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In this paper, by using variational methods, we study the existence and concentration of ground state solutions for the following fractional Schrödinger equation ( − Δ ) α u + V ( x ) u = A ( ϵ x ) f ( u ) , x ∈ R N , where α ∈ ( 0 , 1 ), ϵ is a positive parameter, N > 2 α, ( − Δ ) α stands for the fractional Laplacian, f is a continuous function with subcritical growth, V ∈ C ( R N , R ) is a Z N -periodic function and A ∈ C ( R N , R ) satisfies some appropriate assumptions.

In the current work, we present a variational mechanics framework for the coupled numerical prediction of cavitating turbulent flow and structural motion via a stabilized finite element formulation. To model the finite mass transfer rate in cavitation phenomena, we employ the homogenous mixture-based approach via phenomenological scalar transport differential equations given by the linear and nonlinear mass transfer functions. Stable linearizations of the finite mass transfer terms for the mass continuity equation and the reaction term of the scalar transport equations are derived for the robust and accurate implementation. The linearized matrices for the cavitation equation are imparted a positivity-preserving property to address numerical oscillations arising from high-density gradients typical of two-phase cavitating flows. The proposed formulation is strongly coupled in a partitioned manner with an incompressible 3D Navier-Stokes finite element solver, and the unsteady problem is advanced in time using a fully-implicit generalized-α time integration scheme. We first verify the implementation on the benchmark case of the Rayleigh bubble collapse. We demonstrate the accuracy and convergence of the cavitation solver by comparing the numerical solutions with the analytical solutions of the Rayleigh-Plesset equation for bubble dynamics. We find our solver to be robust for large time steps and the absence of spurious oscillations/spikes in the pressure field. The cavitating flow solver is coupled with a hybrid URANS-LES turbulence model. We validate the coupled solver for a very high Reynolds number turbulent cavitating flow over a NACA0012 hydrofoil section. Finally, the proposed method is applied in an Arbitrary Lagrangian-Eulerian framework to study turbulent cavitating flow over a pitching hydrofoil section and the characteristic features of cavitating flows such as re-entrant jet and periodic cavity shedding are discussed.

In this paper, we study various aspects of the ODE's flow $X$ solution to the equation $\partial_t X(t,x)=b(X(t,x))$, $X(0,x)=x$ in the $d$-dimensional torus $Y_d$, where $b$ is a regular $Z^d$-periodic vector field from $R^d$ in $R^d$. We present an original and complete picture in any dimension of all logical connections between the following seven conditions involving the field $b$: - the everywhere asymptotics of the flow $X$, - the almost-everywhere asymptotics of the flow $X$, - the global rectification of the vector field $b$ in $Y_d$, - the ergodicity of the flow related to an invariant probability measure which is absolutely continuous with respect to Lebesgue's measure, - the unit set condition for Herman's rotation set $C_b$ composed of the means of $b$ related to the invariant probability measures, - the unit set condition for the subset $D_b$ of $C_b$ composed of the means of $b$ related to the invariant probability measures which are absolutely continuous with respect to Lebesgue's measure, - the homogenization of the linear transport equation with oscillating data and the oscillating velocity $b(x/\varepsilon)$ when $b$ is divergence free. The main and surprising result of the paper is that the almost-everywhere asymptotics of the flow $X$ and the unit set condition for $D_b$ are equivalent when $D_b$ is assumed to be non empty, and that the two conditions turn to be equivalent to the homogenization of the transport equation when $b$ is divergence free. In contrast, using an elementary approach based on classical tools of PDE's analysis, we extend the two-dimensional results of Oxtoby and Marchetto to any $d$-dimensional Stepanoff flow: this shows that the ergodicity of the flow may hold without satisfying the everywhere asymptotics of the flow.

In this paper, we study second-order and fourth-order elliptic problems which include not only a Poisson equation in the bulk but also an inhomogeneous Laplace--Beltrami equation on the boundary of the domain. The bulk and the surface PDE are coupled by a boundary condition that is either of Dirichlet or Robin type. We point out that both the Dirichlet and the Robin type boundary condition can be handled simultaneously through our formalism without having to change the framework. Moreover, we investigate the eigenvalue problems associated with these second-order and fourth-order elliptic systems. We further discuss the relation between these elliptic problems and certain parabolic problems, especially the Allen--Cahn equation and the Cahn--Hilliard equation with dynamic boundary conditions.

The initial value problems of bipolar isentropic/non-isentropic compressible Navier-Stokes-Maxwell (CNS-M) systems arising from plasmas in R 3 are studied. The main difficulty of studying the bipolar isentropic/non-isentropic CNS-M systems lies in the appearance of the electromagnetic fields satisfying the hyperbolic Maxwell equations. The large time-decay rates of global smooth solutions with small amplitude in L q ( R 3 ) for 2 ≤ q ≤ ∞ are established. For the bipolar non-isentropic CNS-M system, the difference of velocities of two charged carriers decay at the rate ( 1 + t ) − 3 4 + 1 4 q which is faster than the rate ( 1 + t ) − 3 4 + 1 4 q ( ln ⁡ ( 3 + t ) ) 1 − 2 q of the bipolar isentropic CNS-M system, meanwhile, the magnetic field decay at the rate ( 1 + t ) − 3 4 + 3 4 q ( ln ⁡ ( 3 + t ) ) 1 − 2 q which is slower than the rate ( 1 + t ) − 3 4 + 3 4 q for the bipolar isentropic CNS-M system. The approach adopted is the classical energy method but with some new developments, where the techniques of choosing symmetrizers and the spectrum analysis on the linearized homogeneous system play the crucial roles.

The notion of best approximation mapping (BAM) with respect to a closed affine subspace in finite-dimensional space was introduced by Behling, Bello Cruz and Santos to show the linear convergence of the block-wise circumcentered-reflection method. The best approximation mapping possesses two critical properties of the circumcenter mapping for linear convergence. Because the iteration sequence of BAM linearly converges, the BAM is interesting in its own right. In this paper, we naturally extend the definition of BAM from closed affine subspace to nonempty closed convex set and from $\mathbb{R}^{n}$ to general Hilbert space. We discover that the convex set associated with the BAM must be the fixed point set of the BAM. Hence, the iteration sequence generated by a BAM linearly converges to the nearest fixed point of the BAM. Connections between BAMs and other mappings generating convergent iteration sequences are considered. Behling et al.\ proved that the finite composition of BAMs associated with closed affine subspaces is still a BAM in $\mathbb{R}^{n}$. We generalize their result from $\mathbb{R}^{n}$ to general Hilbert space and also construct a new constant associated with the composition of BAMs. This provides a new proof of the linear convergence of the method of alternating projections. Moreover, compositions of BAMs associated with general convex sets are investigated. In addition, we show that convex combinations of BAMs associated with affine subspaces are BAMs. Last but not least, we connect BAM with circumcenter mapping in Hilbert spaces., 31 pages and 2 figures

This paper develops an efficient and robust solution technique for the steady Boussinesq model of non-isothermal flow using Anderson acceleration applied to a Picard iteration. After analyzing the fixed point operator associated with the nonlinear iteration to prove that certain stability and regularity properties hold, we apply the authors’ recently constructed theory for Anderson acceleration, which yields a convergence result for the Anderson accelerated Picard iteration for the Boussinesq system. The result shows that the leading term in the residual is improved by the gain in the optimization problem, but at the cost of additional higher order terms that can be significant when the residual is large. We perform numerical tests that illustrate the theory, and show that a 2-stage choice of Anderson depth can be advantageous. We also consider Anderson acceleration applied to the Newton iteration for the Boussinesq equations, and observe that the acceleration allows the Newton iteration to converge for significantly higher Rayleigh numbers that it could without acceleration, even with a standard line search.

In (Euzebio et al., 2016 [10] ; Chen and Tang, 2020 [8] ), the bifurcation diagram and all global phase portraits of a degenerate planar piecewise linear differential system x ˙ = F ( x ) − y , y ˙ = g ( x ) − α with three zones were given completely for the non-extreme case. In this paper we deal with the system for the extreme case and find new nonlinear phenomena of bifurcation for this planar piecewise linear system, i.e., a generalized degenerate Hopf bifurcation occurs for points at infinity. Moreover, the bifurcation diagram and all global phase portraits in the Poincare disc are obtained, presenting scabbard bifurcation curves, grazing bifurcation curves for limit cycles, generalized supercritical (or subcritical) Hopf bifurcation curve for points at infinity, generalized degenerate Hopf bifurcation value for points at infinity and double limit cycle bifurcation curve.

The moving particle semi-implicit method (MPS) is a well-known Lagrange method that offers advantageous in addressing complex fluid problems, but particle distribution is an area that requires refinement. For this study, a particle smoothing algorithm was developed and incorporated into the weakly compressible MPS (sWC-MPS). From the definition and derivation of basic MPS operators, uniform particle distribution is critical to numerical accuracy. Within the framework of sWC-MPS, numerical operators were modified by implementing coordinate transformation and smoothing algorithm. Modifying numerical operators significantly improved particle clustering, smoothed pressure distributions, and reduced pressure oscillations. To validate the numerical feasibility of the method, several cases were numerically simulated to compare sWC-MPS to the weakly compressible MPS (WC-MPS): a pre-defined two-dimensional (2-D) analytical function, Poiseuille's flow, Taylor Green vortex, and dam break. The results showed a reduction of errors caused by irregular particle distribution with lower particle clustering and smaller pressure oscillation. In addition, a larger Courant number, which represents a larger time step, was tested. The results showed that the new sWC-MPS algorithm achieves numerical accuracy even using a larger Courant number, indicating improved computational efficiency.

Let Γ be a smooth curve inside a two-dimensional rectangular region Ω. In this paper, we consider the Poisson interface problem − ∇ 2 u = f in Ω ∖ Γ with Dirichlet boundary condition such that f is smooth in Ω ∖ Γ and the jump functions [ u ] and [ ∇ u ⋅ n → ] across Γ are smooth along Γ. This Poisson interface problem includes the weak solution of − ∇ 2 u = f + g δ Γ in Ω as a special case. Because the source term f is possibly discontinuous across the interface curve Γ and contains a delta function singularity along the curve Γ, both the solution u of the Poisson interface problem and its flux ∇ u ⋅ n → are often discontinuous across the interface. To solve the Poisson interface problem with singular sources, in this paper we propose a sixth order compact finite difference scheme on uniform Cartesian grids. Our proposed compact finite difference scheme with explicitly given stencils extends the immersed interface method (IIM) to the highest possible accuracy order six for compact finite difference schemes on uniform Cartesian grids, but without the need to change coordinates into the local coordinates as in most papers on IIM in the literature. Also in contrast with most published papers on IIM, we explicitly provide the formulas for all involved stencils and therefore, our proposed scheme can be easily implemented and is of interest to practitioners dealing with Poisson interface problems. Note that the curve Γ splits Ω into two disjoint subregions Ω + and Ω − . The coefficient matrix A in the resulting linear system A x = b , following from the proposed scheme, is independent of any source term f, jump condition g δ Γ , interface curve Γ and Dirichlet boundary conditions, while only b depends on these factors and is explicitly given, according to the configuration of the nine stencil points in Ω + or Ω − . The constant coefficient matrix A facilitates the parallel implementation of the algorithm in case of a large size matrix and only requires the update of the right hand side vector b for different Poisson interface problems. Due to the flexibility and explicitness of the proposed scheme, it can be generalized to obtain the highest order compact finite difference scheme for non-uniform grids as well. We prove the order 6 convergence for the proposed scheme using the discrete maximum principle. Our numerical experiments confirm the sixth accuracy order of the proposed compact finite difference scheme on uniform meshes for the Poisson interface problems with various singular sources.

In this article, the mixed Fourier Legendre spectral Galerkin (MFLSG) methods are considered to solve the two-dimensional Fredholm integral equations ( fie s) on the Banach spaces with smooth kernel. The same methods are also considered to find the eigenvalues of the eigenvalue problems ( evp s) associated with the two-dimensional fie s. Making use of these methods, we establish the error between the approximated solution as well as iterated approximate solution versus exact solution for two-dimensional fie s in both L 2 and L ∞ norms. We also establish the error between approximated eigen-values, eigen-vectors and iterated eigen-vectors and exact eigen-elements by MFLSG methods in L 2 and L ∞ norms. The numerical illustrations are introduced for the error of these methods.

We prove unexpected decay of the L2-distance from the solution u(t) of a hyperbolic scalar conservation law, to some convex, flow-invariant target sets.

In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space $H^1$. For complete Nevanlinna-Pick spaces $\mathcal H$, we characterize all multipliers of the weak product space $\mathcal H \odot \mathcal H$. In particular, we show that if $\mathcal H$ has the so-called column-row property, then the multipliers of $\mathcal H$ and of $\mathcal H \odot \mathcal H$ coincide. This result applies in particular to the classical Dirichlet space and to the Drury-Arveson space on a finite dimensional ball. As a key device, we exhibit a natural operator space structure on $\mathcal H \odot \mathcal H$, which enables the use of dilations of completely bounded maps., Comment: 21 pages

In this paper we consider a chemotaxis system coupling with the incompressible Euler equations in spatial dimension two, which describing the dynamics of chemotaxis in the inviscid fluid. We establish the regular solutions globally in time under some assumptions on the chemotactic sensitivity.

This work proposes an approach to using the non-cooperative game theory for solving finite element problems. For this, the concept of generalized Nash equilibrium is applied to finite elements implying that each element is treated as a non-cooperative “player” in a larger finite element “game”. The aim of the approach is for all players to reach a Nash equilibrium ensuring that the entire discretization is at a minimum with respect to the decision variables considered. The approach is numerically demonstrated by investigating a nonlinear elasticity problem formulated as a finite element game. It is shown that the approach matches analytical solutions in linear elasticity and is convergent to a prescribed precision for two-player nonlinear problems.

A new fast multipole boundary element method (FM-BEM) is proposed to analyze 2-D elastostatic problems by using linear and three-node quadratic elements. The use of higher-order elements in BEM analysis results in more complex forms of the integrands, in which the direct Gaussian quadrature is difficult to calculate the singular and nearly singular integrals. Herein, the complex notation is first introduced to simplify all integral formulations (including the near-field integrals) in FM-BEM for 2-D elasticity. In direct evaluation of the near-field integrals, the nearly singular integrals on linear elements are calculated by the analytic scheme, and those on quadratic elements are evaluated by a robust semi-analytical algorithm. Numerical examples show that the present method possesses higher accuracy than the FM-BEM with constant elements. The computed efficiency of FM-BEM with higher order elements for analyzing large scale problems is still O(N), where N is the number of linear system of equations. In particular, the proposed FM-BEM is available for solving thin structures.

We present a variant of the immersed boundary method integrated with octree meshes for highly efficient and accurate Large-Eddy Simulations (LES) of flows around complex geometries. We demonstrate the scalability of the proposed method up to $\mathcal{O}(32K)$ processors. This is achieved by (a) rapid in-out tests; (b) adaptive quadrature for an accurate evaluation of forces; (c) tensorized evaluation during matrix assembly. We showcase this method on two non-trivial applications: accurately computing the drag coefficient of a sphere across Reynolds numbers $1-10^6$ encompassing the drag crisis regime; simulating flow features across a semi-truck for investigating the effect of platooning on efficiency., Comment: Accepted for publication at Computer and Mathematics with Applications

This manuscript brings the derivation of influence functions for a three-dimensional full-space under bilinearly-distributed time-harmonic loads. The differential equations describing the medium are decomposed in terms of uncoupled vector fields. A double Fourier transform allows the system of equations to be solved algebraically in the transformed space, where the bilinear-loading boundary conditions are imposed. The resulting displacement and stress solutions are presented in terms of double improper integrals to be evaluated numerically. The manuscript brings selected results from the evaluation of these solutions. These influence functions can be used in boundary element models of elastodynamic problems to yield computationally-efficient solutions and improved representation of sharply-varying contact traction fields.

In this paper, we consider the well-posedness of the inhomogeneous nonlinear biharmonic Schrodinger equation with spatial inhomogeneity coefficient K ( x ) behaves like | x | − b for 0 b min ⁡ { N 2 , 4 } . We show the local well-posedness in the whole H s -subcritical case, with 0 s ≤ 2 . The difficulties of this problem come from the singularity of K ( x ) and the lack of differentiability of the nonlinear term. To resolve this, we derive the bilinear Strichartz's type estimates for the nonlinear biharmonic Schrodinger equations in Besov spaces.

In the current paper we consider an inverse boundary value problem of electromagnetism with nonlinear Second Harmonic Generation (SHG) process. We show the unique determination of the electromagnetic material parameters and the SHG susceptibility parameter of the medium by making electromagnetic measurements on the boundary. We are interested in the case when a frequency is fixed.

In this paper, we introduce a multi-domain Muntz-polynomial spectral collocation method with graded meshes for solving second kind Volterra integral equations with a weakly singular kernel. This method is particularly suitable for problems whose solutions contain non-integer exponent factors. We carry out a rigorous error analysis of hp-version in the L ∞ - and weighted L 2 -norms. Several numerical examples are presented to demonstrate the efficiency and accuracy of the method.

The current article is to study the solvability of Nirenberg problem on S 2 through the so-called Gaussian curvature flow. We aim to propose a unified method to treat the problem for candidate functions without sign restriction and non-degenerate assumption. As a first step, we reproduce the following statement: suppose the critical points of a smooth function f with positive critical values are non-degenerate. Then the required solution exists, if the difference between the number of the local maximum points with positive values and the number of the saddle points with positive critical values as well as negative Laplace is not equal to 1. This statement has been proved for nearly thirty years through different methods.

Through the current article, a numerical technique to obtain an approximate solution of one-dimensional linear hyperbolic partial differential equations is implemented. A certain combination of the shifted Chebyshev polynomials of the fifth-kind is used as basis functions. The main idea behind the proposed technique is established on converting the governed boundary-value problem into a system of linear algebraic equations via the application of the spectral Galerkin method. The resulting linear system can be solved by expedients of the Gaussian elimination procedure. The convergence and error analysis of the shifted Chebyshev expansion are carefully investigated. Various numerical examples are given to demonstrate the power and accuracy of the given method.

In this paper, we propose approximate solvers for two-phase fluids with general equations of state (EOS) in high dimension. The standard finite volume scheme is used for each fluid away from material interface. The two-phase interfaces are captured by the level set method coupled with a multi-resolution algorithm. For the Riemann solver of two-phase fluids with general equations of state, we construct an iterative approximation method by the solver for binomial equations of state. The velocity of the interface and the interface exchange fluxes are obtained precisely. With the help of the adaptive multi-resolution algorithms, we extend the method to three space dimensions conveniently. Numerical examples are carried out to demonstrate the strength and robustness of this method.

In this paper, an adaptive element free Galerkin (EFG) method is presented to solve Poisson equation. In general, element free Galerkin method using moving least square (MLS) approximation needs a background mesh for integration. With the arbitrary polygonal influence domain technique, the shape function of MLS has almost interpolation property, and the Gaussian quadrature points in the background integration element only contribute to the vertices of that element, which enables us to compute the residual based on the background integration element just like the finite element method (FEM). The adaptive procedure based on triangular or tetrahedral background integration elements is then developed for EFG method, in which the residual-based a posteriori error estimation of FEM is used. Numerical examples are provided to illustrate the efficiency of the proposed adaptive EFG method.

By employing the kernel functions with the form of piecewise polynomials in the Sobolev reproducing kernel Hilbert spaces (RKHSs), a globally superconvergent numerical technique is proposed to solve the second kind linear integral equations of Fredholm type. This method has an order of global convergence O ( h 4 ) and O ( h 6 ) based on the kernel functions in the Sobolev RKHSs H 1 and H 2 , respectively. Three linear Fredholm integral equations, one Volterra-Fredholm integral equation and one nonlinear Fredholm integral equation are numerically solved by the present approach to verify the superconvergence and effectiveness.

As a mesoscopic approach, the lattice Boltzmann method has achieved considerable success in simulating fluid flows and associated transport phenomena. The calculation, however, suffers from a massive amount of computing resources. A predictive model, to reduce the computing cost and accelerate the calculations, is proposed in this work. By employing an artificial neural network, composed of convolution layers and convolution long short-term memory layers, the model is an equivalent substitution of multiple time steps. A physical informed training loss function is introduced to improve the model predictive accuracy; and for the two-dimensional driven cavity problem, the mean square error of the prediction is less than 1.5 × 10 − 6 . For non-stationary flow, a time-dependent computing structure based on the current model is established. Nine iterative model calculations are performed consecutively for a two-dimensional driven cavity model, and the results are validated by comparing with the original (serial) lattice Boltzmann algorithm. Generally, in the case of training Reynolds number, for velocity and speed, the mean and the maximum absolute errors are lower than 0.012 and 0.12. Similarly, in the generalizing case, the mean and the maximum absolute errors are lower than 0.017 and 0.012. Besides, the current model's efficiency is about 15 times higher than that of the original lattice Boltzmann method.

In this paper, we study a regularized Lie-Trotter splitting spectral method for a regularized space-fractional logarithmic Schrodinger equation by introducing a small regularized parameter 0 e ≪ 1 . The regularized method can be used to avoid numerical blow-up in the space-fractional logarithmic Schrodinger equation. The regularized space-fractional logarithmic Schrodinger equation is proved to approximate the space-fractional logarithmic Schrodinger equation with linear convergence rate O ( e ) . The proposed numerical scheme can preserve the discrete mass and step-energy for regularized space-fractional logarithmic Schrodinger equation. The first order convergence in time for the numerical method is rigorous proved for the regularized space-fractional logarithmic Schrodinger equation. Due to the appearance of fractional Laplace operator with order α, we can adjust the parameters of order to make the equation show much more dynamic characteristics than the classical logarithmic Schrodinger equation. Numerical simulations for 1D case based on the Fourier spectral approximation in space are presented to validate the theoretical analysis.

In this article, novel linearized implicit difference schemes with energy conservation property for fractional Klein-Gordon-Zakharov system are constructed and analyzed. The important feature of the article is that new auxiliary equations ∂ u ∂ t = − v and ∂ ϕ ∂ t = ∂ 2 ψ ∂ x 2 are introduced to transform the original fractional Klein-Gordon-Zakharov system into an equivalent system of equations exactly. Especially, two kinds of efficacious difference operators, the leap-frog and modified Crank-Nicolson methods are respectively utilized to establish the linearized implicit difference schemes with energy conservation property for simulating the propagation of transformed equations. And above all, by employing the discrete energy method, we have proven that the constructed difference algorithms enjoy the convergence order of O ( Δ t 2 + h 2 ) and O ( Δ t 2 + h 4 ) in L ∞ - and L 2 -norms, without imposing any restrictive conditions on the grid ratio compared with the existing literature. Two numerical examples are carried out to investigate the physical behaviors of the wave propagation and substantiate the effectiveness of the suggested schemes.

Making a rank-one modification on the classical BFGS (Broyden–Fletcher–Goldfarb–Shanno) updating formula, we develop a class of augmented BFGS methods. The suggested formula can be considered as a hybridization of the basic BFGS updating formula for Hessian with an additional rank-one term embedded to guarantee a general modified secant equation. By using the well-known Sherman–Morrison formula, the inverse of a memoryless version of the given updating formula is computed to be applied for solving large-scale problems. Convergence analysis is concisely carried out as well. At last, the practical merits of the method are investigated by numerical tests on a set of CUTEr problems as well as the well-known compressed sensing problem. Results show the computational efficiency of the given method.

The discrete unified gas kinetic scheme (DUGKS) combines the advantages of both the unified gas kinetic scheme (UGKS) and the lattice Boltzmann method. It can adopt the flexible meshes, meanwhile, the flux calculation is simple. However, the original DUGKS is proposed for the compressible flows. When we try to solve a problem governed by the incompressible Navier-Stokes (N-S) equations, the original DUGKS may bring some undesirable errors because of the compressible effect. To eliminate the compressible effect, the DUGKS for incompressible N-S equations is developed in this work. In addition, the Chapman-Enskog analysis ensures that the present DUGKS can solve the incompressible N-S equations exactly, meanwhile, a new non-extrapolation scheme is adopted to treat the Dirichlet boundary conditions. To test the present DUGKS for incompressible N-S equations, four problems are adopted. The first one is a periodic problem driven by an external force, which is used to test the influences of Courant–Friedrichs–Lewy condition number and the M a c h number (Ma). Besides, some comparisons between the present DUGKS and some available results are also conducted. The second problem is Womersley flow, it is also used to test the influence of Ma, and the results show that the compressible effect is reduced obviously. Then, the two-dimensional lid-driven cavity flow is considered. In these simulations, the Reynolds number is varied from 400 to 1000000 to illustrate the accuracy, stability and efficiency of the present DUGKS. Finally, the numerical solutions of the three-dimensional lid-driven cavity flow suggest that the present DUGKS is suitable for the three-dimensional problems.

We show that the stochastic 3D primitive equations with the Neumann boundary condition on the top, the lateral Dirichlet boundary condition and either the Dirichlet or the Neumann boundary condition on the bottom driven by multiplicative gradient-dependent white noise have unique maximal strong solutions both in the stochastic and PDE senses under certain assumptions on the growth of the noise. For the case of the Neumann boundary condition on the bottom, global existence is established by using the decomposition of the vertical velocity to the barotropic and baroclinic modes and an iterated stopping time argument. An explicit example of non-trivial infinite dimensional noise depending on the vertical average of the horizontal gradient of horizontal velocity is presented.

We provide a quantitative lower bound to the Cheeger constant of a set $\Omega$ in both the Euclidean and the Gaussian settings in terms of suitable asymmetry indexes. We provide examples which show that these quantitative estimates are sharp., Comment: 18 pages, 3 figures

In this paper, we generalize the classical Freidlin-Wentzell’s theorem for random perturbations of Hamiltonian systems. In (Probability Theory and Related Fields 128 (2004) 441–466), M.Freidlin and M.Weber generalized the original result in the sense that the coefficient for the noise term is no longer the identity matrix but a state-dependent matrix and taking the drift term into consideration. In this paper, We generalize the result by adding a state-dependent matrix that converges uniformly to 0 on any compact sets as ϵ tends to 0 to a state-dependent noise and considering the drift term which contains two parts, the state-dependent mapping and a state-dependent mapping that converges uniformly to 0 on any compact sets as ϵ tends to 0. In the proof, we adapt a new way to prove the weak convergence inside the edge by constructing an auxiliary process and modify the proof in (Probability Theory and Related Fields 128 (2004) 441–466) when proving gluing condition.

We study a filtered Lie splitting scheme for the cubic nonlinear Schrödinger equation. We establish error estimates at low regularity by using discrete Bourgain spaces. This allows us to handle data in H s H^s with 0 > s > 1 0>s>1 overcoming the standard stability restriction to smooth Sobolev spaces with index s > 1 / 2 s>1/2 . More precisely, we prove convergence rates of order τ s / 2 \tau ^{s/2} in L 2 L^2 at this level of regularity.

PurposeThe purpose of this article is to derive an implicit-explicit local differential transform method (IELDTM) in dealing with the spatial approximation of the stiff advection-diffusion-reaction (ADR) equations.Design/methodology/approachA direction-free numerical approach based on local Taylor series representations is designed for the ADR equations. The differential equations are directly used for determining the local Taylor coefficients and the required degrees of freedom is minimized. The complete system of algebraic equations is constructed with explicit/implicit continuity relations with respect to direction parameter. Time integration of the ADR equations is continuously utilized with the Chebyshev spectral collocation method.FindingsThe IELDTM is proven to be a robust, high order, stability preserved and versatile numerical technique for spatial discretization of the stiff partial differential equations (PDEs). It is here theoretically and numerically shown that the order refinement (p-refinement) procedure of the IELDTM does not affect the degrees of freedom, and thus the IELDTM is an optimum numerical method. A priori error analysis of the proposed algorithm is done, and the order conditions are determined with respect to the direction parameter.Originality/valueThe IELDTM overcomes the known disadvantages of the differential transform-based methods by providing reliable convergence properties. The IELDTM is not only improving the existing Taylor series-based formulations but also provides several advantages over the finite element method (FEM) and finite difference method (FDM). The IELDTM offers better accuracy, even when using far less degrees of freedom, than the FEM and FDM. It is proven that the IELDTM produces solutions for the advection-dominated cases with the optimum degrees of freedom without producing an undesirable oscillation.

In layered media, the solution of the Maxwell equations has discontinuity of the derivative or the function itself at media interfaces. For the first time, finite-difference schemes providing convergence for discontinuous solutions across straight media interfaces are proposed for the one-dimensional formulation of the Maxwell equations. These are bicompact conservative schemes. They are two-point and layer boundaries are taken as mesh nodes. The scheme explicitly accounts for physically correct interface conditions at media interfaces. We propose an essentially new technique which accounts for medium dispersion. All these features provide the second order of accuracy even on discontinuous solutions. Calculation examples, which illustrate these results, are given. The proposed method is verified by comparison with a previously performed experiment on propagation of normally incident plane wave on one-dimensional photonic crystal. Calculated spectrum agrees well with the measured one within experimental error of 2-5%.

In this paper, we make a comprehensive study for the orbital stability of standing waves for the fractional Schrodinger equation with combined power-type nonlinearities (FNLS) i ∂ t ψ − ( − Δ ) s ψ + a | ψ | p 1 ψ + | ψ | p 2 ψ = 0 . We prove that when p 2 = 4 s N and a ( p 1 − 4 s N ) 0 , there exist the standing waves of (FNLS), which are orbitally stable. When a = 0 and 4 s N p 2 4 s N − 2 s , we present a new, simpler method to study the strong instability of standing waves. When a = − 1 , 0 p 1 p 2 and 4 s N ≤ p 2 4 s N − 2 s , or a = 1 and 4 s N ≤ p 1 p 2 4 s N − 2 s , or a = 1 , 0 p 1 4 s N p 2 4 s N − 2 s and ∂ λ 2 S ω ( u ω λ ) | λ = 1 ≤ 0 , we deduce that the ground state standing waves of (FNLS) are strongly unstable by blow-up.

In this work, we study at the L 2 – level global well-posedness as well as long-time stability of an initial-boundary value problem, posed on a bounded interval, for a generalized higher order nonlinear Schrodinger equation, modeling the propagation of pulses in optical fiber, with a localized damping term. In addition, we implement a precise and efficient code to study the energy decay of the higher order nonlinear Schrodinger equation and we prove its convergence and exponential stability of the discrete energy.

The de la Vallée Poussin inequality is a handy tool for the investigation of disconjugacy, and hence, for the oscillation/nonoscillation of differential equations. The results in this paper are extensions of former those of Hartman and Wintner [Quart. Appl. Math. 13 (1955), 330–332] to the impulsive differential equations. Although the inequality first appeared in such an early date for ordinary differential equations, its improved version for differential equations under impulse effect never has been occurred in the literature. In the present study, first, we state and prove a de la Vallée Poussin inequality for impulsive differential equations, then we give some corollaries on disconjugacy. We also mention some open problems and finally, present some examples that support our findings.

In this paper we consider a fluid-fluid system coupled with a friction-type interface condition which is described by a jump of the velocities along the tangential interface. Based on domain decomposition, we propose a Schwarz type iterative method to decouple the different physical process in each subdomain occupied by a single fluid, which allows solving in each subdomain only the physical process described by a Stokes problem. Using energy estimate, we prove that the algorithm converges for any traction coefficient. A more detailed analysis based on Fourier analysis shows that the convergence rate depends on both the fluids' properties, the traction coefficient and the time stepsize, but is independent of the spatial mesh. We finally use numerical examples to illustrate the theoretical results.

The definition of a Green’s function of a Cauchy–Dirichlet problem for the hyperbolic equation in a quarter plane is given. Its existence and uniqueness have been proven. Representation of the Green’s function is given. It is shown that the Green’s function can be represented by the Riemann–Green function.

The paper deals with a special filtered approximation method, which originates interpolation polynomials at Chebyshev zeros by using de la Vallee Poussin filters. In order to get an optimal approximation in spaces of locally continuous functions equipped with weighted uniform norms, the related Lebesgue constants have to be uniformly bounded. In previous works this has already been proved under different sufficient conditions. Here, we complete the study by stating also the necessary conditions to get it. Several numerical experiments are also given to test the theoretical results and make comparisons to Lagrange interpolation at the same nodes.

We study the uniform weighted resolvent estimates of Schrodinger operator with scaling-critical electromagnetic potentials which, in particular, include the Aharonov-Bohm magnetic potential and inverse-square potential. The potentials are critical due to the scaling invariance of the model and the singularities of the potentials, which are not locally integrable. In contrast to the Laplacian −Δ on R 2 , we prove some new uniform weighted resolvent estimates for this 2D Schrodinger operator and, as applications, we show local smoothing estimates for the Schrodinger equation in this setting.

There is a close connection between stability and oscillation of delay differential equations. For the first-order equation x ′ ( t ) + c ( t ) x ( τ ( t ) ) = 0 , t ≥ 0 , where c is locally integrable of any sign, τ ( t ) ≤ t is Lebesgue measurable, lim t → ∞ ⁡ τ ( t ) = ∞ , we obtain sharp results, relating the speed of oscillation and stability. We thus unify the classical results of Myshkis and Lillo. We also generalise the 3/2-stability criterion to the case of measurable parameters, improving 1 + 1 / e to the sharp 3/2 constant.

In this paper, we study the Morse theory on uniform random attractors for non-autonomous random dynamical systems. To handle the non-invariance of uniform random attractors, we construct the Morse sets of uniform random attractor as the projections of the Morse sets on pullback attractor of the skew product semiflow. First, we obtain the Morse decomposition of uniform random attractors in probability. Second, we describe a decaying energy level on such attractor by Lyapunov function with probability one. Furthermore, we study the stability of Morse decompositions of deterministic uniform attractor under a small random disturbance. Finally, we discuss and collect the results on the Morse decomposition under random setting.

This paper provides an effective method for a class of 2-dim nonlinear variable order fractional optimal control problems (2DNVOFOCP). The technique is based on the new class of basis functions namely the generalized shifted Legendre polynomials. The dynamic constraint is described by a nonlinear variable order fractional differential equation where the fractional derivative is in the sense of Caputo. The 2-dim Gauss-Legendre quadrature rule together with the Lagrange multipliers method are utilized to find the solutions of the given 2DNVOFOCP. The convergence analysis of the presented method is investigated. The examined numerical examples manifest highly accurate results.

In this research, the localized singular boundary method (LSBM) is proposed to solve the Laplace and Helmholtz equations in 2D arbitrary domains. In the traditional SBM, the resultant matrix system is a dense matrix, and it is unsuited for solving the large-scale problems. As a localized domain-type meshless method, a local subdomain for every node can be composed by its own and several nearest nodes. To each of the subdomains, the SBM formulation is applied to derive an implicit expression of the variable at each node in conjunction with the moving least-square approximation. To satisfy the boundary conditions at every boundary node and the governing equation at every node, a sparse linear algebraic system can be obtained. Thus, the numerical solutions at all nodes can be achieved by solving it. Compared with the traditional SBM, the LSBM involves only the origin intensity factor on a circular boundary associated with Dirichlet boundary conditions. It can also effectively avoid the boundary layer effect in the conventional SBM. Furthermore, the proposed LSBM requires less memory storage and computational cost due to the sparse and banded matrix system. Several numerical examples are tested to verify the accuracy and stability of the proposed LSBM.

Spherulites are growth patterns of average spherical form which may occur in the polycrystallization of binary mixtures due to misoriented angles at low grain boundaries. The dynamic growth of spherulites can be described by a phase field model where the underlying free energy depends on two phase field variables, namely the local degree of crystallinity and the orientation angle. For the solution of the phase field model we suggest a splitting scheme based on an implicit discretization in time which decouples the model and at each time step requires the successive solution of an evolutionary inclusion in the orientation angle and an evolutionary equation in the local degree of crystallinity. The discretization in space is done by piecewise linear Lagrangian finite elements. The fully discretized splitting scheme amounts to the solution of two systems of nonlinear algebraic equations. For the numerical solution we suggest a predictor-corrector continuation method with the discrete time as a parameter featuring constant continuation as a predictor and a semismooth Newton method for the first system and the classical Newton method for the second system as a corrector. This allows an adaptive choice of the time steps. Numerical results are given for the formation of a Category 1 spherulite.

We find an explicit form of solution of a convolution type integro-differential equation of the first order (1) ∂ Q ( t , z ) ∂ z = α ( t , z ) − ∫ 0 t β ( t − u ) Q ( u , z ) d u , ( t , z ) ∈ ( 0 , T ) × ( 0 , T ) , where T > 0 , and α , β are two given locally integrable functions and Q satisfies a boundary condition Q ( t , 0 ) = γ ( t ) for a locally integrable function γ. To achieve this goal we introduce the notion of a biconvolution algebra of locally integrable functions on R + 2 . We investigate the properties of the biconvolution algebra and study the Volterra integral equations of the second kind associated with the biconvolution operation. Finally, we present an explicit form of solution of integro-differential equation associated with a linear operator (2) Λ n = ∂ n + 1 ∂ z ∂ t n + ∑ i = 0 n − 1 λ i ∂ i ∂ t i , λ i ∈ R , i ≤ n − 1 , n ∈ N .