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In this paper, we consider the high-dimensional two-sided space fractional diffusion equations with monotonic variable diffusion coefficients, which are derived from the fractional Fick's law. We apply the implicit Euler scheme to discretize the temporal derivative and certain difference operator to discretize the space fractional derivatives. For the discretized linear systems, we apply the alternating direction implicit (ADI) scheme to split the linear systems into two sub-systems of linear equations. For both coefficient matrices of the sub-systems of linear equations, we propose the lopsided scaled diagonal and Toeplitz splitting preconditioner. The generalized minimal residual (GMRES) method combined with the proposed preconditioner is applied to solve both sub-systems of linear equations. The spectral distributions of the preconditioned matrices are analyzed, and the theoretical results are given. Numerical results demonstrate that the proposed preconditioner is efficient in accelerating the convergence rate of the GMRES method for solving the discretized system of linear equations. [ABSTRACT FROM AUTHOR]

Transcorneal drug delivery has been used to treat eye diseases of both segments of the eye: anterior and posterior segments. Due to the low corneal permeability, this drug administration route has been shown to be very ineffective. Ultrasound has been used to increase the corneal drug transport. In this paper we consider a system of partial differential equations (PDEs) defined by a wave equation to describe the propagation of the pressure wave generated by a transducer, a convection-diffusion-reaction equation for the drug transport in each layer of the cornea - epithelium, stroma and endothelium. The last equations are linked with the acoustic pressure through the diffusion coefficient and the convective velocity. The system of PDEs is completed with convenient interface conditions, initial and boundary conditions. The initial boundary value problem is studied from analytical point of view and the qualitative behavior is numerically illustrated. The results confirm the effectiveness of ultrasound as enhancer of the drug delivery through the cornea. [ABSTRACT FROM AUTHOR]

In this paper, thermogravitational flow in a square enclosure filled with Cu-water nanofluid which is influenced by inclined magnetic induction, has been mathematically modelled. Isotropic heat flow field and viscous dissipation effects are introduced in the modelling as well. The vertical walls of the domain are subjected to finite temperature gradient while the horizontal walls are thermally insulated. The control equations subject to the flow physics of an incompressible, viscous and electrically conducting fluid are the Navier-Stokes equations along with energy equation. These governing equations are solved numerically by proposing a fourth order compact scheme. The influences of diverse effective parameters including Hartmann number, viscous dissipation parameter (Eckert number) and heat generation or absorption parameter on characteristic features of heat transfer and fluid flow with different concentration distributions of nanoparticles are analyzed and addressed in detail. Results show that the presence of viscous dissipation retards the thermal diffusion process. The average Nusselt number values are decreased with the increasing values of Eckert number. In addition, with the increasing values of Hartmann number, the thermal transport is decreased in spite of incorporation of isotropic heat flow. [ABSTRACT FROM AUTHOR]

This paper presents a sixth-order nonlinear parabolic problem of the square phase-field crystal model. We first demonstrate the time-discrete backward Euler scheme with mass conservation and energy stability. Then, we prove the unconditionally optimal error estimates for the time-discrete backward Euler scheme. In the end, we present 2D and 3D numerical simulations to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

The choice of the shape parameter highly effects the behaviour of radial basis function (RBF) approximations, as it needs to be selected to balance between the ill-conditioning of the interpolation matrix and high accuracy. In this paper, we demonstrate how to use neural networks to determine the shape parameters in RBFs. In particular, we construct a multilayer perceptron (MLP) trained using an unsupervised learning strategy, and use it to predict shape parameters for inverse multiquadric and Gaussian kernels. We test the neural network approach in RBF interpolation tasks and in a RBF-finite difference method in one and two-space dimensions, demonstrating promising results. [ABSTRACT FROM AUTHOR]

This article presents a detailed study of the non-Newtonian transient Casson fluid flow behavior in three different rectangular domains with their upper and bottom parallel walls moving horizontally either in same direction or in opposite direction to each other. The numerical computations are carried out applying the dynamic subscales approach of the Algebraic Sub-Grid multi-Scale (ASGS) stabilized finite element method upto a sufficiently high Reynolds number, Re =10000. The flow patterns, which include the generation of the primary and secondary vortices, are highly impacted by the Re values, the structures of the domains and the directions in which the parallel walls are moving. We have observed the appearances of both the symmetric and stable as well as unstable asymmetric types of primary vortices in the domains of aspect ratios ≤ 1, with walls moving in the same direction. On the other hand, we notice the existence of multiple phases of the solutions for the Casson fluid flowing in the rectangular domains of aspect ratio > 1, with walls moving in the opposite direction. Besides, the evolution of the solutions with time are also carried out here. In most of the cases, the solutions achieve the steady state after a certain point of time, except for a particular type of domain, where the fluid flow solutions show time dependent behavior after reaching a critical value of Re. This paper has reported the ranges, where the critical Reynolds number (R e c) lies, indicating the appearance of the first Hopf bifurcation for the fluid flow problem. [ABSTRACT FROM AUTHOR]

In this paper, a fully discrete finite element scheme is established with L 1 approximation for solving the nonlinear time-fractional wave equation with a variable coefficient. The stability of the scheme is analyzed and the optimal error estimates are derived. To improve the computational efficiency, we propose a two-grid algorithm based on the fully discrete finite element scheme. With the proposed technique, a small-scale nonlinear problem is calculated by iteration in a coarse-grid space with mesh size H , and then a linearized problem is solved in a fine-grid space with mesh size h , H ≫ h. This technique not only maintains the numerical precision, but also saves a lot of computing time. The stability and optimal convergence order are considered. Using the Taylor expansion, the second two-grid algorithm is constructed to improve the optimal convergence order. The similar properties are discussed in detail. Numerical experiments are provided to illustrate the theoretical analysis and test the performance of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper considers the error estimates of a family of Lagrange cubic finite volume methods for solving two dimensional elliptic boundary value problems over triangular meshes. By introducing a novel map of finite volume method, we propose a family of cubic finite volume method schemes with four independent map coefficients. It is detailedly proved that the proposed finite volume method schemes is exactly equivalent to the existing traditional finite volume method schemes of the same boundary value problems. A geometric constraint requirement on the triangular meshes of primary partitions is imposed to guarantee the uniform ellipticity of discrete elliptic bilinear forms resulted from the proposed schemes. The geometric requirements of the primary partitions could be greatly attenuated when the map coefficients are appropriately designed. Once the uniform ellipticity of the discrete elliptic bilinear forms is established, the proposed schemes have the unique solvability, and the optimal H 1 -norm error estimates without any restriction on dual elements of dual partitions of the finite volume methods, which is consistent with the results of literatures. With the help of the introduced map, the proposed schemes have the optimal L 2 -norm error estimates with a mesh restriction for the dual elements of their dual partitions significantly weaker than the existing theoretical results, in which the Aubin-Nitsche duality argument and the orthogonal conditions based on the dual partitions are employed. Finally, we validate our theoretical results by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we study a kinetic model describing the evolution of planetary dust under the action of a planet and its satellites. In particular, we focus our attention on the formation of planetary rings and the role of shepherd moons. The equation describing the considered physical phenomenon is of Vlasov type, posed in an evolutionary domain in time. We first study some theoretical properties of the model. Then, we describe a numerical method, suitable for the study of kinetic equations in evolutionary domains with possibly complicated geometries. Finally, we show and comment some simulations. In particular, our numerical simulations show that shepherd moons play a key role in the formation and maintenance of divisions between rings. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study two spectral Galerkin approximation schemes of the FitzHugh-Nagumo (FHN) neuron model for the nerve impulse propagation, which contains a small parameter perturbation and strong nonlinearity. To this end, we consider two kinds of temporal discretization by a semi-implicit Euler method and a Crank-Nicolson method, and the spatial discretization by the spectral Galerkin method for the perturbed FHN neuron model. In particular, in order to obtain the second-order approximation in time for the numerical scheme, we modify the nonlinear term f (u n) in the Crank-Nicolson spectral Galerkin (CNSG) scheme. We derive the error estimations with optimal convergence rates of the numerical solutions by the first-order semi-implicit spectral Galerkin (SISG) approximation scheme and the second-order modified CNSG (MCNSG) approximation scheme for the perturbed FHN neuron model. Finally, some numerical examples of one-dimensional and two-dimensional nonlinear FHN models with periodic boundary conditions are carried out to verify that the approximate solutions of the proposed SISG and MCNSG schemes satisfy the proved theoretical results. [ABSTRACT FROM AUTHOR]

This paper considers the numerical approximation for the time-fractional diffusion-wave equation with the order α ∈ (1 , 2) , whose solution behaves a weak singularity at t = 0. To construct the high-order scheme, the intermediate variable w = D t α / 2 (u − t ϕ ˜) is introduced, then we can rewrite the original problem as a coupled system, where u is the solution of the time-fractional diffusion-wave equation and ϕ ˜ = u t (x , 0). By using the L1 scheme on graded meshes in temporal direction and the compact difference scheme in spatial direction, a fully discrete scheme is constructed for the coupled system. Furthermore, the stability and the local error estimate of the proposed scheme is given. Finally, numerical examples are presented to verify the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new variational form to simulate the propagation of surface plasmon polaritons on graphene sheets. Here the graphene is treated as a thin sheet of current with an effective conductivity, and modeled as a lower-dimensional interface. A novel time-domain finite element method is proposed for solving this graphene model, which coupled an ordinary differential equation on the interface with Maxwell's equations in the physical domain. Discrete stability and error estimate are proved for our proposed method. Numerical results are presented to demonstrate the effectiveness of this graphene model for simulating the surface plasmon polaritons propagating on graphene sheets. [ABSTRACT FROM AUTHOR]

The main result of the work is the generalization of the concept of the variational splitting from [3] to the higher-order generalized- α schemes proposed in [5] in the context of hyperbolic and elastic wave propagation problems. It is possible to factorize the matrix of the linear system of equations (M + η K) in the linear computational cost. Here M is the mass matrix, and K is the stiffness matrix. We discretize using the isogeometric finite element method. Following the idea proposed in [3] , factorizing the matrix (M + η K) , we obtained an additional error of the order of η 2. This paper introduces a technique to couple the possibility of fast factorization of (M + η K) with higher-order generalized- α schemes proposed in [5] for hyperbolic and elastic wave propagation problems. Thus, we introduce a variationally separable splitting technique for the high-order accuracy generalized- α method. We use tensor-product meshes to develop the splitting method, which results in the linear cost for multi-dimensional problems. We consider standard C 0 finite elements as well as smoother isogeometric analysis for spatial discretization. The direction splitting method requires the mass and the stiffness matrices to have the Kronecker product structure. For other problems, the corresponding Kronecker product solver can be employed as a preconditioner as long as a Kronecker product can sufficiently well approximate the problem matrix. We also study the spectrum of the amplification matrix to establish the unconditional stability of the method. We use various examples to demonstrate the performance of the method and the optimal approximation accuracy. In numerical tests, we compute the L 2 and H 1 norms to show the optimal convergence of the discrete method in space and high-order accuracy in time. [ABSTRACT FROM AUTHOR]

The novel coupling technique applied to standard Finite Element and meshless Finite Difference Methods is presented in this paper. The problem domain is a priori partitioned into a set of disjoint subdomains, which are subsequently discretized using frameworks typical for the finite element method or the meshless finite difference method. The constraints problem, based on the least square approach, is established whose solution yields combinations of degrees of freedom in subdomains that determine the global approximation field, continuous in the entire domain. The constraints problem has the matrix-type form and requires solving the strongly singular system of equations. The appropriate technique is presented to find the complete solution in the form of a hyperplane in the space of degrees of freedom. The same method is applied to enforce the boundary conditions in the meshless subdomains. The proposed approach is illustrated with several 2D examples, including the standard Poisson's problem and selected elasticity and thermo-elasticity problems. In each case, it is shown that the novel method of coupling FEM and MFDM is effective since it provides an accurate and continuous solution in the entire domain. • Domain partitioning does not significantly decrease accuracy of the final solution. • Finite element and meshless finite difference methods are coupled in one domain. • Various discretization and approximation parameters lead to the continuous final solution. • Convergence rates on multi-region domains correspond to those obtained on homogeneous ones. [ABSTRACT FROM AUTHOR]

In this paper, we develop the Virtual Element Method for a system of time-dependent nonlinear convection-diffusion-reaction equations on polygonal meshes. We present a fully nonlinear discrete scheme with a proof of existence and uniqueness of the numerical solution. In order to overcome the complexity of solving nonlinear discrete system, we propose an equivalent iterative linearized system of equations. We perform the stability analysis and then derive the optimal convergence estimates in the H 1 semi-norm. Finally, we investigate the performance of the fully discrete scheme through a set of numerical tests. To improve computational efficiency, we have performed the simulations using a two-grid method. [ABSTRACT FROM AUTHOR]

In this paper, we first design a residual-type error estimator for a modified weak Galerkin (MWG) method of 2D H (curl) elliptic problems. Then, we show that the indicator is reliable with respect to the approximation error measured in terms of a natural energy norm, the main ingredient of our approach is to translate the error between weak solution and MWG solution into the conforming part and nonconforming part. We also prove the efficiency of the error estimator by using standard bubble functions. Finally, we provide several experiments to verify the performance of the error estimator within both uniform meshes and adaptive meshes. [ABSTRACT FROM AUTHOR]

In this paper, two efficient high-order physical property-preserving linearly implicit finite difference schemes are firstly developed and analyzed for nonlinear fourth-order wave equation with damping based on the scalar auxiliary variable (SAV) approach and newly developed Lagrange multiplier (LM) approach, respectively. Then the modified energy preservation property, the priori bounds of the numerical solution and convergence analysis of the former scheme are presented in detail. It is pointed out that the former scheme only conserves/dissipates the modified discrete energy, not the original energy while the latter scheme conserves/dissipates the original discrete energy exactly. Also, the latter scheme only demands to solve a decoupled and linear system with constant coefficients at each time step plus a nonlinear algebraic system. Finally, some numerical simulations are presented to demonstrate the accuracy, efficiency and preservation property of the obtained schemes. • Two efficient physical property-preserving difference methods for the nonlinear fourth-order wave equation with damping are proposed. • The modified preservation property and error estimation of the former scheme are presented in detail. • Both schemes demand to solve a sequence of linear decoupled systems with constant coefficients at each time step. • Some numerical results are performed to demonstrate the accuracy, efficiency and preservation property of both schemes. [ABSTRACT FROM AUTHOR]

In this paper, an efficient p-adaptive numerical algorithm is proposed for solving two- and three-dimensional (2D and 3D) binary fluid surfactant phase field models. The equation is a coupled system of a fourth-order nonlinear Cahn-Hilliard equation and a second-order nonlinear diffusion equation. There are three new ideas about algorithm design. Firstly, the high-dimensional equation is split into an array of one-dimensional sub-problems that could be solved via parallel processes. Secondly, global stability is constructed by guaranteeing the local stability of the sub-problems, and the one-dimensional problems are solved by a stabilized semi-implicit scheme with the fourth-order compact difference discretization. Finally, with the extrapolation method for convergence order improvement, a temporal p-adaptive strategy, which dynamically selects the order of extrapolation according to the speed of energy change, is developed to ensure efficiency and high precision. Besides that, the proposed numerical algorithm gratifies the discrete mass conservation properties and can be easily programmed. Via numerical testing, it has good convergence performance and computational stability. Numerical simulations including the surfactant absorption and coarsening dynamics verify the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, two novel conservative relaxation methods are developed for the space-fractional nonlinear Schrödinger equation. The first type of relaxation scheme adopts the exponential time difference method in time, the scheme can preserve the energy conservation of the space-fractional nonlinear Schrödinger. Furthermore, although the proposed scheme does not preserve the mass conservation, we can also prove that it can conserve the mass boundedness of the space-fractional nonlinear Schrödinger for the case β > 0 , which may be helpful for the analysis of unconditional convergence of the energy conservative scheme. The second type of relaxation scheme adopts the integral factor method in time, this scheme can be proved to inherit the mass conservation of space-fractional nonlinear Schrödinger. All these two schemes are linearly implicit, and thus can avoid costly nonlinear computations. Numerical experiments show that both of the proposed schemes are remarkable efficiency and have good stability comparing with the original relaxation scheme. [ABSTRACT FROM AUTHOR]

Variable-order fractional advection-diffusion equations have always been employed as a powerful tool in complex anomalous diffusion modeling. The proposed paper is devoted to using the meshless method to solve a general variable-order time fractional advection-diffusion-reaction equation (VO-TF-ADRE) with complex geometries. The proposed method is based on the improved backward substitute method (IBSM) in conjunction with the finite difference technique. For temporal derivative, the finite difference technique and for spatial derivatives, the IBSM are utilized to discretize the equation. The newly developed method is an RBF-based meshless approach, whose solution is constructed by the primary approximation and a series of basis functions. The primary approximation is given to satisfy boundary conditions. Each basis function is the sum of radial basis functions and a specific correcting function. Seven different types of numerical experiments are analyzed to validate the efficiency and wide applicability for multidimensional VO-TF-ADREs. [ABSTRACT FROM AUTHOR]

In the present paper, a decoupled second-order time-stepping scheme is proposed and analyzed for solving the evolutionary Peterlin viscoelastic model with finite element spatial discretization. To avoid solving a coupled fully discrete nonlinear system, a second-order extrapolation in time is employed to the nonlinear terms. It only requires sequentially solving the solution of the Navier-Stokes problem and one constitutive equation per time step, which needs smaller memory. Furthermore, optimal error estimates for the velocity, the pressure, and the conformation tensor are proved in suitable norms with using of the Stokes and Ritz projections. Finally, numerical experiments are conducted that confirm our theoretical results. • A second-order decoupled time-stepping scheme is proposed for the time-dependent Peterlin viscoelastic model. • It only requires sequentially solving the solution of the Navier-Stokes problem and one constitutive equation per time step. • The optimal error estimates for the velocity, pressure and conformation tensor are derived. [ABSTRACT FROM AUTHOR]

In this paper, a combination of a time discrete technique on graded meshes with barycentric rational interpolation is employed to solve a class of one- and two-dimensional time-dependent partial integro-differential equations with weakly singular kernels. Equations of this type are also commonly called fractional evolution equations, whose solutions have some singular behaviours near the initial time. This makes that it's difficult to reach an expected convergence result for the numerical treatment concerning time variable on uniform meshes. In the temporal direction, a generalized Crank-Nicolson difference scheme and the product integration formula on graded meshes are used to discretize the differential term and the integral term, respectively. Moreover, it's proved that the time discrete scheme is unconditionally stable and quadratically convergent. Further, a fully discrete numerical approach is structured with spacial directional discretization in terms of barycentric rational interpolation. The results of numerical experiments display the efficiency of the method and confirm the theoretical analyses. In addition, the present scheme can reach second-order convergence in time both for smooth initial value and non-smooth initial value. [ABSTRACT FROM AUTHOR]

In this paper, we develop a Direct Discontinuous Galerkin (DDG) method for solving a time dependent partial differential equation with convection-diffusion terms and a nonlocal term which is a pseudo-differential operator of order α ∈ (1 , 2). This kind of equation was first introduced to describe morphodynamics of dunes and was then used for signal processing methods. We consider the DDG method which is based on the direct weak formulation of the PDE into the DG function space for both numerical solution and test functions. Suitable numerical fluxes for all operators are then introduced. We prove nonlinear stability estimates along with convergence results. Finally numerical experiments are given to illustrate qualitative behaviors of solutions and to confirm convergence results. [ABSTRACT FROM AUTHOR]

This paper proposes an Allen–Cahn Chan–Vese model to settle the multi-phase image segmentation. We first integrate the Allen–Cahn term and the Chan–Vese fitting energy term to establish an energy functional, whose minimum locates the segmentation contour. The subsequent minimization process can be attributed to variational calculation on fitting intensities and the solution approximation of several Allen–Cahn equations, wherein n Allen–Cahn equations are enough to partition m = 2 n segments. The derived Allen–Cahn equations are solved by efficient numerical solvers with exponential time integrations and finite difference space discretization. The discrete maximum bound principle and energy stability of the proposed numerical schemes are proved. Finally, the capability of our segmentation method is verified in various experiments for different types of images. [ABSTRACT FROM AUTHOR]

In this paper, we consider fast numerical methods for linear systems arising from implicit Runge-Kutta temporal discretization methods (based on the fourth-order, 2-stage Gauss method) for one- and two-dimensional Riesz space fractional advection-diffusion equations (RSFADEs). An implicit Runge-Kutta-standard/shifted Grünwald difference scheme for RSFADEs is introduced, and its stability and convergence are also studied. In the one-dimensional case, the coefficient matrix of the discretized linear system is the sum of an identity matrix, a Toeplitz matrix and a square of Toeplitz matrix. We construct a class of Toeplitz times Toeplitz splitting (TTTS) iteration methods to solve the corresponding linear systems. We prove that it converges uniformly to the exact solution without imposing any additional condition, and the optimal parameters for the TTTS iteration method are given. Meanwhile, we design an induced sine transform based preconditioner for two-dimensional problems to accelerate the convergence rate of the conjugate gradient method. Theoretically, we prove that the spectra of the preconditioned matrices of the proposed methods are clustering around 1. Numerical results are presented to illustrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we develop a high-order adaptive virtual element method (VEM) to simulate the polymeric self-consistent field model in arbitrary domains. The VEM is very flexible in handling general polygon elements and can treat hanging nodes as polygon vertices without additional processing. Besides, to effectively simulate the phase separation behavior of strong segregation systems, we present an adaptive method on polygonal mesh equipped with a new Log marking strategy. Compared with the traditional strategies, the Log marking strategy not only marks the elements that need to be refined or coarsened, but also indicates the times of adaptation, which makes full use of the information contained in the numerical results. Using the halfedge data structure, we can apply the adaptive method to the arbitrary polygonal mesh. Numerical results demonstrate that the developed method is efficient in simulating the phase behavior of block polymers on complex geometric domains. The adaptive method can greatly reduce the computational cost to obtain prescribed numerical accuracy for strong segregation systems. [ABSTRACT FROM AUTHOR]

In this paper, we propose a pressure robust staggered discontinuous Galerkin method for the Stokes equations on general polygonal meshes by using piecewise constant approximations. We modify the right-hand side of the body force in the discrete formulation by exploiting a divergence preserving velocity reconstruction operator, which is the crux for pressure-independent velocity error estimates. The optimal convergence for the velocity gradient, velocity, and pressure is proved. In addition, we can establish the superconvergence of the velocity approximation by incorporating a divergence preserving velocity reconstruction operator in the dual problem, which is also an essential contribution of this paper. Finally, several numerical experiments are carried out to confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to extend the versatile mixed methods originally developed by Chen and Williams for isothermal flows in "Versatile Mixed Methods for the Incompressible Navier-Stokes Equations," Computers & Mathematics with Applications, Volume 80, Number 6, 2020, and use them to simulate non-isothermal incompressible flows. These new mixed methods are particularly interesting, as with only minor modifications they can be applied to a much broader range of flows, including non-isothermal weakly-compressible flows, and fully-compressible flows. In the main body of this paper, we carefully develop these mixed methods for solving the Boussinesq model equations. Thereafter, we establish the L2-stability of the discrete temperature and velocity fields. In addition, we obtain rigorous error estimates for the temperature and velocity fields. Finally, we assess the practical behavior of the methods by applying them to a set of well-known convection problems. [ABSTRACT FROM AUTHOR]

In this paper, we want to clarify the Gibbs phenomenon when continuous and discontinuous finite elements are used to approximate discontinuous or nearly discontinuous PDE solutions from the approximation point of view. For a simple step function, we explicitly compute its continuous and discontinuous piecewise constant or linear projections on discontinuity matched or non-matched meshes. For the simple discontinuity-aligned mesh case, piecewise discontinuous approximations are always good. For the general non-matched case, we explain that the piecewise discontinuous constant approximation combined with adaptive mesh refinements is a good choice to achieve accuracy without overshoots. For discontinuous piecewise linear approximations, non-trivial overshoots will be observed unless the mesh is matched with discontinuity. For continuous piecewise linear approximations, the computation is based on a "far-away assumption", and non-trivial overshoots will always be observed under regular meshes. We calculate the explicit overshoot values for several typical cases. Numerical tests are conducted for a singularly-perturbed reaction-diffusion equation and linear hyperbolic equations to verify our findings in the paper. Also, we discuss the L 1 -minimization-based methods and do not recommend such methods due to their similar behavior to L 2 -based methods and more complicated implementations. [ABSTRACT FROM AUTHOR]

This paper presents a fourth-order compact immersed interface method to solve two-dimensional Poisson equations with discontinuous solutions on arbitrary domains divided by an interface. The compact scheme only employs a nine-point stencil for each grid point on the computational domain. The new approach is based on an implicit formulation obtained from generalized Taylor series expansions, and it is constructed from a few modifications to the central finite difference near the interface. The discretization results in a linear system in which the matrix coefficients are the same as the ones for smooth solutions, and the right-hand side system is modified by adding terms known as jump contributions. These contributions are only calculated at those points where the nine-point stencil cuts the interface. However, the contribution formulas require the knowledge of Cartesian jumps up to fourth-order. In this paper, we derived them using only the principal jump conditions and the jumps coming from the known right-hand function of the Poisson equation. We present numerical experiments in two dimensions to verify the feasibility and accuracy of the proposed method. Thus, the implicit immersed interface method results in an attractive fourth-order compact scheme that is easy to be implemented and applied to arbitrary interface shapes. [ABSTRACT FROM AUTHOR]

In this paper, we solve a second-order nonlinear elliptic equation by using the finite volume element method, and give the rigorous error estimates. Firstly, the computational domain is divided into general convex quadrilateral meshes. We choose the isoparametric bilinear element space as the trial function space and the piecewise constant function space as the test function space, and construct the corresponding finite volume element scheme. Secondly, on the h 2 -parallelogram mesh, the boundedness and coercivity of bilinear form are proved. Using the Brouwer fixed point theorem, we give the existence and uniqueness of numerical solution. Thirdly, we derive the estimates of ‖ ∇ (u − u h) ‖ L t with t ≥ 2 and ‖ u − u h ‖ 0 under certain regularity assumptions. At last, we carry out numerical experiments on quadrilateral meshes and calculate the convergence orders in H 1 and L 2 norms, which are consistent with our theoretical results. [ABSTRACT FROM AUTHOR]

Torsional nano-scanners have a variety of applications in nano/micro-electro-mechanical systems. In this paper, two- and three-dimensional numerical solutions based on the lattice Boltzmann are proposed for simulating the dynamic behavior of an electromechanical torsional nano-scanner. The complete governing equations are modified based on the couple stress theory and developed by considering the effect of the Casimir force. Furthermore, the physics of the system is extended by considering the effects of dissolved electric agents on the dynamic performance of the nano-scanner. The passive/active electrolyte flow behavior with the moving solid boundary is described and simulated using Nernst-Plank-Poisson compensation of the electrolyte flow. The proposed numerical results are validated by comparing them with those available in the literature. Finally, the influence of scale dependency and Casimir force on the nano-scanner's dynamic performance and pull-in instability are investigated. Also, as torsional nano-scanners might be immersed in ionic media, the impact of ionic concentration on the pull-in voltage of a nano-scanner is investigated. The obtained results demonstrate that increasing the ionic concentration reduces the instability voltage of the nano-scanner considerably. [ABSTRACT FROM AUTHOR]

This paper proposes a general framework of the transformed differential quadrature method (TDQM) for solving partial differential equations (PDEs). First, the basic mathematical concepts and key steps of the TDQM are presented. Then, several examples including homogeneous/inhomogeneous heat conduction equations and wave propagation equations are solved by the proposed method to show its detailed scheme. The influences of sampling point types, discrete point numbers and numerical Laplace transform inversion on the convergence and error of computational results are also discussed. Besides, we show the accuracy and efficiency of the proposed method by comparing the TDQM with the alternating direction implicit (ADI) scheme of the finite difference method. Compared with the traditional differential quadrature method (DQM), the TDQM introduces the integral transform theorem to decompose the solution process into a small-scale matrix inversion and a transform inversion. The number of algebraic equations is greatly reduced so that the difficulty of large-scale matrix inversion is avoided, and the computational efficiency is improved. Besides, the TDQM is no longer limited to finite entities, and reduces the limitation and influence of boundary conditions on computational results. This strategy is especially suitable for multi-dimensional problems with complex variables such as non-axisymmetric or three-dimensional problems. [ABSTRACT FROM AUTHOR]

Future advanced aero-engine features highly complex internal flow mechanisms, such as small gap blade row interactions and coupled fluid-structure interactions. Simulating these is challenging for convectional body-fitted computation fluid dynamics methods. As a first step towards this ultimate goal, this paper presents applications of an immersed boundary (IB) method for internal-flow problems at high Reynolds numbers. A hybrid mesh strategy, with a single-block and structured mesh to fit the physical domain and an IB method to model the internal walls, is adopted to the internal-flow problems. To achieve this, an adaptive forcing distance (AFD) technique is proposed for determining the forcing point locations when large-aspect-ratio cells are involved near the walls. This extends the IB method to curvilinear grids so that the domain boundaries can be resolved by a body-fitted mesh, which significantly reduces the total cell number needed to resolve the boundaries compared to the usage of a Cartesian grid, and thus improve the computational efficiency. The present methodologies are firstly validated on a Cartesian grid through a 2D NACA0012 test case. Subsequently, the methods are tested for two internal flow problems with increasing complexity: a 3D subsonic compressor cascade and a well-known transonic rotor, NASA Rotor 37. These case studies prove that the present method can achieve the same overall accuracy for the pressure distribution with the traditional body-fitted simulations at the attached-flow regions, and further studies are still necessary to improve the accuracy of predicting the friction coefficient and separated flows. [ABSTRACT FROM AUTHOR]

In this paper, the variable-order (VO) time fractional version of the 2D Wu-Zhang system is introduced. A numerical method based on the spline approximation of the VO fractional terms in this system and the Jacobi polynomials (JPs) approximation of the solution is proposed to solve this system. In the presented method, the VO fractional derivatives are approximated using the spline expansions at first. Then, by employing the expressed spline expansions as well as the θ -weighted method a recurrence relation is obtained. Next, the unknown functions are expanded in terms of the 2D JPs (with unknown coefficients) and substituted into the recurrence relation. Finally, by solving a system of linear algebraic equations, the unknown coefficients and subsequently an approximate solution of the problem is obtained. The accuracy of the established method is investigated by solving some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we focus on using the physics-informed neural networks (PINNs) to design an automatic numerical solver for the Cahn-Hilliard equation, which is a fourth-order phase field model. We directly construct the loss function of the neural network based on the strong form of the equation, boundary conditions, initial conditions and the given solution data. The trained neural network contains the underlying physical information of the equation and it can predict the solution of the next time step. Additionally, a key feature of this approach is that, unlike other commonly used numerical approaches, it is mesh free. As such, it does not suffer from the curse of dimensionality associated with high dimensional partial differential equations (PDEs). Besides, the approximate ability of the neural network is proved by the convergence of the loss function and the convergence of the neural network to the exact solution in L 2 norm. Finally, the effectiveness and accuracy of the proposed method are demonstrated by some numerical examples, especially in the ten dimensional case. Moreover, the predictions remain robust even when the training data are corrupted with 1% uncorrelated Gaussian noise. [ABSTRACT FROM AUTHOR]

This paper presents an original algorithm to simulate quasi-static loading of fluid-saturated porous material for numerical upscaling of the complex fluid-saturated fractured-porous media. The resulting model is anisotropic viscoelastic which means that it is defined by a complex-valued frequency-dependent stiffness tensor that can be recomputed into the frequency-dependent phase velocities and attenuation of seismic waves. Numerical upscaling requires a solution of parabolic approximation of the Biot poroelastic equation for anisotropic media in frequency space for a series of frequencies and multiple right-hand sides. The approach is based on the finite-difference approximation of the quasi-static formulation of the Biot equation with the use of a direct solver to resolve the obtained system of linear algebraic equations. The direct solver allows for efficiently treating multiple right-hand sides which is an essential statement of the upscaling problem. The presented realization of the algorithm allows solving the problems of the size of up to 2000 points in each spatial direction using a single machine, which allows dealing with representative fractured-porous models. To illustrate the applicability of the algorithm we provide several series of experiments illustrating the effects of fracture connectivity and anisotropy of fracture-filling material on seismic waves dispersion and attenuation if propagating in complex fluid-saturated fractured-porous media. • We present a new finite-difference algorithm to upscale fluid-saturated poroelastic media. • The solver allows solving quasi-static Biot equation for hundreds of time frequencies on a single-node machine. • Use of finite differences allows dealing with fractured porous media of almost arbitrary complexity. [ABSTRACT FROM AUTHOR]

The incompressible Navier-Stokes equation can be used for modeling and simulation of fluid flow. One of the important mathematical models for simulating the pollution in the urban street canyons is the pollutant transition equation (PTE) which it is obtained from the incompressible Navier-Stokes equation. The current paper is devoted to propose a new meshless numerical procedure e.g. gradient smoothing method (GSM). First, the time derivative is approximated by the finite difference scheme. Then, the space derivative is discretized by the gradient smoothing method. Also, the proper orthogonal decomposition (POD) approach is utilized to reduce the used CPU time. Several examples with real world application are solved to verify the efficiency of the developed numerical procedure. [ABSTRACT FROM AUTHOR]

This paper first develops a high-efficient accurate mesh-free scheme (CSPH-SFCD) for multi-dimensional time-dependent spatial fractional convection-diffusion (SFCD) equations based on the Riemann-Liouville (RL) derivative in convex domain by coupling a corrected smoothed particle hydrodynamics (CSPH) method with a numerical integral of RL. Subsequently, for the first application, the proposed CSPH-SFCD coupled with up-wind (UW) scheme (CSPH-SFCD-UW) is tentatively extended to investigate the nonlinear shock-wave phenomena in 2D space-fractional Burgers' problem (SFBP) with fractional Neumann boundary at high Reynolds number. The proposed mesh-free scheme for time-dependent SFCD/SFBP with fractional boundary is mainly motived by: (a) the SFCD model with RL fractional calculus is discretized by employing a numerical integral formula and a CSPH scheme, and then a conditionally stable CSPH-SFCD method is first derived; (b) the UW scheme is introduced to stabilize the numerical investigation of convection-dominated SFBP; (c) the fractional Neumann boundary (FNB) condition can be easily enforced in the proposed discretized scheme, and the integer-order NB condition is treated by employing virtual particles interpolation technique; (d) the MPI parallel computing technique is adopted to enhance the computational efficiency. Firstly, the numerical convergence and stability of the proposed scheme are discussed and illustrated by solving several multi-dimensional benchmarks. Secondly, the cases of two-sided and left-sided RL fractional derivatives are also discussed and demonstrated in the numerical tests. The advantages of the proposed method are demonstrated by simulating the 2D/3D SFCD equations in convex domains or under non-uniform point distribution. Finally, the 2D two-component SFBP with FNBC are numerically investigated by using the proposed method, and compared with the FDM results. The complex nonlinear non-local properties governed by SFBP are numerically predicted by the proposed scheme. All the numerical results show that the proposed mesh-free method for time-dependent spatial SFCD problems is high-efficient, flexible and accurate. • A high-efficient accurate CSPH-SFCD scheme is first developed to 2D/3D SFCD equations. • The CSPH-SFCD-UW scheme is first proposed to investigate shock-wave behavior in 2D SFBP. • The FNB condition can be easily enforced in the proposed approximated scheme. • The advantages of the present scheme are shown by numerical tests on convex domain. [ABSTRACT FROM AUTHOR]

In this paper, we present a class of discontinuous Galerkin finite element methods for advection-diffusion-reaction problems and establish a priori error estimates when the solution is only in H 1 + s (Ω) with s ∈ (0 , 1 / 2 ]. [ABSTRACT FROM AUTHOR]

The tumor-growth model, proposed by Oden et al., is a coupling system with high nonlinearity for the surface effects of the diffusion interface. In this paper, scalar auxiliary variable (SAV) method is introduced to handle nonlinear term in gradient flow. At the same time, semi-implicit difference scheme is adopted to discretize the time variable. To approximate the spatial variable, Fourier-spectral method for the first time is employed for the tumor-growth model whose merit is that high precision solution can be obtained without handling too many terms. Generally speaking, an efficient and robust energy stabilization scheme is constructed. It inherits the properties of energy dissipation and mass conservation related to continuous problems. The validity of our proposed numerical scheme is demonstrated by conducting some numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear beam equation with the p (x) -biharmonic operator, where the exponent p (x) is a given function. We transform the problem into a system of two differential equations and prove the existence, uniqueness and regularity of the weak solution. Then we formulate the discrete problem associated with that system using the finite element method and prove the existence, uniqueness and stability of the discrete solution. We also investigate the order of convergence and prove some error estimates. Next, we apply the Lagrange basis to obtain an algebraic system of equations. Finally, we implement the computational codes in Matlab software considering the one and two dimensional cases and we present examples to illustrate the theory. • Study of a nonlinear beam equation with the p (x) -biharmonic operator. • Proof of the existence, uniqueness and regularity of the weak solution. • Discrete formulation with the mixed finite element method. • Demonstration of the existence, uniqueness and regularity of the discrete solution. • Numerical simulations with Matlab software for the one and two dimensional cases. [ABSTRACT FROM AUTHOR]

In this paper, we prove the uniqueness and H 2 -regularity of solution u for a semi-linear elliptical equation with a 2 m − 1 polynomial nonlinearity under the stability and uniqueness conditions. Moreover, we design the finite element method based on P 1 -element for the semi-linear elliptical equation and prove the existence and uniqueness of the finite element solution u h by the finite element Oseen iterative method under the stability and uniqueness condition. Next, we prove the existence and uniqueness of the solution u for the semi-linear elliptical equation by the finite element solution u h under the stability and uniqueness conditions. Furthermore, we provide the H 1 − L 2 optimal error estimates of the finite element solution u h with respect to the exact solution u for the semi-linear elliptical equation under the stability and uniqueness conditions. Finally, some numerical tests are presented to show the efficiency of the proposed method for the semi-linear elliptical equation with a 2 m − 1 polynomial nonlinearity. [ABSTRACT FROM AUTHOR]

In this paper, the splitting-up scheme for rough partial differential equations (RPDE) is introduced. The convergence rate of the splitting-up scheme for weak solutions to semilinear partial differential equations with linear rough noises is given. An application to stochastic partial differential equations with fractional Browian motions is considered. [ABSTRACT FROM AUTHOR]

In this paper, a semi-decoupled marker and cell (MAC) scheme is proposed for the Stokes-Biot system, in which the displacement of structure is split from the whole system using the time-lagging scheme, then the rest system composed by the velocity and pressure is treated in a similar way as the Stokes-Darcy coupling problem. It keeps the desirable qualities such as the mass conservation. The backward Euler scheme is used for the time discretization. We derive the stability and the error estimates for the fully discrete scheme, obtain the second order superconvergence for the discrete L 2 norm for velocity, pressure and displacement and some terms of the H 1 norm for displacement and velocity. Numerical experiments confirm well the theoretical analysis results and the robustness of parameters. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical method to solve the space fractional Allen-Cahn equation. First, to remove the constraint of time step, an efficient scheme is constructed by the stabilization method, where a first order method is used in time direction and a second-order weighted and shifted Grünwald difference formula is applied in spatial direction. Then, the unique solvability of the scheme is given for any time step. Meanwhile, the unconditional discrete maximum principle preserving and energy stability are rigorously proved. And we also give a detailed error analysis. Besides, to reduce computational time and improve computational efficiency, adaptive time step strategy is used when simulating two important properties and coarsening dynamics. Finally, some numerical experiments show that adaptive time step strategy is effective in long time simulations. [ABSTRACT FROM AUTHOR]

In this paper, we design and analyze a modified weak Galerkin (MWG) finite element method for H (curl) -elliptic problem. We first introduce a new discrete weak curl operator and the MWG finite element space. The modified weak Galerkin method does not require the penalty parameter by comparing with traditional DG methods. We prove an optimal error estimate in energy norm. At last, we provide the numerical results to confirm these theoretical results. [ABSTRACT FROM AUTHOR]

Space-time domain decomposition methods have been successfully applied to solve linear systems that arise from the discretization of time-dependent problems. In this paper, we present a space-time pure multiplicative Schwarz method for solving linear parabolic equations. With this method, the systems of equations are first coupled together and then solved by preconditioned GMRES, so the solutions at arbitrary time steps can be obtained simultaneously at a time. Under some mild assumptions, we develop an optimal convergence theory and show that the convergence rate is bounded independently of the mesh sizes, the subdomain partition and the window size. Some numerical results are reported to illustrate the optimality and scalability of the proposed method. Moreover, the numerical comparison also shows that our method has a faster convergence rate. [ABSTRACT FROM AUTHOR]

This paper presents a high order time discretization method by combining the time splitting scheme with semi-implicit spectral deferred correction method for the space-fractional Swift-Hohenberg equation (SFSH). Based on the operator splitting method, the original problem is split into linear and nonlinear subproblems, respectively. The Fourier spectral method is adopted for the linear part, and a first-order accurate finite difference scheme in time together with the Fourier spectral method in space is used for the nonlinear part. The stability and convergence of the obtained numerical scheme are analyzed theoretically. Moreover, the spectral deferred correction (SDC) method is then employed to improve the temporal accuracy. Various two and three dimensional numerical experiments are performed to validate the theoretical results and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, in order to penalize for lack of divergence-free solution, we propose a sparse grad-div stabilized algorithm for the incompressible magnetohydrodynamics equations, which just adds a minimally intrusive module that implements grad-div stabilization with a sparse block structure matrix. Unconditional stability and error estimates of the proposed algorithm are provided and numerical tests are carried out. Compared to other grad-div stabilizations, the sparse grad-div stabilized algorithm is more efficient with some large values of grad-div parameters. [ABSTRACT FROM AUTHOR]