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

Mesh-based numerical methods such as the Finite Element Method (FEM) and Finite Difference Method (FDM) are widely used in various fields of science and engineering. However, the shortcomings of mesh-based methods are becoming increasingly obvious mainly due to the distorted or broken elements when analyzing the problems of large deformation or crack propagation. To overcome the above problems, meshfree methods such as the Meshless Local Petrov–Galerkin (MLPG) and Radial Point Interpolation Method (RPIM) were proposed, which can eliminate or partially eliminate the difficulties triggered by mesh-dependence. In geotechnics, the numerical modeling of deformations and failures of soil and rock masses is one of the most important approaches to understand and prevent geohazards, such as landslides and ground subsidence. In this paper, a local RPIM-based meshfree software package GeoMFree 3D is specifically developed for analyzing the deformations and failures of soil and rock masses. To enhance the computing capability of GeoMFree 3D , (1) effective memory storage strategies are adopted to solve the memory requirement bottleneck; and (2) parallel computing strategies are utilized and heterogeneously implemented on multi-core CPU and many-core GPU to improve the computational efficiency. To demonstrate the performance of GeoMFree 3D , three simple verification examples and one application case are presented in this paper. The verification examples indicate that the computational accuracy and efficiency of the package GeoMFree 3D is acceptable. Furthermore, the application case indicates that GeoMFree 3D is capable of analyzing the displacements of complex rock slopes. GeoMFree 3D is currently under intensive development, plans for future work on the development of GeoMFree 3D are also introduced.

We recently developed the open-source library tIGAr, which extends the FEniCS finite element automation framework to isogeometric analysis. The present contribution demonstrates the utility of tIGAr in complex problems by applying it to immersogeometric fluid–structure interaction (FSI) analysis. This application is implemented as the new open-source library CouDALFISh (Coupling, via Dynamic Augmented Lagrangian, of Fluids with Immersed Shells, pronounced “cuttlefish”), which uses the dynamic augmented Lagrangian (DAL) method to couple fluid and shell structure subproblems. The DAL method was introduced previously, over a series of papers largely focused on heart valve FSI, but an open-source implementation making extensive use of automation to compile numerical routines from high-level mathematical descriptions brings newfound transparency and reproducibility to these earlier developments on immersogeometric FSI analysis. The portions of CouDALFISh that do not use code generation also illustrate how a framework like FEniCS remains useful even when some functionality is outside the scope of its standard workflow. This paper summarizes the workings of CouDALFISh and documents a variety of benchmarks demonstrating its accuracy. Although the implementation emphasizes transparency and extensibility over performance, it is nonetheless demonstrated to be sufficient to simulate 3D FSI of an idealized aortic heart valve. Source code will be maintained at https://github.com/david-kamensky/CouDALFISh .

Solidification heat transfer process of billet is described by nonlinear partial differential equation (PDE). Due to the poor productive environment, the boundary condition of this nonlinear PDE is difficult to be fixed. Therefore, the identification of boundary condition of two-dimensional nonlinear PDE is considered. This paper transforms the identification of boundary condition into a PDE optimization problem. The Lipchitz continuous of the gradient of cost function is proved based on the dual equation. In order to solve this optimization problem, this paper presents a modified conjugate gradient algorithm, and the global convergence of which is analyzed. The results of the simulation experiment show that the modified conjugate gradient algorithm obviously reduces the iterative number and running time. Due to the ill-posedness of the identification of boundary condition, this paper combines regularization method with the modified conjugate gradient algorithm. The simulation experiment illustrates that regularization method can eliminate the ill-posedness of this problem. Finally, the experimental data of a steel plant illustrate the validity of this paper’s method.

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

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

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

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

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

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

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

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

The current paper considers the boundedness of solutions to the following quasilinear Keller–Segel model (with logistic source) (KS) u t = ∇ ⋅ ( D ( u ) ∇ u ) − χ ∇ ⋅ ( u ∇ v ) + μ ( u − u 2 ) , x ∈ Ω , t > 0 , v t − Δ v = u − v , x ∈ Ω , t > 0 , ( D ( u ) ∇ u − χ u ⋅ ∇ v ) ⋅ ν = ∂ v ∂ ν = 0 , x ∈ ∂ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x ∈ Ω , where Ω ⊂ R N ( N ≥ 1 ) is a bounded domain with smooth boundary ∂ Ω , χ > 0 and μ ≥ 0 . One novelty of this paper is that we find a new a-priori estimate ∫ Ω u χ max { 1 , λ 0 } ( χ max { 1 , λ 0 } − μ ) + − e ( x , t ) d x , so that, we develop new L p -estimate techniques and thereby obtain the boundedness results, where C G N and λ 0 ≔ λ 0 ( γ ) are the constants which are corresponding to the Gagliardo–Nirenberg inequality (see Lemma 2.2 ) and the maximal Sobolev regularity (see Lemma 2.3). To our best knowledge, this seems to be the first rigorous mathematical result which indicates the relationship between m and μ χ that yields the boundedness of the solutions, where m is the exponent of diffusion term D ( u ) . The above-mentioned results have significantly improved and extended previous results of several authors.

In this paper, flow of multi-component two-phase fluids in highly heterogeneous anisotropic two-dimensional porous media is studied using computational methods suitable for unstructured triangular and/or quadrilateral grids. The physical model accounts for miscibility and compressibility of fluids while gravity and capillary effects are neglected. The governing equations consist of a pressure equation together with a system of mass conservation equations. For solving pressure equation, a new method called Control Volume Distributed Finite Element Method (CVDFEM) is introduced which uses Control Volume Distributed (CVD) vertex-centered grids. It is shown that the proposed method is able to approximate the pressure field in highly anisotropic and heterogeneous porous media fairly accurately. Moreover the system of mass conservation equations is solved using various upwind and central schemes. These schemes are extended from one-dimensional to two-dimensional unstructured grids. Using a series of numerical test cases, comparison is made between different approaches for approximation of the hyperbolic flux function. Semi one-dimensional high-order data reconstruction procedures are employed to decrease stream-wise numerical diffusion. The results suggest that the Modified Dominant Wave (MDW) scheme outperforms other hyperbolic schemes studied in this paper from both accuracy and computational cost points of view.

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.

In this paper truss layout optimization is used in conjunction with geometry optimization to provide the basis for a powerful conceptual design tool for additively manufactured (AM) components, particularly useful when the degree of design freedom is high. With layout optimization the design domain is discretized using a grid of nodes which are interconnected with discrete line elements, forming a ‘ground structure’. Linear optimization can then be used to identify the subset of elements forming the minimum volume structure required to carry the applied loading. A nonlinear geometry optimization step, which involves adjusting the positions of the nodes, can subsequently be undertaken to simplify and improve the solution. Simple geometrical rules can then be used to automatically transform a line element layout into a 3D continuum, ready for validation and/or manufacture. Various extensions to the basic method are described in the paper, including AM build direction constraints and techniques to permit user-interaction with candidate designs, which has been found to be invaluable at the conceptual design stage. Finally the approach described is applied to a range of design problems, including the redesign of an airbrake hinge for the Bloodhound Supersonic Car.

This paper is devoted to model adaptation and h p q -adaptive finite element methods for modeling and analysis of the problems for which the strong formulation corresponds to Laplace equation. The chosen example of this equation concerns dielectric structures (or media) of electrostatics. The paper addresses hierarchical theories (also called hierarchical models) and hierarchical approximations. In the assessment of the models and approximations, our own and existing a priori error estimation results are applied. The used assessment procedure can be employed to any other applications of Laplace equation in applied sciences. The proposed theories (understood as mathematical formulations) and their numerical approximations are applied to the physical model of linear dielectricity in structures with complex electric description and complex geometry. We take advantage of the 3D and 3D-based theories, hierarchical modeling, and hierarchical approximations within h p q finite element formulation. In our research, the applied theory and discretization parameters, i.e. the element size h , the longitudinal approximation order p , and the transverse order q , differ in each finite element.

In this paper, we consider a final value problem for time-fractional diffusion equation with inhomogeneous source. The main goal of our paper is to determine an approximated initial data from the observation data at final time by constructing a regularized solution using a mollification method. Under appropriate regularity assumptions of the exact solution, we give convergence rate between the reconstructed solution and the exact one. We also provide a numerical example to illustrate the main results.

In this paper, we consider the Marcinkiewicz summability of solutions to the following fractional elliptic problem ( − Δ ) s u = f ( x ) , ∈ Ω , u > 0 , ∈ Ω , u = 0 , x ∈ R N ∖ Ω , where ( − Δ ) s denotes the fractional Laplacian operator, s ∈ ( 0 , 1 ) , Ω ⊂ R N is a bounded domain with Lipschitz boundary, f belongs to some Marcinkiewicz space M m ( Ω ) with m > 1 . The main novelty of this paper is actually the fact that the solutions to the above equation are bounded if m > 2 N N + 2 s 2 , instead of m > N 2 s . The results of this paper are new even for s = 1 .

The aim of this paper is to explore the asymptotic properties of the solutions to the Black–Scholes equation. This paper focuses on the basic properties of options when the volatility σ is sufficiently close to zero. We got an approximate formula for option pricing. This approximate formula is simple and can be applied to financial markets with small volatility.

The paper considers flow modeling of two-phase heterogeneous medium. Each phase of the medium is considered as continuum, which is described by the compressible Euler equations. The phases are separated by the contact surface (interface) and are not mixed on the molecular level. Examples of such medium are: mixtures of solid particles and gas (in dense or dilute concentration of particles), liquids with small bubbles, solid porous materials filled with gas or liquid. The phase can be of connected structure (dense particles, porous solid, gas between dilute particles, etc.) or of non-connected structure (separated inclusions as gas bubbles, dilute particles, closed pores in solid, etc.). The connectivity of the phase is closely related to the propagation of acoustic perturbations. An attempt was made to consider all the cases of the phase connectivity in the framework of a unique approach. The paper presents an approach that couples the models designed for different cases of phase connectivity to the generalized hyperbolic and thermodynamically consistent form. The proposed model is applicable for simulating flows with change of the phase connectivity, e.g. dense-to-dilute two-phase flows. The model is verified on several problems of gas–solid granular medium flows. In numerical simulations the Godunov method with the HLLEM flux approximation on arbitrary moving Euler grids is used.

This paper deals with the blow-up phenomena for the following porous medium equation systems with nonlinear boundary conditions u t = Δ u m + k 1 ( t ) f 1 ( v ) , v t = Δ v n + k 2 ( t ) f 2 ( u ) i n Ω × ( 0 , t ∗ ) , ∂ u ∂ ν = g 1 ( u ) , ∂ v ∂ ν = g 2 ( v ) o n ∂ Ω × ( 0 , t ∗ ) , u ( x , 0 ) = u 0 ( x ) ≥ 0 , v ( x , 0 ) = v 0 ( x ) ≥ 0 i n Ω ¯ , where m , n > 1 , Ω ⊂ R N ( N ≥ 2 ) is bounded convex domain with smooth boundary. Using a differential inequality technique and a Sobolev inequality, we prove that under certain conditions on data, the solution blows up in finite time. We also derive an upper and a lower bound for blow-up time. In addition, as applications of the abstract results obtained in this paper, an example is given.

This paper presents three-dimensional simulations of interactive particles in a fluid. The originality of this paper is the integrated Eulerian–Lagrangian algorithms implemented with high-order methods in both space and time for numerical solutions to Navier–Stokes equations with additional source terms. Implementation of fourth order Stiffly Stable Schemes was tested for time integration before they were used in computations. In order to provide the h p -type mesh refinement, for extra flexibility to achieve higher spatial resolution, a modal spectral element method was used to solve governing equations in three dimensions. Another originality is the efficiency in handling moving particles without adaptive or moving mesh. Simulation results were validated with experimental data and verified with exact solutions. In addition, numerical results were also compared with solutions at higher resolutions and good agreement was accomplished. Results indicate that algorithms and implementations are accurate and appropriate for investigating three dimensional interactive particles in a fluid involving many moving objects.

A NURBS-enhanced boundary element method for 2D elasticity problems with body forces is proposed in this paper. The non-uniform rational B-spline (NURBS) basis functions are applied to construct the geometry and the model can be reproduced exactly at all stages since the refinement will not change the shape of the boundary. Both open curves and closed curves are considered. The fields are approximated by the traditional Lagrangian basis functions in parameter space, rather than by the same NURBS basis functions for geometry approximation. The parametric boundary elements and collocation nodes are defined from the knot vector of the curve and the refinement of the NURBS curve is easy. Boundary conditions can be imposed directly since the Lagrangian basis functions have the property of delta function. In addition, most methods for the treatment of singular integrals in traditional boundary element method can be applied in the proposed method. To overcome the difficulty for evaluation of the domain integrals in problems with body forces, a line integration method is further applied in this paper to compute the domain integrals without additional volume discretizations. Numerical examples have shown the accuracy of the proposed method.

The theoretical part of the paper analyzes discretized Stokes systems with local Coulomb’s slip boundary conditions. Solutions to discrete models are defined by means of fixed-points of an appropriate mapping. We prove the existence of a fixed-point, establish conditions guaranteeing its uniqueness and examine how they depend on the discretization parameter h and the slip coefficient κ . The second part of the paper is devoted to computational aspects. Numerical experiments are presented.

In this paper, we present a phase-field method applied to the fluid-based shape optimization. The fluid flow is governed by the incompressible Navier–Stokes equations. A phase field variable is used to represent material distributions and the optimized shape of the fluid is obtained by minimizing the certain objective functional regularized. The shape sensitivity analysis is presented in terms of phase field variable, which is the main contribution of this paper. It saves considerable amount of computational expense when the meshes are locally refined near the interfaces compared to the case of fixed meshes. Numerical results on some benchmark problems are reported, and it is shown that the phase-field approach for fluid shape optimization is efficient and robust.

In this paper, the N -soliton solution is constructed for the ( 2 + 1 )-dimensional generalized Hirota–Satsuma–Ito equation, from which some localized waves such as line solitons, lumps, periodic solitons and their interactions are obtained by choosing special parameters. Especially, by selecting appropriate parameters on the multi-soliton solutions, the two soliton can reduce to a periodic soliton or a lump soliton, the three soliton can reduce to the elastic interaction solution between a line soliton and a periodic soliton or the elastic interaction between a line soliton and a lump soliton, while the four soliton can reduce to elastic interaction solutions among two line solitons and a periodic soliton or the elastic interaction ones between two periodic solitons. Detailed behaviours of such solutions are illustrated analytically and graphically by analysing the influence of parameters. Finally, an inelastic interaction solution between a lump soliton and a line soliton is constructed via the ansatz method, and the relevant interaction and propagation characteristics are discussed graphically. The results obtained in this paper may be helpful for understanding the interaction phenomena of localized nonlinear waves in two-dimensional nonlinear wave equations.

In this paper, a new exact solution based on the classical shell theory (CST) for free vibration of cross-ply laminated composite cylindrical shells with elastic restraint ends is proposed. The present exact solution can be summed up in the following steps: Firstly, the displacement functions are constructed by the governing differential equations with the exact closed form solutions; Then, the artificial spring technology is introduced to simulate the general boundary conditions of the two end edges of shell; Thirdly, the equation for natural frequencies is obtained by means of the method of reverberation-ray matrix (MRRM); Lastly, the vibration results are presented by the modified golden section search (MGSS) algorithm. By comparing the present method with published papers, the accuracy of present method is verified. On the basis of that, some new exact nature frequencies and mode shapes of the cross-ply laminated composite cylindrical shells with various classical boundary conditions and elastic restraints are performed and they can be served as the benchmark data for the future.

In this paper, a mathematical proof of the existence of a global minimum of Self-Optim (Self-Optimizing Inverse Analysis Method) cost functional is presented based upon weak-solution theory of partial differential equations. The Self-Optim provides single global minimum rather than having multiple global minima corresponding to unrealistic solutions of the inverse problem. Furthermore, discrete approximation of the inverse problem and computational methods for the cost functional are proposed and the proof is numerically verified. This paper provides a rigorous mathematical foundation for applications of the Self-Optim method to various inverse problems in mechanics.

Since being proposed, the HLLEM-type schemes have been widely used because they are with high discontinuity resolutions and can be easily applied to the other system of hyperbolic conservation law. In this paper, we conduct theoretical analyses on the HLLE-type schemes’ performances at low speeds. By realizing that the excessive numerical dissipations corresponding to the velocity-difference terms of the momentum equations make these schemes incapable of obtaining physical solutions at low speeds, we adopt the function g to control such dissipation. Also, we borrow the HLLEMS scheme’s construction and damp the shear waves in the vicinity of the shock to avoid the shock anomaly’s appearance. The moving contact discontinuity case and the Sod shock tube case show that the HLLEMS-AS scheme we propose in this paper can capture contact discontinuities and shocks as sharply as HLLEMS scheme. The Quirk’s odd–even test case and the hypersonic inviscid flow over a cylinder case demonstrate that HLLEMS-AS is robust against the shock anomaly. The inviscid low-speed flow around the NACA0012 airfoil case indicates that HLLEMS-AS is with a high resolution at low speeds. The turbulent flow past a backward facing step case demonstrates the shear wave capturing ability of the HLLEMS-AS scheme. These properties suggest that HLLEMS-AS is promising to be widely used in both cases of low speed and high speed.

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

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

This paper studies two cases of global regularity problems on the 2D magneto-micropolar equations with partial magnetic diffusion and fractional dissipation. For the first case the velocity field is ideal, the micro-rotational velocity is with Laplacian dissipation and the magnetic field has fractional partial diffusion ( − ∂ 22 β b 1 , − ∂ 11 β b 2 ) with β > 1 . In the second case, the velocity has a fractional Laplacian dissipation ( − Δ ) α u with any α > 0 , the micro-rotational velocity is with Laplacian dissipation and the magnetic field has partial diffusion ( − ∂ 22 b 1 , − ∂ 11 b 2 ) . In two cases the global well-posedness of classical solutions is proved in this paper.

In this paper, we prove the existence and multiplicity results of solutions with prescribed L 2 -norm for a class of Kirchhoff type problems − a + b ∫ R 3 | ∇ u | 2 d x Δ u − λ u = f ( u ) in R 3 , where a , b > 0 are constants, λ ∈ R and f ∈ C ( R , R ) . To obtain such solutions, we look into critical points of the energy functional E b ( u ) = a 2 ∫ R 3 | ∇ u | 2 + b 4 ∫ R 3 | ∇ u | 2 2 − ∫ R 3 F ( u ) constrained on the L 2 -spheres S ( c ) = u ∈ H 1 ( R 3 ) : | | u | | 2 2 = c . Here, c > 0 and F ( s ) ≔ ∫ 0 s f ( t ) d t . Under some mild assumptions on f , we show that critical points of E b unbounded from below on S ( c ) exist for c > 0 . In addition, we establish the existence of infinitely many radial critical points { u n b } of E b on S ( c ) provided that f is odd. Finally, the asymptotic behavior of u n b as b ↘ 0 is analyzed. These conclusions extend some known ones in previous papers.

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.

In this paper, we investigate the existence of ground state sign-changing solutions for a class of Choquard equations − △ u + ( 1 + λ f ( x ) ) u = ( I α ∗ k | u | p ) k ( x ) | u | p − 2 u + | u | 2 ∗ − 2 u , x ∈ R N , where k and f are nonnegative functions, N ≥ 3 , 2 ∗ = 2 N N − 2 , p ∈ 2 , N + α N − 2 , − λ 1 λ 0 and λ 1 is the first eigenvalue of the equation − △ u + u = λ f ( x ) u in H 1 ( R N ) . Using the sign-changing Nehari manifold, we prove that the Choquard equation has at least one ground state sign-changing solution. This paper can be regarded as the complementary work of Ghimenti and Van Schaftingen (2016), Van Schaftingen and Xia (2017).

In this paper, the time domain Maxwell’s equations are solved using the discrete exterior calculus (DEC) formalism in the two-dimensional space. To truncate the computational domain, the complex frequency-shifted perfectly matched layer (CFS-PML) concept is applied to create a reflectionless artificial layer. The paper presents a new numerical procedure to easily implement the CFS-PML with curved inner boundary. In order to numerically realize the PML, in a simplicial mesh, this paper proposes to utilize the nearest neighbor algorithm to associate point sets to boundary points. The distance from points to the boundary curve defines the attenuation function inside the PML. The performance of the approach is assessed by measuring the reflection error for three numerical experiments.

In this paper, we consider local multiscale model reduction for problems with multiple scales in space and time. We developed our approaches within the framework of the Generalized Multiscale Finite Element Method (GMsFEM) using space–time coarse cells. The main idea of GMsFEM is to construct a local snapshot space and a local spectral decomposition in the snapshot space. Previous research in developing multiscale spaces within GMsFEM focused on constructing multiscale spaces and relevant ingredients in space only. In this paper, our main objective is to develop a multiscale model reduction framework within GMsFEM that uses space–time coarse cells. We construct space–time snapshot and offline spaces. We compute these snapshot solutions by solving local problems. A complete snapshot space will use all possible boundary conditions; however, this can be very expensive. We propose using randomized boundary conditions and oversampling (cf. Calo et al., 2016). We construct the local spectral decomposition based on our analysis, as presented in the paper. We present numerical results to confirm our theoretical findings and to show that using our proposed approaches, we can obtain an accurate solution with low dimensional coarse spaces. We discuss using online basis functions constructed in the online stage and using the residual information. Online basis functions use global information via the residual and provide fast convergence to the exact solution provided a sufficient number of offline basis functions. We present numerical studies for our proposed online procedures. We remark that the proposed method is a significant extension compared to existing methods, which use coarse cells in space only because of (1) the parabolic nature of cell solutions, (2) extra degrees of freedom associated with space–time cells, and (3) local boundary conditions in space–time cells.

In this paper we consider a family of high-order iterative methods which is more efficient than the Newton method to approximate a solution of symmetric algebraic Riccati equations. In fact, this paper is devoted to the convergence study of a k -steps iterative scheme with low operational cost and high order of convergence. We analyze their accessibility and computational efficiency. We also obtain results about the existence and localization of solution. Numerical experiments confirm the advantageous performance of the iterative scheme analyzed.

In this paper, a semi analytical method is used to investigate the free vibration of doubly-curved shells of revolution with arbitrary boundary conditions. The doubly-curved shells of revolution are divided into their segments in the meridional direction, and the theoretical model for vibration analysis is formulated by applying Flugge’s thin shell theory. Regardless of the boundary conditions, the displacement functions of shell segments are composed by the Jacobi polynomials along the revolution axis direction and the standard Fourier series along the circumferential direction. The boundary conditions at the ends of the doubly-curved shells of revolution and the continuous conditions at two adjacent segments were enforced by the penalty method. Then, the natural frequencies of the doubly-curved shells are obtained by using the Rayleigh–Ritz method. For arbitrary boundary conditions, this method does not require any changes to the mathematical model or the displacement functions, and it is very effective in the analysis of free vibration for doubly-curved shells of revolution. The credibility and exactness of proposed method are compared with the results of finite element method (FEM), and some numerical results are reported for free vibration of the doubly-curved shells of revolution under classical and elastic boundary conditions. Results of this paper can provide reference data for future studies in related field.

This paper deals with the following quasilinear chemotaxis-growth system u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( u ∇ v ) + μ u ( 1 − u ) , x ∈ Ω , t > 0 , v t = Δ v − v + w , x ∈ Ω , t > 0 , τ w t + δ w = u , x ∈ Ω , t > 0 , in a smoothly bounded domain Ω ⊂ R n ( n ≥ 3 ) under zero-flux boundary conditions. The parameters μ , δ and τ are positive and the diffusion function D ( u ) is supposed to generalize the prototype D ( u ) ≥ D 0 u θ with D 0 > 0 and θ ∈ R . Under the assumption θ > 1 − 4 n , it is proved that whenever μ > 0 , τ > 0 and δ > 0 , for any given nonnegative and suitably smooth initial data ( u 0 , v 0 , w 0 ) satisfying u 0 ≢ 0 , the corresponding initial–boundary problem possesses a unique global solution which is uniformly-in-time bounded. The novelty of the paper is that we use the boundedness of the | | v ( ⋅ , t ) | | W 1 , s ( Ω ) with s ∈ [ 1 , 2 n n − 2 ) to estimate the boundedness of | | ∇ v ( ⋅ , t ) | | L 2 q ( Ω ) ( q > 1 ) . Moreover, the result in this paper can be regarded as an extension of a previous consequence on global existence of solutions by Hu et al. (2016) under the condition that D ( u ) ≡ 1 and n = 3 .

This paper deals with the spatial spreading speed and traveling wave solutions of a general epidemic model with nonlocal dispersal in time and space periodic habitats. It should be mentioned that the existence of spreading speed and traveling wave solutions of nonlocal dispersal cooperative system in space–time periodic habitats have been established previously. In this paper, we further show that the epidemic system has a spreading speed c ∗ ( ξ ) and for any c > c ∗ ( ξ ) , there exist a unique, continuous space–time periodic traveling wave solution ( Φ 1 ( x − c t ξ , t , c t ξ ) , Φ 2 ( x − c t ξ , t , c t ξ ) ) of epidemic model in the direction of ξ with speed c , and there is no such solution for c c ∗ ( ξ ) .

The split quaternionic Schrodinger equation ∂ ∂ t | f 〉 = − A | f 〉 plays an important role in split quaternionic mechanics, in which A a split quaternion matrix. This paper, by means of a real representation of split quaternion matrices, studies problems of split quaternionic Schrodinger equation, and gives an algebraic technique for the split quaternionic Schrodinger equation. This paper also derives an algebraic technique for finding eigenvalues and eigenvectors of a split quaternion matrix in split quaternionic mechanics.

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

This paper deals with a new fifth-order weighted essentially non-oscillatory (WENO) scheme improving the WENO-NS and WENO-P methods which are introduced in Ha et al. J. Comput. Phys. (2013) and Kim et al., J. Sci. Comput. (2016) respectively. These two schemes provide the fifth-order accuracy at the critical points where the first derivatives vanish but the second derivatives are non-zero. In this paper, we have presented a scheme by defining a new global-smoothness indicator which shows an improved behavior over the solution to the WENO-NS and WENO-P schemes and the proposed scheme attains optimal approximation order, even at the critical points where the first and second derivatives vanish but the third derivatives are non-zero., 23 pages, 14 figures

This paper deals with a fully parabolic chemotaxis-growth system with singular sensitivity u t = Δ u − χ ∇ ⋅ u ∇ ln v + r u − μ u 2 , ( x , t ) ∈ Ω × ( 0 , ∞ ) , v t = Δ v − v + u , ( x , t ) ∈ Ω × ( 0 , ∞ ) , under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R 2 , where the parameters χ , μ > 0 and r ∈ R . Global existence and boundedness of solutions to the above system were established under some suitable conditions by Zhao and Zheng (2017). The main aim of this paper is further to show the large time behavior of global solutions which cannot be derived in the previous work.

We investigate a novel method for the numerical solution of two-dimensional time-dependent convection–diffusion–reaction equations with nonhomogeneous boundary conditions. We first approximate the equation in space by a stable Gaussian radial basis function (RBF) method and obtain a matrix system of ODEs. The advantage of our method is that, by avoiding Kronecker products, this system can be solved using one of the standard methods for ODEs. For the linear case, we show that the matrix system of ODEs becomes a Sylvester-type equation, and for the nonlinear case we solve it using predictor–corrector schemes such as Adams–Bashforth and implicit–explicit (IMEX) methods. This work is based on the idea proposed in our previous paper (2016), in which we enhanced the expansion approach based on Hermite polynomials for evaluating Gaussian radial basis function interpolants. In the present paper the eigenfunction expansions are rebuilt based on Chebyshev polynomials which are more suitable in numerical computations. The accuracy, robustness and computational efficiency of the method are presented by numerically solving several problems.

In this paper, we present a modification of the Somigliana identity for the 3D Navier–Lame equation in order to analytically include in its mathematical formalism the boundary represented by Coons and Bezier parametric surface patches. As a result, the equations called the parametric integral equation system (PIES) with integrated boundary shape are obtained. The PIES formulation is independent from the boundary shape representation and it is always, for any shape, defined in the parametric domain and not on the physical boundary as in the traditional boundary integral equations (BIE). This feature is also helpful during numerical solving of PIES, as from a formal point of view, a separation between the approximation of the boundary and the boundary functions is obtained. In this paper, the generalized Chebyshev series are used to approximate the boundary functions. Numerical examples demonstrate the effectiveness of the presented strategy for boundary representation and indicate the high accuracy of the obtained results.

The paper shows application of the method of fundamental solutions in combination with the radial basis functions for analysis of fluid flow and heat transfer in an internally corrugated tube. Cross-section of such a tube is mathematically described by a cosine function and it can potentially represent a natural duct with internal corrugations, e.g. inside arteries. The boundary value problem is described by two partial differential equations (one for fluid flow problem and one for heat transfer problem) and appropriate boundary conditions. During solving this boundary value problem the average fluid velocity and average fluid temperature are calculated numerically. In the paper the Nusselt number and the product of friction factor and Reynolds number are presented for some selected geometrical parameters (the number and amplitude of corrugations). It is shown that for a given number of corrugations a minimal value of the product of friction factor and Reynolds number can be found. As it was expected the Nusselt number increases with increasing amplitude and number of corrugations.

This work is focused on the application of functional-type a posteriori error estimates and corresponding indicators to a class of time-dependent problems. We consider the algorithmic part of their derivation and implementation and also discuss the numerical properties of these bounds that comply with obtained numerical results. This paper examines two different methods of solution approximation for evolutionary models, i.e., a time-marching technique and a space–time approach. The first part of the study presents an algorithm for global minimisation of the majorant on each of discretisation time-cylinders (time-slabs), the effectiveness of this approach to error estimation is confirmed by extensive numerical tests. In the second part of the publication, the application of functional error estimates is discussed with respect to a space–time approach. It is followed by a set of extensive numerical tests that demonstrates the efficiency of proposed error control method. The numerical results obtained in this paper rely on the implementation carried out using open source software, which allows formulating the problem in a weak setting. To work in a variational framework appears to be very natural for functional error estimates due to their derivation method. The search for the optimal parameters for the majorant is done by a global functional minimisation, which to the authors’ knowledge is the first work using this technique in an evolutionary framework.