodic solution is obtained. • Derive the algorithm of normal form of double Hopf bifurcation for reaction-diffusion system with delay and spatial average. • Investigate the dynamics near the double Hopf bifurcation point for two biological models. • Find bistability of homogeneous and inhomogeneous periodic solutions and pattern transitions for pollen tube tip growth model. • Find stable inhomogeneous quasi-periodic solutions and pattern transitions of periodic and quasi-periodic solutions for Lotka-Volterra model. [ABSTRACT FROM AUTHOR]

This paper is concerned with the development and application of optimally efficient numerical methods for the simulation of vascular tumour growth. This model used involves the flow and interaction of four different, but coupled, phases which are each treated as incompressible fluids, Hubbard and Byrne (2013). A finite volume scheme is used to approximate mass conservation, with conforming finite element schemes to approximate momentum conservation and an associated equation. The principal contribution of this paper is the development of a novel block preconditioner for solving the linear systems arising from the discrete momentum equations at each time step. In particular, the preconditioned system has both a solution time and a memory requirement that is shown to scale almost linearly with the problem size. [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]

The paper presents a study aimed at understanding the characteristics of an internal explosion within a chamber with limited venting. The study includes numerical simulations and analytical derivations. An in-house 3D code employing an improved weighted essentially non-oscillatory (WENO) conservative finite difference scheme was used to carry out the simulations. It is indicated that the proposed improved WENO scheme can resolve the shock waves with higher accuracy and resolution. Further, a simplified analytical model to predict the quasi-static overpressure was developed based on the conservation law of total energy and dimensional analysis theory. It is demonstrated that the proposed simplified approach for prediction of the quasi-static overpressure agrees well with simulation results for a wide range of explosive weights and venting hole sizes. The studies in this paper provide an efficient method to predict the blast load inside a partially confined chamber for the analysis of the consequences of explosion. [ABSTRACT FROM AUTHOR]

In this paper we consider the numerical solution of the Hamiltonian wave equation in two spatial dimensions. We construct a two step procedure in which we first discretize the space by the Mimetic Finite Difference (MFD) method and then we employ a standard symplectic scheme to integrate the semi-discrete Hamiltonian system derived. The main characteristic of the MFD methods, when applied to stationary problems, is to mimic important properties of the continuous system. This approach yields a full numerical procedure suitable to integrate Hamiltonian problems. A complete theoretical analysis of the method and some numerical simulations are developed in the paper. [ABSTRACT FROM AUTHOR]

The present paper is the third of a series of papers devoted to the study of h - p spectral element methods for three dimensional elliptic problems on non-smooth domains using parallel computers. In this paper we provide error estimates, preconditioners and numerical results. The spectral element functions are fully non-conforming. We propose preconditioners on non-smooth domains which can be diagonalized using separation of variables technique. Optimal error estimates in terms of number of layers in the geometrical mesh and in terms of number of degrees of freedom are obtained. The method is easy to implement on a parallel computer and we briefly outline computational techniques. We give results of numerical simulations to confirm the theoretical estimates. Theoretical results have been also validated by computational experiments which are published independently in Dutt et al. (2014). [ABSTRACT FROM AUTHOR]

The present study focuses on the receptor driven endocytosis typical of viral entry into a cell. A locally increased density of receptors at the time of contact between the cell and the virus is necessary in this case. The virus is considered as a substrate with fixed receptors on its surface, whereas the receptors of the host cell are free to move over its membrane, allowing a local change in their concentration. In the contact zone the membrane inflects and forms an envelope around the virus. The created vesicle imports its cargo into the cell. This paper assumes the diffusion equation accompanied by boundary conditions requiring the conservation of binders to describe the process. Moreover, it introduces a condition defining the energy balance at the front of the adhesion zone. The latter yields the upper limit for the size of virus which can be engulfed by the cell membrane. The described moving boundary problem in terms of the binder density and the velocity of the adhesion front is well posed and numerically solved by using the finite difference method. The illustrative examples have been chosen to show the influence of the process parameters on the initiation and the duration of the process. [ABSTRACT FROM AUTHOR]

The deflagration-to-detonation transition via the interaction of a weak shock with a series of discrete laminar flames is analyzed computationally based on the unsteady reactive Navier–Stokes equations with one-step Arrhenius chemistry. For comparison, simulations with the Euler equations are also performed. The numerical setup aims to mimic an array of laminar flames ignited at different spark times, artificially inducing chemical activity to stimulate the coupling between the gas dynamics and the chemical energy release for the deflagration-to-detonation transition. The interaction of the weak shock with the first cylindrical flame demonstrates a very good agreement with the results in the literature and that a single weak shock–flame interaction is insufficient to cause a prompt DDT. However, a high degree of Richtmyer–Meshkov instabilities induced by repetitive shock–flame and shock–boundary interactions generate turbulence that accelerates the flame surface, referred to as the flame brush, until eventually a hot spot ignition in the unreacted material develops into a multi-headed detonation wave. In the absence of physical diffusion in the Euler simulation, the enhanced burning rate of the turbulent flame brush is suppressed. Nevertheless, the intense flow fluctuations generated by the interactions of shocks, boundary and flames create the conditions under which a deflagration-to-detonation transition can potentially occur at later times. A parametric study is also reported in this paper to assess the influence of various physical parameters on the transition event and to explore scaling relationships among these parameters. [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]

Percolation flow problems are discussed in many research fields, such as seepage hydraulics, groundwater hydraulics, groundwater dynamics and fluid dynamics in porous media. Many physical processes appear to exhibit fractional-order behavior that may vary with time, or space, or space and time. The theory of pseudodifferential operators and equations has been used to deal with this situation. In this paper we use a fractional Darcy’s law with variable order Riemann–Liouville fractional derivatives; this leads to a new variable-order fractional percolation equation. In this paper, a new two-dimensional variable-order fractional percolation equation is considered. A new implicit numerical method and an alternating direct method for the two-dimensional variable-order fractional model is proposed. Consistency, stability and convergence of the implicit finite difference method are established. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of the methods. This technique can be used to simulate a three-dimensional variable-order fractional percolation equation. [ABSTRACT FROM AUTHOR]

Excessive synchronizations of neurons in the brain networks can be a reason for some episodic disorders such as epilepsy. In this paper, we simulate neural dynamic models and their synchronizations numerically by a new numerical algorithm. In order to overcome the complexity of the problem, a suitable combination of the Legendre spectral element method and operator splitting technique is proposed to convert a complex system of partial differential equations into sparse linear algebraic systems. The simulations are explored in different aspects such as accuracy, computing the wavefront and angular speeds, computing time, the condition number of obtained algebraic systems to demonstrate the proposed scheme effectiveness. The numerical simulations and comparisons confirm the method is suitable in terms of accuracy and speed. [ABSTRACT FROM AUTHOR]

The nucleation phenomenon and its consequence formation of droplets are destructive effects that occur as a result of the sudden expansion of the flow through the steam turbine blades. The formation of these droplets causes blade corrosion and severe thermodynamic losses, thereby reducing the turbine efficiencyg.In this paper, two holes on the pressure and suction surfaces of the turbine blade were improvised in order to control and optimize the turbine performance. These holes transfer the steam from the high-pressure to the low-pressure part of the cascade; accordingly, the steam is moved from within the cascade itself without using the external steam. For this purpose, an in-house code is written in two-dimensional, compressible, viscous, and turbulent, based on Navier–Stokes equations. To simulate the turbulent flow, the k-k L - ω turbulence model is used. The governing equations of the dry flow are solved by the Eulerian approach using a semi-implicit density-based method. Also, in order to avoid the oscillations caused by high accuracy numerical solution, a limiter method called Normalized Variable Diagram (NVD) is employed. For the wet part of the solution field, the equations corresponding to the Lagrangian form are solved. To validate the present numerical code, the results are compared with the experimental data and good agreement is obtained. The results showed that as the hole inlet comes to close the leading edge, further improvement is achieved. Constructing the holes on the suction and pressure surfaces in the case where the holes inlet were closer to the leading edge resulted in complete removal of the wetness from suction and pressure surfaces, and on the middle passage at the outlet of the turbine blade was reduced by 6.7% compared to the baseline geometry. The droplet radius size at the turbine blade outlet on the suction, pressure and mid-pass surface decreased by 99%, 95%, and 29%, respectively, compared to the baseline geometry. Since the hole construction is a passive flow control method without the need to consume energy, using this method to control wet steam is an optimal method. • The condensation flow of a steam turbine blade was numerically investigated. • The code was based on the Eulerian-Lagrange perspective. • To eliminate the fluctuations Normalized Variables Diagram limiter method was used. • The effects of hole creation on the condensation flow control were​ investigated. [ABSTRACT FROM AUTHOR]

This paper deals with a mutualism system in random environments, in which the cooperation of two species is guaranteed depending on increasing the carrying capacity of each other. Mathematically, we obtain the existence and uniqueness of a stable stationary distribution by means of Markov semigroup theory and Fokker–Planck equation. In addition, we prove that densities of the distributions for the solutions converge in L 1 to an invariant density under appropriate conditions. Numerical simulations are also presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The current paper is devoted to the time–space fractional Ginzburg–Landau equation driven by fractional Brownian motion. The spatial–temporal regularity of the nonlocal stochastic convolution is firstly established, and then the existence and uniqueness of the global mild solution are obtained by the Banach fixed point theorem and Mittag-Leffler functions. Finally, the numerical simulations for the time-fractional stochastic Ginzburg–Landau equation are provided to verify the mathematical analysis results. [ABSTRACT FROM AUTHOR]

Cognitive disorders especially epilepsy are closely linked with synchronization/desynchronization of neurons in the brain. In this paper, the dynamical modeling and behavior analysis of a FitzHugh–Nagumo neuron and also synchronization control of a network of FitzHugh–Nagumo neurons which promise the understanding of cognitive processing, are studied. To numerically simulate these models and in order to overcome their difficulties such as computational complexity, multi-dimensionality, non-linearity, having large spatial and temporal domains and also having a discrete initial data, we propose the use of a strongly meshless technique based on Wendland compactly supported radial basis function in conjunction with an operator splitting algorithm. The main advantage of the proposed algorithm is its computational complexity (for example O (N x 1 3 + N x 2 3) of the inversion of all matrices) which is much lower than the complexity of pure radial basis functions method (O (N x 1 3 N x 2 3) of the inversion of matrices). Numerical experiments are presented showing that the proposed approaches are extremely accurate and fast. [ABSTRACT FROM AUTHOR]

In this paper, the chemotaxis partial differential equations models which describe the movement by one community in reaction to one chemical or biological signal are given and solved numerically on surfaces. The models and the numerical method concerned are in terms of a generic form of chemotaxis models. We develop a new semi-implicit finite element scheme based on the gradient and Laplacian recoveries to linearize the equations. Meanwhile, we modify the proposed semi-implicit scheme as a characteristic form and present a novel strategy for the discretization of characteristic derivative on surface. The lumped mass modification is employed for positivity preservation. And the related analysis results are provided. We investigate the accuracy and convergence of the proposed method by numerical tests. And the simulations of blowing-up solution, pattern formulations and aggregations of bacteria demonstrate the applicability of the proposed methods. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the problem of nonlinear wave tomography is formulated as a coefficient inverse problem for a hyperbolic equation in the time domain. Efficient methods for solving the inverse problems of wave tomography for the case of transparent boundary conditions are presented. The algorithms are designed for supercomputers. We prove the Fréchet differentiability theorem for the residual functional and derive an exact expression for the Fréchet derivative in the case of a transparent boundary in the direct and conjugate problems. The expression for the Fréchet derivative of the residual functional remains valid if the experimental data are provided for only a part of the boundary. The effectiveness of the proposed method is illustrated by the numerical solution of a model problem of low-frequency wave tomography. The model problem is tailored to apply to the differential diagnosis of breast cancer. [ABSTRACT FROM AUTHOR]

Abstract Two-dimensional Kelvin–Helmholtz instability problems are popular examples for assessing discretizations for incompressible flows at high Reynolds number. Unfortunately, the results in the literature differ considerably. This paper presents computational studies of a Kelvin–Helmholtz instability problem with high order divergence-free finite element methods. Reference results in several quantities of interest are obtained for three different Reynolds numbers up to the beginning of the final vortex pairing. A mesh-independent prediction of the final pairing is not achieved due to the sensitivity of the considered problem with respect to small perturbations. A theoretical explanation of this sensitivity to small perturbations is provided based on the theory of self-organization of 2D turbulence. Possible sources of perturbations that arise in almost any numerical simulation are discussed. [ABSTRACT FROM AUTHOR]

Abstract A cantilever with an open surface crack excited by the pedestal movement with the form of the Gaussian white noise was modeled in this paper. The open crack was simplified as an elastic hinge and the modal functions of the cracked cantilever were obtained. The dynamical equation governing the transverse motion was gained with the Kane's method after sufficient consideration of the geometrical nonlinearity and the axial inertia. Both quasi-conservative stochastic averaging method and the improved energy envelop stochastic averaging method which is effective on strong nonlinearity were applied to analyze the responses of the cantilever model theoretically. The stationary probability density function (PDF) of the amplitude, as well as the joint stationary PDF of the displacement and velocity, was investigated. For the purpose of exploring the reliability of the cantilever, the reliability function was established to avoid the potential fatigue or damage. Numerical simulations were carried out to vindicate the theoretical analysis. The advantage of the improved energy envelop stochastic averaging method was explicated by the digital results of both the stationary PDF of the amplitude and the joint PDF of displacement and velocity. Furthermore, the accuracy of the prediction on the reliability given by the theoretical analysis was also vindicated by the numerical simulation (Monte Carlo method) of the reliability. [ABSTRACT FROM AUTHOR]

Abstract The lattice Boltzmann method (LBM) has been widely considered as a distinctive and reliable approach for simulating the complex particulate flows. As an intrinsic kinetic scheme, it is quite convenient for LBM to apply the bounce-back (BB) type methods to handle the moving boundaries with complicated geometry, which is a tricky task for general interface-resolved methods. However, the two major schemes in LBM, i.e., the simple BB rule and the curved boundary condition (CBC) are presently encountered by the problems of low precision and loss of local computation, respectively. To overcome those two deficiencies in the boundary treatment simultaneously, a one-point second-order CBC is proposed in this paper. Information of only a single node is required in the present scheme, and the second-order accuracy is validated in the channel flow and cylindrical Couette flow. Applications to the particulate flows are further implemented to verify the present scheme. Numerical results are in good agreement with those in the literature. [ABSTRACT FROM AUTHOR]

In the present paper we derive novel (non)linear PDE models for pricing European options and the associated total value adjustment (XVA), when incorporating the counterparty risk. The main innovative aspect is the consideration of stochastic spreads instead of less realistic constant spreads previously used in the literature. For the nonlinear model, a rigorous mathematical analysis based on sectorial differential operators allows to state the existence and uniqueness of a solution. Moreover, for the numerical solution we propose an appropriate set of techniques based on the method of characteristics for time discretization, finite element for spatial discretization and fixed point iteration for the nonlinear terms. Finally, numerical examples illustrate the expected behaviour of the option prices and the total value adjustment. [ABSTRACT FROM AUTHOR]

We present in this paper a numerical simulation to the problem of triple flame propagation in a porous-walls channel in the presence of a temperature gradient across the channel. The problem has been formulated using the thermo-diffusive approximation and then solved numerically using finite elements method. The study showed that temperature gradient plays a crucial role on the existence and propagation of triple flames. More precisely, the effect of temperature gradient on the flame propagation was found to: (i) cause the flame to exist only for a limited range of values of the temperature gradient parameter, (ii) establish multiplicity of solutions each of them characterizes unique combustion regimes, (iii) modifies the flame shape from the usual triple flame shape when the temperature gradient is large, and finally (iv) enhance the reactivity of the underlying mixture, but on the other hand will have serious implications on the safety of the combustion chamber. [ABSTRACT FROM AUTHOR]

In this paper, based on the LU decomposition, we propose three non-orthogonal Jacobi-like alternating iterative algorithms with two strategies for solving the joint diagonalization problem of a set of Hermitian matrices. In this kind of algorithm, each transformation includes one upper triangular iterative step and one lower triangular iterative step, and each step involves one parameter. The optimal parameter of each step is derived analytically. The convergence of our proposed algorithms is proven. According to this convergence analysis, the existing GNJD algorithm is revisited. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed algorithms in comparison with existing ones. [ABSTRACT FROM AUTHOR]

A new Hirota’s bilinear form of the (4+1)-dimensional Fokas equation is obtained by an appropriate wave transformation. Applying the parameter limit method introduced in this paper, we obtain two homoclinic solutions and a rogue wave solution of (4+1)-dimensional Fokas equation. Besides, we also discuss the interaction between rogue wave solution and different forms of stripe solitons (one stripe soliton and double stripe solitons), and investigate the annihilation and emergence of rogue waves. Finally, the parametric reasons of soliton fusion and fission are discussed, furthermore, the fusion and fission phenomena of soliton are simulated by three-dimensional plots. [ABSTRACT FROM AUTHOR]

In this paper, we present an effective low-Mach number improvement for upwind schemes. The artificial viscosity of upwind schemes scales with 1 ∕ M a incurring a loss of accuracy for the Mach number approaching zero. The remedy is based on three steps: (i) the jump of the left and right states is split into the density diffusion part and velocity diffusion part; (ii) the velocity diffusion part is rescaled by multiplying a scaling function; (iii) the scaling function is only related to the local Mach number without the cut-off reference Mach number and meantime restricted by a shock senor. The resulting modification is very easily implemented and applied within Roe, HLL and Rusanov, etc. Then, asymptotic analysis and numerical experiments for a wide Mach number demonstrate that this novel approach is equipped with these attractive properties: (1) free from the cut-off global problem and damping checkerboard modes; (2) satisfying the correct M a 2 scaling of pressure fluctuations, the divergence constraint and a Poisson equation; (3) independent of Mach number in terms of accuracy; (4) better accurate and higher resolution in low Mach number regimes when compared with existing upwind schemes, and applicable for moderate or high Mach numbers. Thus, the proposed modification is expected to provide as an excellent and reliable candidate to simulate turbulent flows at all Mach numbers. [ABSTRACT FROM AUTHOR]

With the wide applications of social networks, government and individuals increasingly emphasize information networks security. This paper is devoted to investigating a reaction–diffusion malware propagation model with mixed delays to describe the process of social networks. Applying matrix theory for characteristic values, we establish the local stability conditions of a positive equilibrium point. Based on the linear approximation method of nonlinear systems, the Hopf bifurcation at the positive equilibrium point is considered. Additionally, we identify some sensitive parameters in the process of malware propagation that are significant for control theory. Finally, numerical simulations are performed to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is an important class of LB model with lots of advantages over the traditional single-relaxation-time (SRT) LB model. Generally, the computation of strain rate tensor is crucial for the MRT-LB simulations of some complex flows. At present, only two formulae are available to compute the strain rate tensor in the MRT LB model. One is to compute the strain rate tensor using the non-equilibrium parts of macroscopic moments (Yu formula). The other is to compute the strain rate tensor using the non-equilibrium parts of density distribution functions (Chai formula). The mathematical expressions of these two formulae are so different that we do not know which formula to choose for computing the strain rate tensor in the MRT LB model. To overcome this problem, this paper presents a theoretical study of the relationship between Chai and Yu formulae. The results show that the Yu formula can be deduced from the Chai formula, although they have their own advantages and disadvantages. In particular, the Yu formula is computationally more efficient, while the Chai formula is applicable to more lattice patterns of the MRT LB models. Furthermore, the derivation of the Yu formula in a particular lattice pattern from the Chai formula is more convenient than that proposed by Yu et al. [ABSTRACT FROM AUTHOR]

The particle-in-cell (PIC) method has been widely used for plasma simulation, because of its noise-reduction capability and moderate computational cost. The immersed finite element (IFE) method is efficient for solving interface problems on Cartesian meshes, which is desirable for the PIC method. The combination of these two methods provides an effective tool for plasma simulation with complex interface/boundary. This paper introduces an improved IFE–PIC method that enhances the performance in both IFE and PIC aspects. For the electric field solver, we adopt the newly developed partially penalized IFE method with enhanced accuracy. For PIC implementation, we introduce a new interpolation technique to ensure the conservation of the charge. Numerical examples are provided to demonstrate the features of the improved IFE–PIC method. [ABSTRACT FROM AUTHOR]

In this paper we propose a multigrid optimization algorithm (MG/OPT) for the numerical solution of a class of quasilinear variational inequalities of the second kind. This approach is enabled by the fact that the solution of the variational inequality is given by the minimizer of a nonsmooth energy functional, involving the p -Laplace operator. We propose a Huber regularization of the functional and a finite element discretization for the problem. Further, we analyze the regularity of the discretized energy functional, and we are able to prove that its Jacobian is slantly differentiable. This regularity property is useful to analyze the convergence of the MG/OPT algorithm. In fact, we demonstrate that the algorithm is globally convergent by using a mean value theorem for semismooth functions. Finally, we apply the MG/OPT algorithm to the numerical simulation of the viscoplastic flow of Bingham, Casson and Herschel–Bulkley fluids in a pipe. Several experiments are carried out to show the efficiency of the proposed algorithm when solving this kind of fluid mechanics problems. [ABSTRACT FROM AUTHOR]

In this paper, we study stability and convergence of fully discrete finite element method on large timestep which used Crank–Nicolson extrapolation scheme for the nonstationary Navier–Stokes equations. This approach bases on a finite element approximation for the space discretization and the Crank–Nicolson extrapolation scheme for the time discretization. It reduces nonlinear equations to linear equations, thus can greatly increase the computational efficiency. We prove that this method is unconditionally stable and unconditionally convergent. Moreover, taking the negative norm technique, we derive the L 2 , H 1 -unconditionally optimal error estimates for the velocity, and the L 2 -unconditionally optimal error estimate for the pressure. Also, numerical simulations on unconditional L 2 -stability and convergent rates of this method are shown. [ABSTRACT FROM AUTHOR]

In this paper, we discuss problems arising when computing resonances with a finite element method. In the pre-asymptotic regime, we detect for the one dimensional case, spurious solutions in finite element computations of resonances when the computational domain is truncated with a perfectly matched layer (PML) as well as with a Dirichlet-to-Neumann map (DtN). The new test is based on the Lippmann–Schwinger equation and we use computations of the pseudospectrum to show that this is a suitable choice. Numerical simulations indicate that the presented test can distinguish between spurious eigenvalues and true eigenvalues also in difficult cases. [ABSTRACT FROM AUTHOR]

Suppression of spiral wave attracts more and more attention in nonlinear systems. In this paper, a spiral wave local feedback control approach based on the FitzHugh–Nagumo (FHN) model is studied with lattice Boltzmann method. Numerical simulations are performed to investigate the effects of initial conditions for the spiral wave formation, model parameters, size and position of the feedback control region, and feedback control parameters on the behavior of spiral wave. The results show that there are three characteristics of spiral wave elimination. The first is that initial conditions of the spiral wave formation have little influence on feedback control of spiral wave. Secondly, the model parameters are related to the time needed for the elimination of spiral wave, for example, the larger the mutual time scales, the faster the elimination of spiral wave. Finally, through selecting the size and position of the feedback region, spiral wave can be effectively removed with weak feedback signal. [ABSTRACT FROM AUTHOR]

The Swift–Hohenberg (SH) equation has been widely used as a model for the study of pattern formation. The SH equation is a fourth-order nonlinear partial differential equation and cannot generally be solved analytically. Therefore, computer simulations play an essential role in understanding of nonequilibrium processing. The aim of this paper is to present first- and second-order semi-analytical Fourier spectral methods as an accurate and efficient approach for solving the SH equation. The methods are based on the operator splitting method and are to split the SH equation into linear and nonlinear subequations. The linear and nonlinear subequations have closed-form solutions in the Fourier and physical spaces, respectively. The methods are simple to implement and computationally cheap to achieve high-order time accuracy. Numerical experiments are presented demonstrating the accuracy and efficiency of proposed methods. [ABSTRACT FROM AUTHOR]

Numerical simulation of the propagation of electrical signals in the heart is a very demanding application. In fact, very fine meshes and small time steps are currently required to capture the phenomena. In this paper, we propose and explore a very high-order scheme specifically designed for this application. Its numerical properties are detailed and the different choices on both the scheme’s definition and implementation are discussed and justified. Numerical results show the importance of considering very high-order schemes, even for classical tests such as the propagation of planar or spiral waves. [ABSTRACT FROM AUTHOR]

This paper is concerned with traveling wave solutions of a nonlocal dispersal Susceptible–Infective–Removal–Healing (for short SIRH ) model with relapse. It is found that the existence and nonexistence of traveling waves of the system are not only determined by the critical wave speed c ∗ , but also by the basic reproduction number R 0 of the corresponding system of ordinary differential equations. More precisely, we use Schauder’s fixed-point theorem to obtain the existence of traveling waves for R 0 > 1 and c > c ∗ , and the nonexistence of traveling waves for R 0 > 1 and 0 < c < c ∗ . Some numerical simulations and discussions are also provided to illustrate our analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we develop a fictitious domain method with Distributed Lagrange Multipliers for simulating 2D unsteady shear-thinning non-newtonian incompressible flow in a single-screw and twin-screw extruder. The advantage of the fictitious domain method is that one can use a fixed mesh even as the fluid domain changes in time (as the screws are rotating), eliminating the need for repeated remeshing and projection. The method uses a finite element discretization in space and an operator-splitting technique for discretization in time. Numerical results are given for the flow inside a single-screw extruder or twin-screw extruder, which show that the fictitious domain method is efficient for the problem. [ABSTRACT FROM AUTHOR]

In this paper we discuss Turing instability of a generalized Keller–Segel chemotaxis model involving n -species and m -chemoattractants. By using combinatorial matrix theory we extend the result on the instability of a single-species and n -chemicals Keller–Segel system (Leenheer et al., 2012) to a general ( n + m ) × ( n + m ) system. A sufficient condition for chemotaxis driven Turing instability of the multi-species model is obtained. Moreover, some examples and numerical simulations are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the n -species predator–prey models in a bounded domain in two cases, one of which involves n − 1 cooperative preys and the other involves n − 1 competitive preys. In these models, the diffusion coefficients may be density dependent, and the reaction functions include ratio-dependent terms. Some simple conditions are obtained to ensure the dynamical behaviors of the time-dependent solutions in relation to some positive solutions or quasi-solutions of the steady-state problems, which include the existence of these solutions. This dynamical behaviors lead to the coexistence and global attractors of the predator–prey systems. Some numerical simulations are given in the end. [ABSTRACT FROM AUTHOR]

Pattern structures are indicators of populations distribution and may provide some signals for population protection. In this paper, we present a predator–prey system with spatial diffusion and ratio-dependent functional response. Based on both mathematical analysis and numerical simulations, we find that predation rate of predator plays an important role in pattern formation of populations. As it varies, transition from stationary patterns to no-stationary patterns emerges. Th e obtained results show that predation behavior of predator on prey is a key determinant in the persistence of populations. [ABSTRACT FROM AUTHOR]