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We design a virtual element method for the numerical treatment of the two-dimensional parabolic variational inequality problem on unstructured polygonal meshes. Due to the expected low regularity of the exact solution, the virtual element method is based on the lowest-order virtual element space that contains the subspace of the linear polynomials defined on each element. The connection between the nonnegativity of the virtual element functions and the nonnegativity of the degrees of freedom, i.e., the values at the mesh vertices, is established by applying the Maximum and Minimum Principle Theorem. The mass matrix is computed through an approximate L 2 polynomial projection, whose properties are carefully investigated in the paper. We prove the well-posedness of the resulting scheme in two different ways that reveal the contractive nature of the VEM and its connection with the minimization of quadratic functionals. The convergence analysis requires the existence of a nonnegative quasi-interpolation operator, whose construction is also discussed in the paper. The variational crime introduced by the virtual element setting produces five error terms that we control by estimating a suitable upper bound. Numerical experiments confirm the theoretical convergence rate for the refinement in space and time on three different mesh families including distorted squares, nonconvex elements, and Voronoi tesselations.

The understanding of nonlinear, high dimensional flows, e.g, atmospheric and ocean flows, is critical to address the impacts of global climate change. Data Assimilation techniques combine physical models and observational data, often in a Bayesian framework, to predict the future state of the model and the uncertainty in this prediction. Inherent in these systems are noise (Gaussian and non-Gaussian), nonlinearity, and high dimensionality that pose challenges to making accurate predictions. To address these issues we investigate the use of both model and data dimension reduction based on techniques including Assimilation in Unstable Subspaces, Proper Orthogonal Decomposition, and Dynamic Mode Decomposition. Algorithms that take advantage of projected physical and data models may be combined with Data Analysis techniques such as Ensemble Kalman Filter and Particle Filter variants. The projected Data Assimilation techniques are developed for the optimal proposal particle filter and applied to the Lorenz'96 and Shallow Water Equations to test the efficacy of our techniques in high dimensional, nonlinear systems., 30 pages, 13 figures, 3 tables To appear in Computers & Mathematics with Applications, 2021,ISSN 0898-1221

An efficient strategy for the numerical solution of time-fractional diffusion-reaction problems is devised. A standard finite difference discretization of the space derivative is initially applied which results in a linear stiff term. Then a rectangular product-integration (PI) rule is implemented in an implicit-explicit (IMEX) framework in order to implicitly treat this linear stiff term and handle in an explicit way the non-linear, and usually non-stiff, term. The kernel compression scheme (KCS) is successively adopted to reduce the overload of computation and storage need for the persistent memory term. To reduce the computational effort the semidiscretized problem is described in a matrix-form, so as to require the solution of Sylvester equations only with small matrices. Theoretical results on the accuracy, together with strategies for the optimal selection of the main parameters of the whole method, are derived and verified by means of numerical experiments carried out in two-dimensional domains. The computational advantages with respect to other approaches are also shown and some applications to the detection of pattern formation are illustrated at the end of the paper.

We address the issue of designing robust stabilization terms for the nonconforming virtual element method. To this end, we transfer the problem of defining the stabilizing bilinear form from the elemental nonconforming virtual element space, whose functions are not known in closed form, to the dual space spanned by the known functionals providing the degrees of freedom. By this approach, we manage to construct different bilinear forms yielding optimal or quasi-optimal stability bounds and error estimates, under weaker assumptions on the tessellation than the ones usually considered in this framework. In particular, we prove optimality under geometrical assumptions allowing a mesh to have a very large number of arbitrarily small edges per element. Finally, we numerically assess the performance of the VEM for several different stabilizations fitting with our new framework on a set of representative test cases.

Sparse direct solvers using Block Low-Rank compression have been proven efficient to solve problems arising in many real-life applications. Improving those solvers is crucial for being able to 1) solve larger problems and 2) speed up computations. A main characteristic of a sparse direct solver using low-rank compression is when compression is performed. There are two distinct approaches: (1) all blocks are compressed before starting the factorization, which reduces the memory as much as possible, or (2) each block is compressed as late as possible, which usually leads to better speedup. The objective of this paper is to design a composite approach, to speedup computations while staying under a given memory limit. This should allow to solve large problems that cannot be solved with Approach 2 while reducing the execution time compared to Approach 1.We propose a memory-aware strategy where each block can be compressed either at the beginning or as late as possible. We first consider the problem of choosing when to compress each block, under the assumption that all information on blocks is perfectly known, i.e., memory requirement and execution time of a block when compressed or not. We show that this problem is a variant of the NP-complete Knapsack problem, and adapt an existing 2-approximation algorithm for our problem. Unfortunately, the required information on blocks depends on numerical properties and in practice cannot be known in advance. We thus introduce models to estimate those values. Experiments on the PaStiX solver demonstrate that our new approach can achieve an excellent trade-off between memory consumption and computational cost. For instance on matrix Geo1438, Approach 2 uses three times as much memory as Approach 1 while being three times faster. Our new approach leads to an execution time only 30% larger than Approach 2 when given a memory 30% larger than the one needed by Approach 1., Les solveurs directs creux utilisant de la compression low-rank ont montré leur efficacité pour résoudre de grands systèmes. Les améliorer permetde 1) résoudre de plus grands problèmes et 2) accélérer la résolution. Une caractéristique principale de ces solveurs est quand la compression est réalisée. Il existe deux approches: (1) tous les blocs sont compressés avant le début de la factorisation, ce qui minimise la consommation mémoire, ou (2) chaque bloc est compressé au plus tard, ce qui permet d’avoir de meilleures performances. L’objectif de cet article est de proposer une approche intermédiaire, qui accélère les calculs au maximum tout en restant en dessous d’une limite mémoire donnée. Cela devrait permettre de résoudre de grands problèmes que l’approche 2 ne peut pas résoudre ou de réduire le temps d’exécution de l’approche 1. Nous proposonsune stratégie memory-aware où chaque bloc peut ˆêtre compressé au début ou au plus tard. Nous commençons par considérer le problème où toutes les informations (consommation mémoire et temps d’exécution en version compressée ou non) sont parfaitement connues. Nous montrons que ce problème est une variante du problème NP-complet Knapsack et adaptons une 2-approximation pour notre problème. Malheureusement, toutes ces informations dépendent de propriétés numériques et ne sont pas connues `a l’avance. Des modèles sont donc introduits afin d’estimer ces valeurs. Des expérimentations avec le solveur PaStiX montrent que notre approche permet d’avoir un excellent compromis entre consommation mémoire et temps d’exécution. Par exemple, pour la matrice Geo1438, l’approche 2 utilise trois fois plus de mémoire que l’approche 1, qui est trois fois plus lente. Notre nouvelle méthode permet d’obtenir à la fois un temps d’exécution seulement 30% supérieur à celui de l’approche 2 tout en ayant une consommation mémoire seulement supérieure de 30% à celle de l’approche 1.

Beam-like compliant elements have found wide-ranging application in many fields of engineering and science often where 3D large deflections can be of concern such as soft robotics, DNA mechanics and helicopter/wind turbine rotor blades. The homotopy analysis method (HAM) is used for the first time to obtain a novel analytical solution in converged series form for the arbitrary large deflection of geometrically exact beams subject to both conservative and follower loading scenarios. The homotopy analysis method, which offers desirable characteristics such as being free from small or large parameters, coupled with auxiliary parameters controlling convergence, is applied directly to the intrinsic governing equations of a geometrically exact beam theory. The system of first-order differential governing equations of geometrically exact beams with intrinsic formulation is free from rotation and displacement variables, and offers a low degree of nonlinearity (quadratic at most) and compact mathematical form, making it suitable for analytical solutions. Due to the relatively poor convergence of the original HAM algorithm, the iterative HAM technique is employed which is known to accelerate convergence and to improve the computational efficiency of the homotopy analysis method. The obtained homotopy series offers a number of novel features in the context of the analytical solutions for the large deflection of beams, including (a) the direct calculation of internal forces and moments which is significant for engineering design purposes, (b) being able to capture 3D deflections, (c) considering transverse shear effects which can be important for thicker beams or when the Young’s modulus to shear modulus ratio is significant (such as composite materials) and (d) considering conservative and follower tip and distributed loads, in a unified framework. In order to investigate the efficacy, applicability and accuracy of the proposed method, a number of numerical examples are considered in which a cantilever beam subject to tip or distributed loads undergoes large deflection. Large deflection results for both conservative and follower loads are compared against those of less comprehensive analytical solutions as well as against numerical methods including finite element and Chebyshev collocation methods where good agreement is observed. These results demonstrate the applicability and effectiveness of HAM for the large deflection analysis of geometrically exact beams.

In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fr\'echet--Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method., Comment: v3: final version, incorporated referee comments, to appear in Indagationes Mathematicae, 27 pages

We present a construction of a multi-scale Gaussian beam parametrix for the Dirichlet boundary value problem associated with the wave equation, and study its convergence rate to the true solution in the highly oscillatory regime. The construction elaborates on the wave-atom parametrix of Bao, Qian, Ying, and Zhang and extends to a multi-scale setting the technique of Gaussian beam propagation from a boundary of Katchalov, Kurylev and Lassas., Comment: 64 pages, 7 figures, minor update

The susceptible-infected-recovered (SIR) model is perhaps the most basic epidemiological model for the evolution of disease spread within a population. Because of its direct representation of fundamental physical quantities, a true solution to an SIR model possesses a number of qualitative properties, such as conservation of the total population or positivity or monotonicity of its constituent populations, that may only be guaranteed to hold numerically under step-size restrictions on the solver. Operator-splitting methods with order greater than two require backward sub-steps in each operator, and the effects of these backward sub-steps on the step-size restrictions for guarantees of qualitative correctness of numerical solutions are not well studied. In this study, we analyze the impact of backward steps on step-size restrictions for guaranteed qualitative properties by applying third- and fourth-order operator-splitting methods to the SIR epidemic model. We find that it is possible to provide step-size restrictions that guarantee qualitative property preservation of the numerical solution despite the negative sub-steps, but care must be taken in the choice of the method. Results such as this open the door for the design and application of high-order operator-splitting methods to other mathematical models in general for which qualitative property preservation is important.

The fractional Laplacian, ( − △ ) s , s ∈ ( 0 , 1 ) , appears in a wide range of physical systems, including Levy flights, some stochastic interfaces, and theoretical physics in connection to the problem of stability of the matter. In this paper, a matrix transfer technique (MTT) is employed combining with spectral/element method to solve fractional diffusion equations involving the fractional Laplacian. The convergence of the MTT method is analyzed by the abstract operator theory. Our method can be applied to solve various fractional equation involving fractional Laplacian on some complex domains. Numerical results indicate exponential convergence in the spatial discretization which is in good agreement with the theoretical analysis.

The Hilbert L-matrix A s = [ a i j ( s ) ] , where a i j ( s ) = 1 / ( max ⁡ { i , j } + s ) with i , j ≥ 0 , was introduced in [3] . As a surprising property, we showed that its 2-norm is constant for s ≥ s 0 , where the critical point s 0 is unknown but relies in the interval ( 1 / 4 , 1 / 2 ) . In this note, using some delicate calculations we sharpen this result by improving the upper and lower bounds of the interval surrounding s 0 . Moreover, we establish that the same property persists for the p-norm of A s matrices.

In this article, we investigate some of the fine properties of the value function associated to an optimal control problem in the Wasserstein space of probability measures. Building on new interpolation and linearisation formulas for non-local flows, we prove semiconcavity estimates for the value function, and establish several variants of the so-called sensitivity relations which provide connections between its superdifferential and the adjoint curves stemming from the maximum principle. We subsequently make use of these results to study the propagation of regularity for the value function along optimal trajectories, as well as to investigate sufficient optimality conditions and optimal feedbacks for mean-field optimal control problems., Comment: 55 pages

Nonlinear contaminant transports through porous media are important in many scientific and engineering applications. In this paper, we develop and analyze fourth-order block-centered compact difference scheme (BCCDS) for the nonlinear contaminant transport equations with adsorption process in porous media. Based on block-centered mesh, a fourth order compact difference scheme of solution and its flux is derived. We prove the mass conservation of the proposed scheme and its unconditional stability. We analyze the convergence and obtain the fourth-order error estimate under the smooth regularity of exact solution. Numerical experiments are presented to show the performance of the schemes.

Waiting cost is usually considered as the highest cost imposed to passengers in transit systems. Efficient planning and operations of transit systems require accurate estimation of passengers’ waiting time. While the assumption of half the headway, as the mean waiting time experienced by transit users, has been extensively used in waiting time cost estimation, it is not always a realistic assumption considering heterogeneous passengers, service irregularities, and different types of transit services. In addition, many transit studies considered the waiting times of passengers only at the origin, while waiting times can also be incurred at transfer points and the destination, the latter for passengers with fixed arrival time at their destination. In this paper, we developed new mean waiting time formulations for different transit systems, including feeder-trunk service, Dial-a-Ride service, and single route with unreliable service. All possible combinations of types of passenger (planning and non-planning, with fixed and flexible arrival time at destination) and service types (schedule-based/frequency-based and high-frequency/low-frequency) are considered to capture the underlying dynamics of transit systems. The developed approach in this paper could be utilised in transit studies to better model these underlying dynamics and to achieve better designs and more efficient operations.

We consider a class of cooperative reaction-diffusion systems with free boundaries in one space dimension, where the diffusion terms are nonlocal, given by integral operators involving suitable kernel functions, and they are allowed not to appear in some of the equations in the system. Such a system covers various models arising from mathematical biology, in particular a West Nile virus model and an epidemic model considered recently in [16] and [44] , respectively, where a “spreading-vanishing” dichotomy is known to govern the long time dynamical behaviour, but the question on spreading speed was left open. In this paper, we develop a systematic approach to determine the spreading profile of the system, and obtain threshold conditions on the kernel functions which decide exactly when the spreading has finite speed, or infinite speed (accelerated spreading). This relies on a rather complete understanding of both the associated semi-waves and travelling waves. When the spreading speed is finite, we show that the speed is determined by a particular semi-wave. This is Part 1 of a two part series. In Part 2, for some typical classes of kernel functions, we will obtain sharp estimates of the spreading rate for both the finite speed case, and the infinite speed case.

This paper is concerned with a class of nonlocal reaction-diffusion equations with time-delay and degenerate diffusion. Affected by the degeneracy of diffusion, it is proved that, the Cauchy problem of the equation possesses the Holder-continuous solution. Furthermore, the non-critical traveling waves are proved to be globally L 1 -stable, which is the first frame work on L 1 -wavefront-stability for the degenerate diffusion equations. The time-exponential convergence rate is also derived. The adopted approach for the proof is the technical L 1 -weighted energy estimates combining the compactness analysis, but with some new development.

In this paper, we considered a time-optimal control problem for a new type of linear parameter varying (LPV) system which is obtained through data identification in the process of dealing with actual problems. The addition of non-linear terms is compensation for the method that does not require linear expansion at the equilibrium point. Since the objective function is the terminal time which is an implicit function concerning decision variables, it is a non-standard optimal control problem with uncertain terminal time. To find the global optimal solution to this problem, firstly, the control parameterization method is used to transform it into a nonlinear optimization problem of parameter selection, and then the modifed particle swarm optimization (PSO) algorithm is combined to solve the equivalent nonlinear programming problem. Numerical examples are used to illustrate the effectiveness of the proposed algorithm.

Method of asymptotic partial decomposition of a domain (MAPDD) proposed and justified earlier for thin domains (rod structures, tube structures consisting of a set of thin cylinders) generates some specific interface conditions between three-dimensional and one-dimensional parts. In the case of the heat equation these conditions ensure the continuity of the solution and continuity in average of the normal flux. However for some computational reasons the strong continuity condition for the solution may be undesirable. It may be preferable to replace it by more flexible condition of continuity in average over the interface cross section. In the present paper we introduce and justify this alternative junction condition allowing such discontinuity of the solution of both stationary and non-stationary problems. The closeness of solutions of these hybrid dimension problems to the solution of the fully three-dimensional setting is proved. At the discrete level, finite volume schemes are considered, an error estimate is established. The new version of the MAPDD is compared to the classical one via numerical tests. It shows better stability of the new scheme.

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, a general class of nonlinear initial value problems involving a Riemann-Liouville fractional derivative and a variable-order fractional derivative is investigated. An existence result of the exact solution is established by using Weissinger's fixed point theorem and Gronwall-Bellman lemma. An hp-version spectral collocation method is presented to solve the problem in numerical frames. The collocation method employs the Legendre-Gauss interpolations to conquer the influence of the nonlinear term and variable-order fractional derivative. The most remarkable feature of the method is its capability to achieve higher accuracy by refining the mesh and/or increasing the degree of the polynomial. The error estimates under the H 1 -norm for smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes are derived. Numerical results are given to support the theoretical conclusions.

We analyze a system of differential equations with state-dependent delay (SD-DDE) from cell biology, in which the delay is implicitly defined as the time when the solution of an ODE, parametrized by the SD-DDE state, meets a threshold. We show that the system is well-posed and that the solutions define a continuous semiflow on a state space of Lipschitz functions. Moreover we establish for an associated system a convex and compact set that is invariant under the time-t-map for a finite time. It is known that, due to the state dependence of the delay, necessary and sufficient conditions for well-posedness can be related to functionals being almost locally Lipschitz, which roughly means locally Lipschitz on the restriction of the domain to Lipschitz functions, and our methodology involves such conditions. To achieve transparency and wider applicability, we elaborate a general class of two component functional differential equation systems, that contains the SD-DDE from cell biology and formulate our results also for this class.

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.

Let ( g n ) n ⩾ 1 be a sequence of independent and identically distributed random elements of the general linear group G L d ( R ) , with law μ . Consider the random walk G n : = g n … g 1 . Denote respectively by ‖ G n ‖ and ρ ( G n ) the operator norm and the spectral radius of G n . For log ‖ G n ‖ and log ρ ( G n ) , we prove moderate deviation principles under exponential moment and strong irreducibility conditions on μ ; we also establish moderate deviation expansions in the normal range [ 0 , o ( n 1 / 6 ) ] and Berry–Esseen bounds under the additional proximality condition on μ . Similar results are found for the couples ( X n x , log ‖ G n ‖ ) and ( X n x , log ρ ( G n ) ) with target functions, where X n x : = G n ⋅ x is a Markov chain and x is a starting point on the projective space P ( R d ) .

In this paper, energy-preserving time high-order average vector field (AVF) compact finite difference scheme is proposed and analyzed for solving two-dimensional nonlinear wave equations including the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation. We first present the corresponding Hamiltonian system to the two-dimensional nonlinear wave equations, and further apply the compact finite difference (CFD) operator and AVF method to develop an energy conservative high-order scheme in two dimensions. The L p -norm boundedness of two-dimensional numerical solution is obtained from the energy conservation property, which plays an important role in the analysis of the scheme for the two-dimensional nonlinear wave equations in which the nonlinear term satisfies local Lipschitz continuity condition. We prove that the proposed scheme is energy conservative and uniquely solvable. Furthermore, optimal error estimate for the developed scheme is derived for the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation in two dimensions. Numerical experiments are carried out to confirm the theoretical findings and to show the performance of the proposed method for simulating the propagation of nonlinear waves in layered media.

Let Ω be a C 2 bounded domain in R N ( N ≥ 3 ), δ ( x ) = dist ( x , ∂ Ω ) and C H ( Ω ) be the best constant in the Hardy inequality with respect to Ω. We investigate positive solutions to a boundary value problem for Lane-Emden equations with Hardy potential of the form − Δ u − μ δ 2 u = u p in Ω , u = ρ ν on ∂ Ω , ( P ρ ) where 0 μ C H ( Ω ) , ρ is a positive parameter, ν is a positive Radon measure on ∂Ω with norm 1 and 1 p N μ , with N μ being a critical exponent depending on N and μ. It is known from [22] that there exists a threshold value ρ ⁎ such that problem ( P ρ ) admits a positive solution if 0 ρ ≤ ρ ⁎ , and no positive solution if ρ > ρ ⁎ . In this paper, we go further in the study of the solution set of ( P ρ ) . We show that the problem admits at least two positive solutions if 0 ρ ρ ⁎ and a unique positive solution if ρ = ρ ⁎ . We also prove the existence of at least two positive solutions for Lane-Emden systems { − Δ u − μ δ 2 u = v p in Ω , − Δ v − μ δ 2 v = u q in Ω , u = ρ ν , v = σ τ on ∂ Ω , under the smallness condition on the positive parameters ρ and σ.

This note relies mainly on a refined version of the main results of the paper by F. Catrina and D. Costa (J. Differential Equations 2009). We provide very short and self-contained proofs. Our results are sharp and minimizers are obtained in suitable functional spaces. As main tools we use the so-called \textit{expand of squares} method to establish sharp weighted $L^{2}$-Caffarelli-Kohn-Nirenberg (CKN) inequalities and density arguments., Comment: 13 pages

A proper orthogonal decomposition-based B-splines B\'ezier elements method (POD-BSBEM) is proposed as a non-intrusive reduced-order model for uncertainty propagation analysis for stochastic time-dependent problems. The method uses a two-step proper orthogonal decomposition (POD) technique to extract the reduced basis from a collection of high-fidelity solutions called snapshots. A third POD level is then applied on the data of the projection coefficients associated with the reduced basis to separate the time-dependent modes from the stochastic parametrized coefficients. These are approximated in the stochastic parameter space using B-splines basis functions defined in the corresponding B\'ezier element. The accuracy and the efficiency of the proposed method are assessed using benchmark steady-state and time-dependent problems and compared to the reduced order model-based artificial neural network (POD-ANN) and to the full-order model-based polynomial chaos expansion (Full-PCE). The POD-BSBEM is then applied to analyze the uncertainty propagation through a flood wave flow stemming from a hypothetical dam-break in a river with a complex bathymetry. The results confirm the ability of the POD-BSBEM to accurately predict the statistical moments of the output quantities of interest with a substantial speed-up for both offline and online stages compared to other techniques., Comment: 45 pages, 15 figures

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.

Solving Burgers' equation always posses challenges for a small value of viscosity. Here we present a numerical method based on the Haar wavelet collocation method coupled with a nonstandard finite difference (NSFD) scheme for a class of generalized Burgers' equation. In the solution process, the time derivative is discretized by the NSFD scheme and the spatial derivatives are approximated by the Haar wavelets series. The nonlinear terms are linearized with the help of the quasilinearisation process. We illustrate the efficiency of the proposed method by solving several test problems and report their L 2 -error and L ∞ -error norms. The derived method is quite easy to implement compared to the other methods. Also, the error analysis of the current method is discussed. It is also observed that for the small number of grid points, the current method produces results that are in great agreement with the analytical solutions.

In this paper, a weak Galerkin finite element method for solving the time fractional reaction-convection diffusion problem is proposed. We use the well known L 1 discretization in time and a weak Galerkin finite element method on uniform mesh in space. Both continuous and discrete time weak Galerkin finite element method are considered and analyzed. The stability of the discrete time scheme is proved. The error estimates for both schemes are given. Finally, we give some numerical experiments to show the efficiency of the proposed method.

It was proved by Karch and Pilarczyk that Landau solutions are asymptotically stable under any L 2 -perturbation. In our earlier work with L. Li, we have classified all ( − 1 ) -homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south and north poles. In this paper, we study the asymptotic stability of the least singular solutions among these solutions other than Landau solutions, and prove that such solutions are asymptotically stable under any L 2 -perturbation.

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.

In the context of time-domain simulation of integrated circuits, one often encounters large systems of coupled differential-algebraic equations. Simulation costs of these systems can become prohibitively large as the number of components keeps increasing. In an effort to reduce these simulation costs a twofold approach is presented in this paper. We combine maximum entropy snapshot sampling method and a nonlinear model order reduction technique, with multirate time integration. The obtained model order reduction basis is applied using the Gaus-Newton method with approximated tensors reduction. This reduction framework is then integrated using a coupled-slowest-first multirate integration scheme. The convergence of this combined method is verified numerically. Lastly it is shown that the new method results in a reduction of the computational effort without significant loss of accuracy.

Second derivative general linear methods (SGLMs) have been already implemented in a variable stepsize environment using Nordsieck technique. In this paper, we introduce variable stepsize SGLMs directly on nonuniform grid. By deriving the order conditions of the proposed methods of order p and stage order q = p , some explicit examples of these methods up to order four are given. By some numerical experiments, we show the efficiency of the proposed methods in solving nonstiff problems and confirm the theoretical order of convergence.

In this paper, based on the implicit Euler scheme in the temporal direction, the superconvergence property is investigated by using the special property of the bilinear element on the rectangular mesh for the Sobolev equation with Burgers' nonlinearity. The existence and uniqueness of the fully-discrete solution is proved. Further, the superconvergence error estimate in L ∞ ( H 1 ) -norm is established in terms of a novel approach, i.e., the technique of the combination of the interpolation operator and projection operator. Finally, a numerical experiment is carried out to confirm the theoretical analysis.

We partly extend the localisation technique from convex geometry to the multiple constraints setting. For a given $1$-Lipschitz map $u\colon\mathbb{R}^n\to\mathbb{R}^m$, $m\leq n$, we define and prove the existence of a partition of $\mathbb{R}^n$, up to a set of Lebesgue measure zero, into maximal closed convex sets such that restriction of $u$ is an isometry on these sets. We consider a disintegration, with respect to this partition, of a log-concave measure. We prove that for almost every set of the partition of dimension $m$, the associated conditional measure is log-concave. This result is proven also in the context of the curvature-dimension condition for weighted Riemannian manifolds. This partially confirms a conjecture of Klartag., accepted in Journal de Math\'ematiques Pures et Appliqu\'ees; the present preprint is formed from arXiv:1905.02182, which has been split; 28 pages

The Double Absorbing Boundary (DAB) is a recently proposed absorbing layer used to truncate an unbounded domain with high-order accuracy. While it was originally designed for time-dependent acoustics and elastodynamics, here the DAB construction is adapted and applied to the 2D Helmholtz equation. Both wave-guide and corner configurations are considered. A high-order spectral finite element scheme is used in order to match the discretization accuracy to the accuracy of the DAB. The DAB scheme is analyzed, and numerical experiments demonstrate its performance.

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.

We give an effective method for solving fractional Riccati differential equations. We first define the fractional-order Boubaker wavelets (FOBW). Using the hypergeometric functions, we determine the exact values for the Riemann-Liouville fractional integral operator of the FOBW. The properties of FOBW, the exact formula, and the collocation method are used to transform the problem of solving fractional Riccati differential equations to the solution of a set of algebraic equations. These equations are solved via Newton's iterative method. The error estimation for the present method is also determined. The performance of the developed numerical schemes is assessed through several examples. This method yields very accurate results. The given numerical examples support this claim.

In this paper, we investigate adaptive streamline upwind/Petrov Galerkin (SUPG) methods for singularly perturbed convection-diffusion-reaction equations in a new dual norm presented in [Du and Zhang, J. Sci. Comput. (2015)]. The flux is recovered by either local averaging in conforming $H({\rm div})$ spaces or weighted global $L^2$ projection onto conforming $H({\rm div})$ spaces. We further introduce a recovery stabilization procedure, and develop completely robust a posteriori error estimators with respect to the singular perturbation parameter $\varepsilon$. Numerical experiments are reported to support the theoretical results and to show that the estimated errors depend on the degrees of freedom uniformly in $\varepsilon$., 20 pages, 14 figures

We study the simplest parabolic-elliptic model of chemotaxis in space dimensions N ≥ 3 , and show the optimal conditions on the initial data for the finite time blow-up and the global existence of solutions in terms of stationary solutions. Our argument is based on the study of the Cauchy problem for the transformed equation involving the averaged mass of the solution.

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.

In this paper we introduce and study several mean forms of sensitive tuples. It is shown that the topological or measure-theoretical entropy tuples are correspondingly mean sensitive tuples under certain conditions (minimal in the topological setting or ergodic in the measure-theoretical setting). Characterizations of the question when every non-diagonal tuple is mean sensitive are presented. Among other results we show that under minimality assumption a topological dynamical system is weakly mixing if and only if every non-diagonal tuple is mean sensitive and so as a consequence every minimal weakly mixing topological dynamical system is mean n-sensitive for any integer n ≥ 2 . Moreover, the notion of weakly sensitive in the mean tuple is introduced and it turns out that this property has some special lift property. As an application we obtain that the maximal mean equicontinuous factor for any topological dynamical system can be induced by the smallest closed invariant equivalence relation containing all weakly sensitive in the mean pairs.

We study the convergence of the method of reflections for the Stokes equations in domains perforated by countably many spherical particles with boundary conditions typical for the suspension of rigid particles. We prove that a relaxed version of the method is always convergent in H ˙ 1 under a mild separation condition on the particles. Moreover, we prove optimal convergence rates of the method in W ˙ 1 , q , 1 q ∞ and in L ∞ in terms of the particle volume fraction under a stronger separation condition of the particles.

International audience; In the framework of the development of new passive safety systems for the second and third generations of nuclear reactors, the numerical simulations, involving complex turbulent two-phase flows around thin or massive inflow obstacles, are privileged tools to model, optimize and assess new design shapes. In order to match industrial demands, computational fluid dynamics tools must be the fastest, most accurate and most robust possible. To face this issue, we have chosen to solve the Navier-Stokes equations using a projection scheme for a mixture fluid coupled with an Immersed Boundary (IB) approach: the penalized direct forcing method - a technique whose characteristics inherit from both penalty and immersed boundary methods - adapted to infinitely thin obstacles and to a Finite Element (FE) formulation. Various IB conditions (slip, no-slip or Neumann) for the velocity on the IB can be managed by imposing Dirichlet values in the vicinity of the thin obstacles. To deal with these imposed Dirichlet velocities, we investigated two variants: one in which we use the obstacle velocity and another one in which we use linear interpolations based on discrete geometrical properties of the IB (barycenters and normal vectors) and the FE basis functions. This last variant is motivated by an increase of the accuracy/computation time ratio for coarse meshes. As a first step, concerning academic test cases for one-phase dilatable-fluid laminar flows, the results obtained via those two variants are in good agreement with analytical and experimental data. Moreover, when compared to each other, the linear interpolation variant increases the spatial order of convergence as expected. An industrial test case illustrates the advantages and drawbacks of this approach. In a shortcoming second step, to face two-phase turbulent fluid simulations, some methodology modifications will be considered such as adapting the projection scheme to low-compressible fluid and immersed wall-law boundary conditions.

A fully discrete ADI scheme is proposed for solving the two-dimensional time-fractional diffusion equation with weakly singular solutions, where L1 scheme on graded mesh is adopted to tackle the initial singularity. An improved discrete fractional Gronwall inequality is employed to give an α-robust H 1 -norm convergence analysis of the fully discrete ADI scheme, where the error bound does not blow up when the order of fractional derivative α → 1 − . Numerical results show that the theoretical analysis is sharp.

This paper is dedicated to solving the iterative solution to the discrete-time periodic Sylvester matrix equations. Inspired by Jacobi iterative algorithm and hierarchical identification principle, the Jacobi gradient based iterative (JGI) algorithm and the accelerated Jacobi gradient based iterative (AJGI) algorithm are proposed. It is verified that the proposed algorithms are convergent for any initial matrix when the parameter factor μ satisfies certain condition. The necessary and sufficient conditions are provided for the presented new algorithms. Moreover, we also apply the JGI algorithm and AJGI algorithm to study a more generalized discrete-time periodic matrix equations and give the convergence conditions of the algorithms. Finally, two numerical examples are given to illustrate the effectiveness, accuracy and superiority of the proposed algorithms.

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.