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This paper is devoted to the finite element analysis for the Bean–Kim model governed by the full 3D Maxwell equations. Describing type-II superconductivity at the macroscopic level, this model leads to a challenging coupled system consisting of the Faraday equation and a hyperbolic quasi-variational inequality (QVI) of the second kind with L1-type nonlinearity, that arises explicitly from the magnetic field dependency in the critical current. With the involved Maxwell coupling in the 3D H(curl)-setting, the hyperbolic QVI character poses the primary challenge in the numerical investigation. Two mixed finite element methods based on implicit Euler and leapfrog time-stepping are proposed. On the one hand, the implicit Euler method results in a nonstandard system of curl-curl elliptic QVI with a first-order curl-type nonlinearity. Though the well-posedness of this system is guaranteed, its numerical realization is not straightforward and requires the use of a two-stage iteration process of high computational complexity. On the other hand, by approximating the electric and magnetic fields at two different time step levels, the leapfrog method turns out to be more suitable as it naturally eliminates the notorious QVI structure. More importantly, utilizing suited subdifferential and optimization techniques, we are able to prove an efficiently computable explicit formula for its exact solution in terms of the electric field, which makes its numerical computation substantially more favorable than the Euler method. As further advantages, the leapfrog method applies to broad scenarios involving low regular data of bounded variation (BV) in time for both the applied current source and the temperature distribution. Through nonstandard technical arguments tailored to the BV data, our analysis proves the conditional stability and, eventually, the uniform convergence of the proposed leapfrog method. This paper is closed by 3D numerical tests showcasing the reasonable and efficient performance of the proposed numerical solution. [ABSTRACT FROM AUTHOR]

In this paper, we derive explicit second-order necessary and sufficient optimality conditions of a local minimizer to an optimal control problem for a quasilinear second-order partial differential equation with a piecewise smooth but not differentiable nonlinearity in the leading term. The key argument rests on the analysis of level sets of the state. Specifically, we show that if a function vanishes on the boundary and its the gradient is different from zero on a level set, then this set decomposes into finitely many closed simple curves. Moreover, the level sets depend continuously on the functions defining these sets. We also prove the continuity of the integrals on the level sets. In particular, Green's first identity is shown to be applicable on an open set determined by two functions with nonvanishing gradients. In the second part to this paper, the explicit sufficient second-order conditions will be used to derive error estimates for a finite-element discretization of the control problem. [ABSTRACT FROM AUTHOR]

In this paper, we study three numerical schemes for the McKean-Vlasov equation {dXt=b(t,Xt,μt)dt+σ(t,Xt,μt)dBt,∀t∈[0,T],μt is the probability distribution of Xt, { d X t = b (t , X t , μ t) d t + σ (t , X t , μ t) d B t , ∀ t ∈ [ 0 , T ] , μ t  is the probability distribution of   X t , $ \{ \begin{array}{l}dX_t=b{(t,X_t,\mu_t)}dt+\sigma{(t,X_t,\mu_t)}dB_t,\\ \forall t\in\lbrack 0,T\rbrack,\mu_t\text{is the probability distribution of }X_t,\end{array} $ where X0 : (Ω, F, ℙ) → (ℝd, B(ℝd)) is a known random variable. Under the assumption on the Lipschitz continuity of the coefficients b and σ, our first result proves the convergence rate of the particle method with respect to the Wasserstein distance, which extends previous work [M. Bossy and D. Talay, Math. Comput. 66 (1997) 157–192.] established in a one-dimensional setting. In the second part, we present and analyse two quantization-based schemes, including the recursive quantization scheme (deterministic scheme) in the Vlasov setting, and the hybrid particle-quantization scheme (random scheme inspired by the K-means clustering). Two simulations are presented at the end of this paper: Burgers equation introduced in [M. Bossy and D. Talay, Math. Comput. 66 (1997) 157–192.] and the network of FitzHugh- Nagumo neurons (see [J. Baladron, D. Fasoli, O. Faugeras and J. Touboul, J. Math. Neurosci. 2 (2012) 1–50.] and [M. Bossy, O. Faugeras and D. Talay, J. Math. Neurosci. 5 (2015) 1–23.]) in dimension 3. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of accelerating the numerical simulation of time dependent problems involving a multi-step time scheme by the parareal algorithm. The parareal method is based on combining predictions made by a coarse and cheap propagator, with corrections computed with two propagators: the previous coarse and a precise and expensive one used in a parallel way over the time windows. A multi-step time scheme can potentially bring higher approximation orders than plain one-step methods but the initialisation of each time window needs to be appropriately chosen. Our main contribution is the design and analysis of an algorithm adapted to this type of discretisation without being too much intrusive in the coarse or fine propagators. At convergence, the parareal algorithm provides a solution that coincides with the solution of the fine solver. In the classical version of parareal, the local initial condition of each time window is corrected at every iteration. When the fine and/or coarse propagators is a multi-step time scheme, we need to choose a consistent approximation of the solutions involved in the initialisation of the fine solver at each time windows. Otherwise, the initialisation error will prevent the parareal algorithm to converge towards the solution with fine solver's accuracy. In this paper, we develop a variant of the algorithm that overcome this obstacle. Thanks to this, the parareal algorithm is more coherent with the underlying time scheme and we recover the properties of the original version. We show both theoretically and numerically that the accuracy and convergence of the multi-step variant of parareal algorithm are preserved when we choose carefully the initialisation of each time window. [ABSTRACT FROM AUTHOR]

Superpositions of plane waves are known to approximate well the solutions of the Helmholtz equation. Their use in discretizations is typical of Trefftz methods for Helmholtz problems, aiming to achieve high accuracy with a small number of degrees of freedom. However, Trefftz methods lead to ill-conditioned linear systems, and it is often impossible to obtain the desired accuracy in floating-point arithmetic. In this paper we show that a judicious choice of plane waves can ensure high-accuracy solutions in a numerically stable way, in spite of having to solve such ill-conditioned systems. Numerical accuracy of plane wave methods is linked not only to the approximation space, but also to the size of the coefficients in the plane wave expansion. We show that the use of plane waves can lead to exponentially large coefficients, regardless of the orientations and the number of plane waves, and this causes numerical instability. We prove that all Helmholtz fields are continuous superposition of evanescent plane waves, i.e., plane waves with complex propagation vectors associated with exponential decay, and show that this leads to bounded representations. We provide a constructive scheme to select a set of real and complex-valued propagation vectors numerically. This results in an explicit selection of plane waves and an associated Trefftz method that achieves accuracy and stability. The theoretical analysis is provided for a two-dimensional domain with circular shape. However, the principles are general and we conclude the paper with a numerical experiment demonstrating practical applicability also for polygonal domains. [ABSTRACT FROM AUTHOR]

In this paper we present a hybridizable discontinuous Galerkin method for the time-dependent Navier–Stokes equations coupled to the quasi-static poroelasticity equations via interface conditions. We determine a bound on the data that guarantees stability and well-posedness of the fully discrete problem and prove a priori error estimates. A numerical example confirms our analysis. [ABSTRACT FROM AUTHOR]

Multi-layer structures are ubiquitous in constructing metamaterial devices to realise various frontier applications including super-resolution imaging and invisibility cloaking. In this paper, we develop a general mathematical framework for studying elastostatics within multi-layer material structures in Rd, d = 2, 3. The multi-layer structure is formed by concentric balls and each layer is filled by either a regular elastic material or an elastic metamaterial. The number of layers can be arbitrary and the material parameters in each layer may be different from one another. In practice, the multi-layer structure can serve as the building block for various material devices. Considering the impingement of an incident field on the multi-layer structure, we first derive the exact perturbed field in terms of an elastic momentum matrix, whose dimension is the same as the number of layers. By highly intricate and delicate analysis, we derive a comprehensive study of the spectral properties of the elastic momentum matrix. This enables us to establish a handy algebraic framework for studying polariton resonances associated with multi-layer metamaterial structures, which forms the fundamental basis for many metamaterial applications. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a specific modification of the numerical fluxes in order to insure the well-balanced property of schemes on staggered grids for the Euler equations. This property is crucial for the numerical representation of equilibrium solutions of balance laws with source terms, like when describing flows subjected to gravity and a complex topography. We propose first and second order versions of the well-balanced scheme. The performances of the method are evaluated through a series of 1D and 2D benchmarks. [ABSTRACT FROM AUTHOR]

This paper is the second of the series of two papers, which focuses on the derivation of an averaged 1D model for compressible bubbly flows. For this, we start from a microscopic description of the interactions between a large but finite number of small bubbles with a surrounding compressible fluid. This microscopic model has been derived and analysed in the first paper. In the present one, provided physical parameters scale according to the number of bubbles, we prove that solutions to the microscopic model exist on a timespan independent of the number of bubbles. Considering then that we have a large number of bubbles, we propose a construction of the macroscopic variables and derive the averaged system satisfied by these quantities. Our method is based on a compactness approach in a strong-solution setting. In the last section, we propose the derivation of the Williams–Boltzmann equation corresponding to our setting. [ABSTRACT FROM AUTHOR]

We prove that there exists a unique solution to the initial value problem describing the motion of a particle subject to a unilateral constraint and Coulomb friction, if the external force acting on the particle is an analytic function of time and of the particle's position and velocity. Previous work claimed that this problem has a local series solution that corresponds to an analytic function, after any impacts have been resolved. However, a counterexample to that claim was recently discovered, involving a particle starting to slide, in which the series solution is divergent, and thus does not correspond to an analytic function. This paper corrects previous arguments by considering a general formal series solution for a particle that is starting to slide. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a numerical method combining the Ciarlet-Raviart mixed finite element formulation and an iterative algorithm for the Maxwell's transmission eigenvalue problem. The eigenvalue problem is first written as a nonlinear quad-curl eigenvalue problem. Then the real transmission eigenvalues are proved to be the roots of a non-linear function. They are the generalized eigenvalues of a related linear self-adjoint quad-curl eigenvalue problem. These generalized eigenvalues are computed by a mixed finite element method. We derive the error estimates using the spectral approximation of compact operators, the theory of mixed finite element method for quad-curl problems, and the derivatives of eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper, we construct, analyze, and numerically validate a linearized Crank-Nicolson virtual element method (VEM) for solving quasilinear parabolic problems on general polygonal meshes. In particular, we consider the more general nonlinear term a(x, u), which does not require Lipschitz continuity or uniform ellipticity conditions. To ensure that the fully discrete solution remains bounded in L∞-norm, we construct two novel elliptic projections and apply a new error splitting technique. With the help of the boundedness of numerical solution and delicate analysis of the nonlinear term, we derive the optimal error estimates for any k-order VEMs without any time-step restrictions. Numerical experiments on various polygonal meshes validate the accuracy of theoretical analysis and the unconditional convergence of the proposed scheme. [ABSTRACT FROM AUTHOR]

Maxwell's equations describe the propagation of electromagnetic waves and are therefore fundamental to understanding many problems encountered in the study of antennas and electromagnetics. The aim of this paper is to propose and analyse an efficient fully discrete scheme for solving three-dimensional Maxwell's equations. This is accomplished by combining Fourier pseudospectral methods in space and exact formulation in time. Fast computation is efficiently implemented in the scheme by using the matrix diagonalisation method and fast Fourier transform algorithm which are well known in scientific computations. An optimal error estimate which is not encumbered by the CFL condition is established and the resulting scheme is proved to be of spectral accuracy in space and exact in time. Furthermore, the scheme is shown to have multiple conservation laws including discrete energy, helicity, momentum, symplecticity, and divergence-free field conservations. All the theoretical results of the accuracy and conservations are numerically illustrated by two numerical tests. [ABSTRACT FROM AUTHOR]

In this paper a technique is given to recover the classical order of the method when explicit exponential Runge–Kutta methods integrate reaction-diffusion problems. In the literature, methods of high enough stiff order for problems with vanishing boundary conditions have been constructed, but that implies restricting the coefficients and thus, the number of stages and the computational cost may significantly increase with respect to other methods without those restrictions. In contrast, the technique which is suggested here is cheaper because it just needs, for any method, to add some terms with information only on the boundaries. Moreover, time-dependent boundary conditions are directly tackled here. [ABSTRACT FROM AUTHOR]

This paper is concerned with optimal design problems in the setting of the Kirchhoff–Love model for pure bending of a thin solid symmetric plate under a transverse load. For two isotropic elastic materials with a prescribed amount, the goal is to find their rearrangement within the domain that forms a least compliant structure. The homogenization method is used as a relaxation tool to overcome the lack of a classical solution of optimal design problem. Neccessary conditions of optimality were derived and an optimality criteria method for the single state compliance minimization problems is developed and tested on several examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider a semiconducting device with an active zone made of a single-layer material. The associated Poisson equation for the electrostatic potential (to be solved in order to perform self-consistent computations) is characterized by a surface particle density and an out-of-plane dielectric permittivity in the region surrounding the single-layer. To avoid mesh refinements in such a region, we propose an interface problem based on the natural domain decomposition suggested by the physical device. Two different interface continuity conditions are discussed. Then, we write the corresponding variational formulations adapting the so called three-fields formulation for domain decomposition and we approximate them using a proper finite element method. Finally, numerical experiments are performed to illustrate some specific features of this interface approach. [ABSTRACT FROM AUTHOR]

The shallow water flow model is widely used to describe water flows in rivers, lakes, and coastal areas. Accounting for uncertainty in the corresponding transport-dominated nonlinear PDE models presents theoretical and numerical challenges that motivate the central advances of this paper. Starting with a spatially one-dimensional hyperbolicity-preserving, positivity-preserving stochastic Galerkin formulation of the parametric/uncertain shallow water equations, we derive an entropy-entropy flux pair for the system. We exploit this entropy-entropy flux pair to construct structure-preserving second-order energy conservative, and first- and second-order energy stable finite volume schemes for the stochastic Galerkin shallow water system. The performance of the methods is illustrated on several numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we use the Bayesian inversion approach to study the data assimilation problem for a family of tumor growth models described by porous-medium type equations. The models contain uncertain parameters and are indexed by a physical parameter m, which characterizes the constitutive relation between density and pressure. Based on these models, we employ the Bayesian inversion framework to infer parametric and nonparametric unknowns that affect tumor growth from noisy observations of tumor cell density. We establish the well-posedness and the stability theories for the Bayesian inversion problem and further prove the convergence of the posterior distribution in the so-called incompressible limit, m → ∞. Since the posterior distribution across the index regime m ∈ [2, ∞) can thus be treated in a unified manner, such theoretical results also guide the design of the numerical inference for the unknown. We propose a generic computational framework for such inverse problems, which consists of a typical sampling algorithm and an asymptotic preserving solver for the forward problem. With extensive numerical tests, we demonstrate that the proposed method achieves satisfactory accuracy in the Bayesian inference of the tumor growth models, which is uniform with respect to the constitutive relation. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze the least squares finite element methods for the linear elasticity interface problem in the stress-displacement system on unfitted meshes. We consider the cases that the interface is C2 or polygonal, and the exact solution (σ, u) belongs to Hs(div; Ω0 ∪ Ω1) × H1+s(Ω0 ∪ Ω1) with s > 1/2. Two types of least squares functionals are defined to seek the numerical solutions. The first is defined by simply applying the L2 norm least squares principle, and requires the condition s ≥ 1. The second is defined with a discrete minus norm, which is related to the inner product in H−1/2(Γ). The use of this discrete minus norm results in a method of optimal convergence rates and allows the exact solution has the regularity of any s > 1/2. The stability near the interface for both methods is guaranteed by the ghost penalty bilinear forms and we can derive the robust condition number estimates. The convergence rates under L2 norm and the energy norm are derived for both methods. We illustrate the accuracy and the robustness of the proposed methods by a series of numerical experiments for test problems in two and three dimensions. [ABSTRACT FROM AUTHOR]

Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula with variable temporal stepsize in time. With the help of discrete orthogonal convolution kernels, temporal-spatial error splitting idea and a cut-off numerical technique, the unique solvability, maximum-norm stability and corresponding error estimate of the high-order nonlinear difference scheme are established under assumption that the temporal stepsize ratio satisfies rk := τk/τk−1 < 4.8645. Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction rk < 4.8645 on the variable temporal stepsize, unconditional and optimal fourth-order in space and second-order in time maximum-norm error estimates of the two-grid difference scheme is established. Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

This paper studies the collective motion of self-driven agents in a one-dimensional space with periodic boundaries, using a stochastic Port-Hamiltonian system (PHS) with symmetric nearest-neighbor interactions and additive Brownian noise as an external input. In the case of a quadratic potential the PHS is an Ornstein-Uhlenbeck process for which we explicitly determine the distribution for any time t ≥ 0 and in the limit t → ∞. In particular, we characterize the collective motion by showing that the agents' positions tend to build exactly one cluster. This is confirmed in simulations that show rapid and coordinated motion among agents, driven by noise, despite the absence of a preferred direction of motion in the model. Remarkably, the theoretical properties observed in the Ornstein-Uhlenbeck process also emerge in simulations of the nonlinear model incorporating a general interaction potential. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse electromagnetic scattering problem for anisotropic media. We use the interior resonant modes to develop an inverse scattering scheme for imaging the scatterer. The whole procedure consists of three phases. First, we determine the interior Maxwell transmission eigenvalues of the scatterer from a family of far-field data by the mechanism of the linear sampling method. Next, we determine the corresponding transmission eigenfunctions by solving a constrained optimization problem. Finally, based on both global and local geometric properties of the transmission eigenfunctions, we design an imaging functional which can be used to determine the shape of the medium scatterer. We provide rigorous theoretical basis for our method. Numerical experiments verify the effectiveness, better accuracy and super-resolution results of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we present a numerical approach to solve the McKean–Vlasov equations, which are distribution-dependent stochastic differential equations, under some non-globally Lipschitz conditions for both the drift and diffusion coefficients. We establish a propagation of chaos result, based on which the McKean–Vlasov equation is approximated by an interacting particle system. A truncated Euler scheme is then proposed for the interacting particle system allowing for a Khasminskii- type condition on the coefficients. To reduce the computational cost, the random batch approximation proposed in [S. Jin, L. Li and J. Liu, J. Comput. Phys. 400 (2020) 108877.] is extended to the interacting particle system where the interaction could take place in the diffusion term. An almost half order of convergence is proved in Lp sense. Numerical tests are performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with plane wave discontinuous Galerkin (PWDG) methods for time-harmonic Maxwell equations in three-dimensional anisotropic media, for which the coefficients of the equations are piecewise constant symmetric matrices, where each constant symmetric matrix is defined on a medium (subdomain). By using suitable scaling transformations and coordinate (complex) transformations on every subdomain, the original Maxwell equation in anisotropic media is transformed into a Maxwell equation in isotropic media occupying a union domain of specific subdomains of complex Euclidean space. Based on these transformations, we define anisotropic plane wave basis functions and discretize the considered Maxwell equations by PWDG method with the proposed plane wave basis functions. We derive error estimates of the resulting approximate solutions, and further introduce a practically feasible local hp-refinement algorithm, which substantially improves accuracies of the approximate solutions. Numerical results indicate that the approximate solutions generated by the proposed PWDG methods possess high accuracy for the case of strong discontinuity media. [ABSTRACT FROM AUTHOR]

In this paper, we study numerical methods for solving a class of nonlinear backward stochastic partial differential equations. By utilizing finite element methods in space and θ-scheme in time, the proposed scheme forms a generalized spatio-temporal full discrete scheme, which can be solved in parallel. We rigorously prove the boundedness and error estimates, and obtain the optimal convergence rates in both time (first order/second order) and space (k + 1, k in L2 and H1, respectively). Numerical results are finally provided to demonstrate the effectiveness of the proposed scheme and validate the theoretical analyses. [ABSTRACT FROM AUTHOR]

The solutions of hyperbolic systems may contain discontinuities. These weak solutions verify not only the original PDEs, but also an entropy inequality that acts as a selection criterion determining whether a discontinuity is physical or not. Obtaining a discrete version of these entropy inequalities when approximating the solutions numerically is crucial to avoid convergence to unphysical solutions or even unstability. However such a task is difficult in general, if not impossible for schemes of order 2 or more. In this paper, we introduce an optimization framework that enables us to quantify a posteriori the decrease or increase of entropy of a given scheme, locally in space and time. We use it to obtain maps of numerical diffusion and to prove that some schemes do not have a discrete entropy inequality. A special attention is devoted to the widely used second order MUSCL scheme for which almost no theoretical results are known. [ABSTRACT FROM AUTHOR]

Although the stabilization technique is favorable in designing unconditionally energy stable or maximum-principle-preserving schemes for gradient flow systems, the induced time delay is intractable in computations. In this paper, we propose a class of delay-free stabilization schemes for the Allen–Cahn gradient flow system. Considering the Fourier pseudo-spectral spatial discretization for the Allen–Cahn equation with either the polynomial or the logarithmic potential, we establish a semi-discrete, mesh-dependent maximum principle by adopting a stabilization technique. To unconditionally preserve the mesh-dependent maximum principle and energy stability, we investigate a family of exponential time differencing Runge–Kutta (ETDRK) integrators up to the second-order. After reformulating the ETDRK schemes as a class of parametric Runge–Kutta integrators, we quantify the lagging effect brought by stabilization, and eliminate delayed convergence using a relaxation technique. The temporal error estimate of the relaxation ETDRK integrators in the maximum norm topology is analyzed under a fixed spatial mesh. Numerical experiments demonstrate the delay-free and structure-preserving properties of the proposed schemes. [ABSTRACT FROM AUTHOR]

One of the main difficulties that has to be faced with fictitious domain approximation of incompressible flows with immersed interfaces is related to the potential lack of mass conservation across the interfaces. In this paper, we propose and analyze a low order fictitious domain stabilized finite element method which mitigates this issue with the addition of a single velocity constraint. We provide a complete a priori numerical analysis of the method under minimal regularity assumptions. A comprehensive numerical study illustrates the capabilities of the proposed method, including comparisons with alternative fitted and unfitted mesh methods. [ABSTRACT FROM AUTHOR]

We consider the problem of attaining either the maximal increase or reduction of the robustness of a complex network by means of a bounded modification of a subset of the edge weights. We propose two novel strategies combining Krylov subspace approximations with a greedy scheme and an interior point method employing either the Hessian or its approximation computed via the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm (L-BFGS). The paper discusses the computational and modeling aspects of our methodology and illustrates the various optimization problems on networks that can be addressed within the proposed framework. Finally, in the numerical experiments we compare the performances of our algorithms with state-of-the-art techniques on synthetic and real-world networks. [ABSTRACT FROM AUTHOR]

In this paper we propose a stable and robust strategy to approximate the 3D incompressible hydrostatic Euler and Navier–Stokes systems with free surface. Compared to shallow water approximation of the Navier–Stokes system, the idea is to use a Galerkin type approximation of the velocity field with piecewise constant basis functions in order to obtain an accurate description of the vertical profile of the horizontal velocity. Such a strategy has several advantages. It allows to rewrite the Navier–Stokes equations under the form of a system of conservation laws with source terms, the easy handling of the free surface, which does not require moving meshes, the possibility to take advantage of robust and accurate numerical techniques developed in extensive amount for Shallow Water type systems. Compared to previous works of some of the authors, the three dimensional case is studied in this paper. We show that the model admits a kinetic interpretation including the vertical exchanges terms, and we use this result to formulate a robust finite volume scheme for its numerical approximation. All the aspects of the discrete scheme (fluxes, boundary conditions,...) are completely described and the stability properties of the proposed numerical scheme (well-balancing, positivity of the water depth,...) are discussed. We validate the model and the discrete scheme with some numerical academic examples (3D non stationary analytical solutions) and illustrate the capability of the discrete model to reproduce realistic tsunami waves propagation, tsunami runup and complex 3D hydrodynamics in a raceway. [ABSTRACT FROM AUTHOR]

This paper presents the lowest-order nonconforming immersed virtual element method for solving elliptic interface problems on unfitted polygonal meshes. The local discrete space on each interface mesh element consists of the solutions of local interface problems with Neumann boundary conditions, and the elliptic projection is modified so that its range is the space of broken linear polynomials satisfying the interface conditions. We derive optimal error estimates in the broken H1-norm and L2-norm, under the piecewise H2-regulartiy assumption. In our scheme, the mesh assumptions for error analysis allow small cut elements. Several numerical experiments are provided to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Allowing for space- and time-dependence of mass in Klein–Gordon equations resolves the problem of negative probability density and of violation of Lorenz covariance of interaction in quantum mechanics. Moreover it extends their applicability to the domain of quantum cosmology, where the variation in mass may be accompanied by high oscillations. In this paper we propose a third-order exponential integrator, where the main idea lies in embedding the oscillations triggered by the possibly highly oscillatory component intrinsically into the numerical discretisation. While typically high oscillation requires appropriately small time steps, an application of Filon methods allows implementation with large time steps even in the presence of very high oscillation. This greatly improves the efficiency of the time-stepping algorithm. Proof of the convergence and its rate are nontrivial and require alternative representation of the equation under consideration. We derive careful bounds on the growth of global error in time discretisation and prove that, contrary to standard intuition, the error of time integration does not grow once the frequency of oscillations increases. Several numerical simulations are presented to confirm the theoretical investigations and the robustness of the method in all oscillatory regimes. [ABSTRACT FROM AUTHOR]

The paper deals with the numerical modeling of two-phase flows while using Baer–Nunziato type models. Focus is given here on the numerical treatment of source terms that involve three (or four) relaxation time scales. A new coupled approach relying on the continuous analysis of the system of ODEs is compared with a more widely used strategy grounded on the fractional step approach. Properties of schemes are given in both cases. Several numerical applications show that the coupled approach should be preferred for both stability and accuracy reasons. [ABSTRACT FROM AUTHOR]

In this paper, we develop a unified framework for the a priori and a posteriori error control of different lowest-order finite element methods for approximating the regular solutions of systems of partial differential equations under a set of hypotheses. The systems involve cubic nonlinearities in lower order terms, non-homogeneous Dirichlet boundary conditions, and the results are established under minimal regularity assumptions on the exact solution. The key contributions include (i) results for existence and local uniqueness of the discrete solutions using Newton–Kantorovich theorem, (ii) a priori error estimates in the energy norm, and (iii) a posteriori error estimates that steer the adaptive refinement process. The results are applied to conforming, Nitsche, discontinuous Galerkin, and weakly over penalized symmetric interior penalty schemes for variational models of ferronematics and nematic liquid crystals. The theoretical estimates are corroborated by substantive numerical results. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a semi-discrete finite volume scheme for the three-dimensional barotropic compressible Euler equations driven by a multiplicative Brownian noise. We derive necessary a priori estimates for numerical approximations, and show that the Young measure generated by the numerical approximations converge to a dissipative measure-valued martingale solution to the stochastic compressible Euler system. These solutions are probabilistically weak in the sense that the driving noise and associated filtration are integral part of the solution. Moreover, we demonstrate strong convergence of numerical solutions to the regular solution of the limit systems at least on the lifespan of the latter, thanks to the weak (measure-valued)–strong uniqueness principle for the underlying system. To the best of our knowledge, this is the first attempt to prove the convergence of numerical approximations for the underlying system. [ABSTRACT FROM AUTHOR]

In the first part of the paper, we study the discontinuous Galerkin (DG) and C0 interior penalty (C0-IP) finite element approximation of the periodic strong solution to the fully nonlinear second-order Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation with coefficients satisfying the Cordes condition. We prove well-posedness and perform abstract a posteriori and a priori analyses which apply to a wide family of numerical schemes. These periodic problems arise as the corrector problems in the homogenization of HJBI equations. The second part of the paper focuses on the numerical approximation to the effective Hamiltonian of ergodic HJBI operators via DG/C0-IP finite element approximations to approximate corrector problems. Finally, we provide numerical experiments demonstrating the performance of the numerical schemes. [ABSTRACT FROM AUTHOR]

We consider an equilibrium version of a common two-fluid model for pipe flow, containing one mixture mass equation and one mixture energy equation. This model can be derived from a five-equation model with instantaneous thermal equilibrium, to which additional phase relaxation terms are added. An original contribution of this paper is a quasilinear formulation of the instantaneous phase relaxation limit. From this, the mixture sound speed intrinsic to the model can be extracted. This allows us to directly prove some subcharacteristic conditions with respect to a previously established model hierarchy of different relaxation processes. These subcharacteristic conditions reveal the fundamental insight of this paper; in the hierarchy, thermodynamic versus velocity relaxation both reduce the mixture sound velocity with a factor that is independent of whether the other type of relaxation has been performed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop numerical analysis for the two-dimensional peridynamics which depicts nonlocal phenomena with artificial boundary conditions (ABCs). To this end, the artificial boundary conditions for the fully discretized peridynamics are proposed. Then, the numerical analysis of the fully discretized scheme is developed such that the ABCs solve the corner reflection problem with second-order accuracy. Finally numerical examples are given to verify theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we analyze a homogeneous parabolic problem with initial data in the space of regular Borel measures. The problem is discretized in time with a discontinuous Galerkin scheme of arbitrary degree and in space with continuous finite elements of orders one or two. We show parabolic smoothing results for the continuous, semidiscrete and fully discrete problems. Our main results are interior L∞ error estimates for the evaluation at the endtime, in cases where the initial data is supported in a subdomain. In order to obtain these, we additionally show interior L∞ error estimates for L2 initial data and quadratic finite elements, which extends the corresponding result previously established by the authors for linear finite elements. [ABSTRACT FROM AUTHOR]

In this paper, we study the pattern formation of the Cucker–Smale type kinetic models. Two distributed Cucker–Smale type kinetic models for formation control are introduced based on gradient flow. We provide rigorous proof to prove that the above two kinetic models will achieve the desired position with the same velocity over a long time. In particular, the exponential convergence rate of the pattern formation on the corresponding particle models is obtained. Our analysis shows the gradient flow structure of the velocity field is important for establishing the convergence rate results of distributed control kinetic models. Finally, some numerical simulations are performed to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we analyze a lowest order virtual element method for the classic load reaction–convection–diffusion problem and the convection–diffusion spectral problem, where the assumptions on the polygonal meshes allow to consider small edges for the polygons. Under well defined seminorms depending on a suitable stabilization for this geometrical approach, we derive the well posedness of the numerical scheme and error estimates for the load problem, whereas for the spectral problem we derive convergence and error estimates fo the eigenvalues and eigenfunctions. We report numerical tests to asses the performance of the small edges on our numerical method for both problems under consideration. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a fully discrete finite element method for a constrained transport (CT) model of the incompressible magnetohydrodynamic (MHD) equations. The spatial discretization is based on mixed finite elements, where the hydrodynamic unknowns are approximated by stable finite element pairs, the magnetic field and magnetic vector potential are discretized by H(curl)-conforming edge element. The time marching is combining a backward Euler scheme and some subtle implicit-explicit treatments for nonlinear and coupling terms. With these treatments, the fully discrete scheme is linear in the implementation and the computation of the magnetic vector potential is decoupled from the whole coupled system. The most attractive feature of this scheme that it can yield the exactly divergence-free magnetic field and current density on the discrete level. The unique solvability and unconditional stability of the scheme are also proved rigorously. By utilizing the energy argument, error estimates for the velocity, magnetic field and magnetic vector potential are further demonstrated under the low regularity hypothesis for the exact solutions. Numerical results are provided to verify the theoretical analysis and to show the effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this note, we introduce a microscopic model for the motion of gas bubbles in a viscous fluid. By interpreting a bubble as a compressible fluid with infinite shear viscosity, we derive a pde/ode system coupling the density/velocity/pressure in the surrounding fluid with the linear/angular velocities and radii of the bubbles. We provide a 1D analogue of the system and construct an existence theory for this simplified system in a natural regularity framework. The second part of the paper is a preparatory work for the derivation of an averaged or macroscopic model. [ABSTRACT FROM AUTHOR]

In this paper we introduce and analyze a model of sedimentation based on a solid velocity formulation. A particular feature of the governing equations is given by the fact that the velocity field is non-divergence free. We introduce extra variables such as the pseudostress tensor relating the velocity gradient with the pressure, thus leading to a mixed variational formulation consisting of two systems of equations coupled through their source terms. A result of existence and uniqueness of solutions is shown by means of a fixed-point strategy and the help of the Babuška–Brezzi theory and Banach theorem. Additionally, we employ suitable finite dimensional subspaces to approximate both systems of equations via associated mixed finite element methods. The well-posedness of the resulting coupled scheme is also treated via a fixed-point approach, and hence the discrete version of the existence and uniqueness result is derived analogously to the continuous case. The above is then combined with a finite volume method for the transport equation. Finally, several numerical results illustrating the performance of the proposed model and the full numerical scheme, and confirming the theoretical rates of convergence, are presented. [ABSTRACT FROM AUTHOR]

In this paper, an important discovery has been found for nonconforming immersed finite element (IFE) methods using the integral values on edges as degrees of freedom for solving elliptic interface problems. We show that those IFE methods without penalties are not guaranteed to converge optimally if the tangential derivative of the exact solution and the jump of the coefficient are not zero on the interface. A nontrivial counter example is also provided to support our theoretical analysis. To recover the optimal convergence rates, we develop a new nonconforming IFE method with additional terms locally on interface edges. The new method is parameter-free which removes the limitation of the conventional partially penalized IFE method. We show the IFE basis functions are unisolvent on arbitrary triangles which is not considered in the literature. Furthermore, different from multipoint Taylor expansions, we derive the optimal approximation capabilities of both the Crouzeix–Raviart and the rotated-Q1 IFE spaces via a unified approach which can handle the case of variable coefficients easily. Finally, optimal error estimates in both H1- and L2-norms are proved and confirmed with numerical experiments. [ABSTRACT FROM AUTHOR]

The aim of this paper is to formulate and analyze numerical discretizations of charged-particle dynamics (CPD) in a strong nonuniform magnetic field. A strategy is firstly performed for the two dimensional CPD to construct the semi-discretization and full-discretization which have improved accuracy. This accuracy is improved in the position and in the velocity when the strength of the magnetic field becomes stronger. This is a better feature than the usual so called ''uniformly accurate methods". To obtain this refined accuracy, some reformulations of the problem and two-scale exponential integrators are incorporated, and the improved accuracy is derived from this new procedure. Then based on the strategy given for the two dimensional case, a new class of uniformly accurate methods with simple scheme is formulated for the three dimensional CPD in maximal ordering case. All the theoretical results of the accuracy are numerically illustrated by some numerical tests. [ABSTRACT FROM AUTHOR]

In this paper, we study a cancer invasion model both theoretically and numerically. The model is a nonstationary, nonlinear system of three coupled partial differential equations modeling the motion of cancer cells, degradation of the extracellular matrix, and certain enzymes. We first establish existence of global classical solutions in both two- and three-dimensional bounded domains, despite the lack of diffusion of the matrix-degrading enzymes and corresponding regularizing effects in the analytical treatment. Next, we give a weak formulation and apply finite differences in time and a Galerkin finite element scheme for spatial discretization. The overall algorithm is based on a fixed-point iteration scheme. Our theory and numerical developments are accompanied by some simulations in two and three spatial dimensions. [ABSTRACT FROM AUTHOR]

Schwarz waveform relaxation methods provide space-time parallelism for the solution of time dependent partial differential equations. The algorithms are differentiated by the choice of the transmission conditions enforced at the introduced space-time boundaries. Early results considered the theoretical analysis of these algorithms in the continuous and semi-discrete (in space) settings for various families of linear partial differential equations. Later, fully discrete results were obtained under the simplifying assumption of an infinite spatial domain. In this paper, we provide a first analysis of a fully discrete classical Schwarz Waveform algorithm for the one-dimensional heat equation on an arbitrary but finite number of bounded subdomains. The θ-method is chosen as the time integrator. Convergence results are given in both the infinity norm and two norm, with an explicit contraction given in the case of a uniform partitioning. The results are compared to the numerics and to the earlier theoretical results. [ABSTRACT FROM AUTHOR]

Starting from a reference partial differential equation model of the complex transverse water proton magnetization in a voxel due to diffusion-encoding magnetic field gradient pulses, one can use periodic homogenization theory to establish macroscopic models. A previous work introduced an asymptotic model that accounted for permeable interfaces in the imaging medium. In this paper we formulate a higher order asymptotic model to treat higher values of permeability. We explicitly solved this new asymptotic model to obtain a system of ordinary differential equations that can model the diffusion MRI signal and we present numerical results showing the improved accuracy of the new model in the regime of higher permeability. [ABSTRACT FROM AUTHOR]

The extracellular-membrane-intracellular (EMI) model consists in a set of Poisson equations in two adjacent domains, coupled on interfaces with nonlinear transmission conditions involving a system of ODEs. The unusual coupling of PDEs and ODEs on the boundary makes the EMI models challenging to solve numerically. In this paper, we reformulate the problem on the interface using a Steklov–Poincaré operator. We then discretize the model in space using a finite element method (FEM). We prove the existence of a semi-discrete solution using a reformulation as an ODE system on the interface. We derive stability and error estimates for the FEM. Finally, we propose a manufactured solution and use it to perform numerical tests. The order of convergence of the numerical method agrees with what is expected on the basis of the theoretical analysis of the convergence. [ABSTRACT FROM AUTHOR]