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It is demonstrated that anomalous transport at the plasma edge in tokamaks is essentially affected by poloidal inhomogeneities in the plasma temperature and density arising, e.g., by the formation of multifaceted asymmetric radiation from the edge at the density limit. [ABSTRACT FROM AUTHOR]

The transport of charged particles, which can be described by the Maxwell–Ampère Nernst–Planck (MANP) framework, is essential in various applications including ion channels and semiconductors. We propose a decoupled structure-preserving numerical scheme for the MANP model in this work. The Nernst-Planck equations are treated by the implicit exponential time differencing method associated with the Slotboom transform to preserve the positivity of the concentrations. In order to be effective with the Fast Fourier Transform, additional diffusive terms are introduced into Nernst–Planck equations. Meanwhile, the correction is introduced in the Maxwell–Ampère equation to fulfill Gauss’s law. The curl-free condition for electric displacement is realized by a local curl-free relaxation algorithm whose complexity is O(N). We present sufficient restrictions on the time and spatial steps to satisfy the positivity and energy dissipation law at a discrete level. Numerical experiments are conducted to validate the expected numerical accuracy and demonstrate the structure-preserving properties of the proposed method. [ABSTRACT FROM AUTHOR]

Dynamical low-rank approximation has become a valuable tool to perform an on-the-fly model order reduction for prohibitively large matrix differential equations. A core ingredient is the construction of integrators that are robust to the presence of small singular values and the resulting large time derivatives of the orthogonal factors in the low-rank matrix representation. Recently, the robust basis-update & Galerkin (BUG) class of integrators has been introduced. These methods require no steps that evolve the solution backward in time, often have favourable structure-preserving properties, and allow for parallel time-updates of the low-rank factors. The BUG framework is flexible enough to allow for adaptations to these and further requirements. However, the BUG methods presented so far have only first-order robust error bounds. This work proposes a second-order BUG integrator for dynamical low-rank approximation based on the midpoint quadrature rule. The integrator first performs a half-step with a first-order BUG integrator, followed by a Galerkin update with a suitably augmented basis. We prove a robust second-order error bound which in addition shows an improved dependence on the normal component of the vector field. These rigorous results are illustrated and complemented by a number of numerical experiments. [ABSTRACT FROM AUTHOR]

We propose an operator-splitting scheme to approximate scalar conservation equations with stiff source terms having multiple (at least two) stable equilibrium points. The scheme combines a (reaction-free) transport substep followed by a (transport-free) reaction substep. The transport substep is approximated using the forward Euler method with continuous finite elements and graph viscosity. The reaction substep is approximated using an exponential integrator. The crucial idea of the paper is to use a mesh-dependent cutoff of the reaction time-scale in the reaction substep. We establish a bound on the entropy residual motivating the design of the scheme. We show that the proposed scheme is invariant-domain preserving under the same CFL restriction on the time step as in the nonreactive case. Numerical experiments in one and two space dimensions using linear, convex, and nonconvex fluxes with smooth and nonsmooth initial data in various regimes show that the proposed scheme is asymptotic preserving. [ABSTRACT FROM AUTHOR]

We analyze the spectral and dynamical stability of solitary wave solutions to the Lugiato–Lefever equation on R . Our interest lies in solutions that arise through bifurcations from the phase-shifted bright soliton of the nonlinear Schrödinger equation. These solutions are highly nonlinear, localized, far-from-equilibrium waves, and are the physical relevant solutions to model Kerr frequency combs. We show that bifurcating solitary waves are spectrally stable when the phase angle satisfies θ ∈ (0 , π) , while unstable waves are found for angles θ ∈ (π , 2 π) . Furthermore, we establish asymptotic orbital stability of spectrally stable solitary waves against localized perturbations. Our analysis exploits the Lyapunov–Schmidt reduction method, the instability index count developed for linear Hamiltonian systems, and resolvent estimates. [ABSTRACT FROM AUTHOR]

For the charged-particle dynamics under a strong and constant magnetic field, we consider and analyze exponential integrators in this paper. We derive two kinds of exponential integrators and study their long-time behavior by using the technique of modulated Fourier expansions. It is shown that a symmetric two-stage exponential integrator and some one-stage symplectic exponential integrators approximately conserve the energy and magnetic moment of the charged-particle dynamics over long times. [ABSTRACT FROM AUTHOR]

Numerical methods for computing the solutions of Markov backward stochastic differential equations (BSDEs) driven by continuous-time Markov chains (CTMCs) are explored. The main contributions of this paper are as follows: (1) we observe that Euler-Maruyama temporal discretization methods for solving Markov BSDEs driven by CTMCs are equivalent to exponential integrators for solving the associated systems of ordinary differential equations (ODEs); (2) we introduce multi-stage Euler–Maruyama methods for effectively solving "stiff" Markov BSDEs driven by CTMCs; these BSDEs typically arise from the spatial discretization of Markov BSDEs driven by Brownian motion; (3) we propose a multilevel spatial discretization method on sparse grids that efficiently approximates high-dimensional Markov BSDEs driven by Brownian motion with a combination of multiple Markov BSDEs driven by CTMCs on grids with different resolutions. We also illustrate the effectiveness of the presented methods with a number of numerical experiments in which we treat nonlinear BSDEs arising from option pricing problems in finance. [ABSTRACT FROM AUTHOR]

In this article, we propose and analyze an energy stable, linear, second-order in time, exponential time differencing multi-step (ETD-MS) method for solving the Swift–Hohenberg equation with quadratic–cubic nonlinear term. The ETD-based explicit multi-step approximations and Fourier collocation spectral method are applied in time integration and spatial discretization of the corresponding equation, respectively. In particular, a second-order artificial stabilizing term, in the form of A τ 2 ∂ (Δ 2 + 1) u ∂ t , is added to ensure the energy stability. The long-time unconditional energy stability of the algorithm is established rigorously. In addition, error estimates in ℓ ∞ (0 , T ; ℓ 2) -norm are derived, with a careful estimate of the aliasing error. Numerical examples are carried out to verify the theoretical results. The long-time simulation demonstrates the stability and the efficiency of the numerical method. [ABSTRACT FROM AUTHOR]

We propose a new parallel discontinuous Galerkin method for the approximation of hyperbolic systems of conservation laws. The method remains stable with large time steps, while keeping the complexity of an explicit scheme: it does not require the assembly and resolution of large linear systems for the time iterations. The approach is based on a kinetic representation of the system of conservation laws previously investigated by the authors. Coulette et al. (in: Springer proceedings in mathematics & statistics, Springer, Berlin, pp 171–178, 2017. ), Badwaik et al. (ESAIM Proc Surv; 63:60–77, 2018. ), Coulette et al. (Comput Fluids; 190:485–502, 2019. ), Drui et al. (CR Mécanique; 347(3):259–269, 2019. ) and Gerhard et al. (Comput Math Appl; 112:116–137, 2022. . ). In this paper, the approach is extended with a subdomain strategy that improves the parallel scaling of the method on computers with distributed memory. [ABSTRACT FROM AUTHOR]

We consider the Davey–Stewartson system in the hyperbolic–elliptic case (DS-II) in two dimensional case. It is a mass-critical equation, and was proved recently by Nachman et al. (Invent Math 220(2):395–451, 2020) the global well-posedness and scattering in L 2 . In this paper, we give the numerical study on this model and construct a first order low-regularity integrator for the DS-II in the periodic case. It only requires the boundedness of one additional derivative of the solution to get the first order convergence. The Fast Fourier Transform is exploited to speed up the numerical implementation. By rigorous error analysis, we prove that the numerical scheme provides first order convergence in H γ (T 2) for rough initial data in H γ + 1 (T 2) with γ > 1 . The optimality of the convergence is conformed by numerical experience. [ABSTRACT FROM AUTHOR]

Parameter identification problems in PDEs are special class of nonlinear inverse problems which has many applications in science and technology. One such application is the Electrical Impedance Tomography (EIT) problem. Although many methods are available in literature to tackle nonlinear problems, the computation of Fréchet derivative is often a bottle neck for deriving the solution. Moreover, many assumptions are required to establish the convergence of such methods. In this paper, we propose a modified form of Levenberg–Marquardt scheme which does not require the knowledge of exact Fréchet derivative, instead, uses an approximate form of it and at the same time, no additional assumptions are required to establish the convergence of the scheme. We illustrate the theoretical result through numerical examples. In order to ensure that the proposed scheme can be applied to practical problems, we have applied the scheme to EIT problem and the reconstruction process clearly demonstrates that the method can be successfully applied to practical problems. [ABSTRACT FROM AUTHOR]

Betatron x-ray sources from laser-plasma interaction are characterized by compactness, ultrashort duration, broadband spectrum and micron source size. However, high-quality measurements with good statistics, especially in a single shot, require fluxes and energies beyond the current capabilities. Here, we propose a method to enhance the flux and brightness of the betatron sources without increasing the laser energy. By irradiating an edge of a microtape target with a femtosecond laser, a strong surface plasma wave (SPW) is excited at the edge and travels along the lateral plasma-vacuum interfaces. Tens of nC of electrons are peeled off and accelerated to superponderomotive energies by the longitudinal field of the SPW, whilst undergoing transverse betatron oscillations, leading to emission of hard x-rays. Via three-dimensional particle-in-cell simulations, we demonstrate that a tabletop 100 TW class femtosecond laser can produce an ultrabright hard x-ray pulse with flux up to 107 photons eV−1 and brilliance about 1023 photons s−1 mm−2 mrad−2 0.1%BW−1, paving the way for single-shot x-ray measurements in ultrafast science and high-energy-density physics. High-quality measurements with good statistics, especially in a single shot, require fluxes and energies beyond the current capabilities of laserbased betatron x-ray sources. Here, the authors propose a method to enhance the flux and brightness of such sources without increasing the laser energy, paving the way for single-shot x-ray measurements in ultrafast science. [ABSTRACT FROM AUTHOR]

Multiscale high-contrast media can cause astonishing wave propagation phenomena through resonance effects. For instance, waves could be exponentially damped independent of the incident angle or waves could be re-focused as through a lense. In this review article, we discuss the numerical treatment of wave propagation through multiscale high-contrast media at the example of the Helmholtz equation. First, we briefly summarize the findings of analytical homogenization theory, which inspire the design of numerical methods and indicate interesting regimes for simulation. In the main part, we discuss two different classes of numerical multiscale methods and focus on how to treat especially high-contrast media. Some elements of a priori error analysis are discussed as well. Various numerical simulations showcase the applicability of the numerical methods to explore unusual wave phenomena, for instance exponential damping and lensing with flat interfaces. [ABSTRACT FROM AUTHOR]

For the well known parareal algorithm, we formulate and analyse a novel class of parareal exponential schemes with improved uniform accuracy for highly oscillatory system q ¨ + 1 ε 2 M q = 1 ε μ f (q) with μ = 0 or 1. The solution of this considered system propagates waves with wavelength at O (ε) in time and the value of μ corresponds to the strength of nonlinearity. This brings significantly numerical burden in scientific computation for highly oscillatory systems with 0 < ε ≪ 1 . The new proposed algorithm is formulated by using some reformulation approaches to the problem, Fourier pseudo-spectral methods, and parareal exponential integrators. The fast Fourier transform is incorporated in the implementation. We rigorously study the convergence, showing that for nonlinear systems, the algorithm has improved uniform accuracy O (ε (2 k + 3) (1 - μ) Δ t 2 k + 2 + ε 5 (1 - μ) δ t 4) in the position and O (ε (2 k + 3) (1 - μ) - 1 Δ t 2 k + 2 + ε 4 - 5 μ δ t 4) in the momenta, where k is the number of parareal iterations, and Δ t and δ t are two time stepsizes used in the algorithm. The energy conservation is also discussed and the algorithm is shown to have an improved energy conservation. Numerical experiments are provided and the numerical results demonstrate the improved uniform accuracy and improved energy conservation of the obtained integrator through four Hamiltonian differential equations including nonlinear wave equations. [ABSTRACT FROM AUTHOR]

|$L^2$| norm error estimates of semi- and full discretizations of wave equations with dynamic boundary conditions, using bulk–surface finite elements and Runge–Kutta methods, are studied. The analysis rests on an abstract formulation and error estimates, via energy techniques, within this abstract setting. Four prototypical linear wave equations with dynamic boundary conditions are analysed, which fit into the abstract framework. For problems with velocity terms or with acoustic boundary conditions we prove surprising results: for such problems the spatial convergence order is shown to be less than 2. These can also be observed in the presented numerical experiments. [ABSTRACT FROM AUTHOR]

We present a family of novel explicit numerical methods for the diffusion or heat equation with Fisher, Huxley and Nagumo-type reaction terms. After discretizing the space variables as in conventional method of lines, our methods do not apply a finite difference approximation for the time derivatives, they instead combine constant- linear- and quadratic-neighbour approximations, which decouple the ordinary differential equations. In the obtained methods, the time step size appears in exponential form in the final expression with negative coefficients. In the case of the pure heat equation, the new values of the variable are convex combinations of the old values, which guarantees unconditional positivity and stability. We analytically prove that the convergence of the methods is fourth order in the time step size for linear ODE systems. We also prove that the concentration values in the case of Fisher's and Nagumo's equations lie within the unit interval regardless of the time step size. We construct an adaptive time step size time integrator with an extremely cheap embedded error control method. Several numerical examples are provided to demonstrate that the proposed methods work for nonlinear equations in stiff cases as well. According to the comparisons with other solvers, the new methods can have a significant advantage. [ABSTRACT FROM AUTHOR]

Exponential integrators have been applied successfully in several physics-related differential equations. However, their application in hyperbolic systems with absorbing boundaries, like the ones arising in seismic imaging, still lacks theoretical and experimental investigations. The present work conducts an in-depth study of exponential integration using Faber polynomials, consisting of a generalization of a well-known exponential method that uses Chebyshev polynomials. This allows solving non-symmetric operators that emerge from classic seismic wave propagation problems with absorbing boundaries. Theoretical as well as numerical results are presented for Faber approximations. One of the theoretical contributions is the proposal of a sharp bound for the approximation error of the exponential of a normal matrix. We also show the practical importance of determining an optimal ellipse encompassing the full spectrum of the discrete operator to ensure and enhance the convergence of the Faber exponential series. Furthermore, based on estimates of the spectrum of the discrete operator of the wave equations with a widely used absorbing boundary method, we numerically investigate the stability, dispersion, convergence, and computational efficiency of the Faber exponential scheme. Overall, we conclude that the method is suitable for seismic wave problems and can provide accurate results with large time step sizes, with computational efficiency increasing with the increase of the approximation degree. [ABSTRACT FROM AUTHOR]

We formulate variational material modeling in a space-time context. The starting point is the description of the space-time cylinder and the definition of a thermodynamically consistent Hamilton functional which accounts for all boundary conditions on the cylinder surface. From the mechanical perspective, the Hamilton principle then yields thermo-mechanically coupled models by evaluation of the stationarity conditions for all thermodynamic state variables which are displacements, internal variables, and temperature. Exemplary, we investigate in this contribution elastic wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage. Therein, one key novel aspect are initial and end time velocity conditions for the wave equation, replacing classical initial conditions for the displacements and the velocities. The motivation is intensively discussed and illustrated with the help of a prototype numerical simulation. From the mathematical perspective, the space-time formulations are formulated within suitable function spaces and convex sets. The unified presentation merges engineering and applied mathematics due to their mutual interactions. Specifically, the chosen models are of high interest in many state-of-the art developments in modeling and we show the impact of this holistic physical description on space-time Galerkin finite element discretization schemes. Finally, we study a specific discrete realization and show that the resulting system using initial and end time conditions is well-posed. [ABSTRACT FROM AUTHOR]

Hamiltonian Monte Carlo (HMC) algorithms, which combine numerical approximation of Hamiltonian dynamics on finite intervals with stochastic refreshment and Metropolis correction, are popular sampling schemes, but it is known that they may suffer from slow convergence in the continuous time limit. A recent paper of Bou-Rabee and Sanz-Serna (Ann Appl Prob, 27:2159-2194, 2017) demonstrated that this issue can be addressed by simply randomizing the duration parameter of the Hamiltonian paths. In this article, we use the same idea to enhance the sampling efficiency of a constrained version of HMC, with potential benefits in a variety of application settings. We demonstrate both the conservation of the stationary distribution and the ergodicity of the method. We also compare the performance of various schemes in numerical studies of model problems, including an application to high-dimensional covariance estimation. [ABSTRACT FROM AUTHOR]

This study examines the relationship between the concentration of atmospheric sodium and its meteoric input function (MIF). We use the measurements from the Colorado State University (CSU) and the Andes Lidar Observatory (ALO) lidar instruments with a new numerical model that includes sodium chemistry in the mesosphere and lower-thermosphere (MLT) region. The model is based on the continuity equation to treat all sodium-bearing species and runs at a high temporal resolution. The model simulation employs data assimilation to compare the MIF inferred from the meteor radiant distribution and the MIF derived from the new sodium chemistry model. The simulation captures the seasonal variability in the sodium number density compared with lidar observations over the CSU site. However, there were discrepancies for the ALO site, which is close to the South Atlantic Anomaly (SAA) region, indicating that it is challenging for the model to capture the observed sodium over the ALO. The CSU site had significantly more lidar observations (27 930 h) than the ALO site (1872 h). The simulation revealed that the uptake of the sodium species on meteoric smoke particles was a critical factor in determining the sodium concentration in the MLT, with the sodium removal rate by uptake found to be approximately 3 times that of the NaHCO 3 dimerization. Overall, the study's findings provide valuable information on the correlation between the MIF and the sodium concentration in the MLT region, contributing to a better understanding of the complex dynamics of this region. This knowledge can inform future research and guide the development of more accurate models to enhance our comprehension of the MLT region's behavior. [ABSTRACT FROM AUTHOR]

This paper analyses the long-time behaviour of one-stage symplectic or symmetric trigonometric integrators when applied to nonlinear wave equations. It is shown that energy, momentum, and all harmonic actions are approximately preserved over a long time for one-stage explicit symplectic or symmetric trigonometric integrators when applied to nonlinear wave equations via spectral semi-discretisations. For the long-term analysis of symplectic or symmetric trigonometric integrators, we derive a multi-frequency modulated Fourier expansion of the trigonometric integrator and show three almost-invariants of the modulation system. In the analysis of this paper, we neither assume symmetry for symplectic methods, nor assume symplecticity for symmetric methods. The results for symplectic and symmetric methods are obtained as a byproduct of the above analysis. We also give another proof by establishing a relationship between symplectic and symmetric trigonometric integrators and trigonometric integrators which have been researched for wave equations in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we present a computationally efficient technique based on the Method of Lines for the approximation of the Bermudan option values via the associated partial differential equations. The method of lines converts the Black Scholes partial differential equation to a system of ordinary differential equations. The solution of the system of ordinary differential equations so obtained only requires spatial discretization and avoids discretization in time. Additionally, the exact solution of the ordinary differential equations can be obtained efficiently using the exponential matrix operation, making the method computationally attractive and straightforward to implement. An essential advantage of the proposed approach is that the associated Greeks can be computed with minimal additional computations. We illustrate, through numerical experiments, the efficacy of the proposed method in pricing and computation of the sensitivities for a European call, cash-or-nothing, powered option, and Bermudan put option. [ABSTRACT FROM AUTHOR]

We develop two new sets of stable, rank-adaptive dynamically orthogonal Runge--Kutta (DORK) schemes that capture the high-order curvature of the nonlinear low-rank manifold. The DORK schemes asymptotically approximate the truncated singular value decomposition at a greatly reduced cost while preserving mode continuity using newly derived retractions. We show that arbitrarily high-order optimal perturbative retractions can be obtained, and we prove that these new retractions are stable. In addition, we demonstrate that repeatedly applying retractions yields a gradient-descent algorithm on the low-rank manifold that converges superlinearly when approximating a low-rank matrix. When approximating a higher-rank matrix, iterations converge linearly to the best low-rank approximation. We then develop a rank-adaptive retraction that is robust to overapproximation. Building off of these retractions, we derive two rank-adaptive integration schemes that dynamically update the subspace upon which the system dynamics are projected within each time step: the stable, optimal dynamically orthogonal Runge--Kutta (so-DORK) and gradient-descent dynamically orthogonal Runge--Kutta (gd-DORK) schemes. These integration schemes are numerically evaluated and compared on an ill-conditioned matrix differential equation, an advection-diffusion partial differential equation, and a nonlinear, stochastic reaction-diffusion partial differential equation. Results show a reduced error accumulation rate with the new stable, optimal and gradient-descent integrators. In addition, we find that rank adaptation allows for highly accurate solutions while preserving computational efficiency. [ABSTRACT FROM AUTHOR]

Materials can now be designed and architectured like structural components for targeted mechanical and physical properties. Structures and microstructures should not be studied independently and their design will benefit from a multiscale approach combining nonlinear continuum mechanics approaches and physical descriptions of elasticity, viscoplasticity, phase transformations and damage of microstructures, at various scales. The aim of the workshop was to gather outstanding junior and senior researchers in the various branches of mathematics, physics and engineering sciences suited to address the question of design of materials and structures by means of multiscale discrete and continuum approaches to their constitutive behavior. Examples include atomic or macroscopic lattices, random or periodic cellular materials, smart materials like shape memory alloys, 3D woven composites, acoustic and electromagnetic metamaterials, etc. Modern continuum mechanics relies on sophisticated constitutive laws for anisotropic materials exhibiting elastoviscoplastic behavior, still a field of intense research with new mathematical concepts. In particular size-dependent properties are addressed by resorting to generalized continua such as gradient or micromorphic and phase field models. The latter are attractive for the simulation of microstructure evolution coupled with mechanics, due to thermodynamic and metallurgical processes and damage. Scale transition and homogenization methods for continuous and discrete systems are required for the determination of effective material and structural behavior. Metamaterials are architectured materials specifically designed to achieve certain propagation and dispersion properties of elastic and plastic waves. Optimization strategies for the design of optimal architectures are involved in the design process. Target functions for optimization are now based on multicriteria (stiffness, strength, thermal expansion, transport properties, anisotropy etc.). [ABSTRACT FROM AUTHOR]

We propose explicit stochastic Runge–Kutta (RK) methods for high-dimensional Itô stochastic differential equations. By providing a linear error analysis and utilizing a Strang splitting-type approach, we construct them on the basis of orthogonal Runge–Kutta–Chebyshev methods of order 2. Our methods are of weak order 2 and have high computational accuracy for relatively large time-step size, as well as good stability properties. In addition, we take stochastic exponential RK methods of weak order 2 as competitors, and deal with implementation issues on Krylov subspace projection techniques for them. We carry out numerical experiments on a variety of linear and nonlinear problems to check the computational performance of the methods. As a result, it is shown that the proposed methods can be very effective on high-dimensional problems whose drift term has eigenvalues lying near the negative real axis and whose diffusion term does not have very large noise. [ABSTRACT FROM AUTHOR]

Computing has revolutionised the study of complex nonlinear systems, both by allowing us to solve previously intractable models and through the ability to visualise solutions in different ways. Using ubiquitous computing infrastructure, we provide a means to go one step further in using computers to understand complex models through instantaneous and interactive exploration. This ubiquitous infrastructure has enormous potential in education, outreach and research. Here, we present VisualPDE, an online, interactive solver for a broad class of 1D and 2D partial differential equation (PDE) systems. Abstract dynamical systems concepts such as symmetry-breaking instabilities, subcritical bifurcations and the role of initial data in multistable nonlinear models become much more intuitive when you can play with these models yourself, and immediately answer questions about how the system responds to changes in parameters, initial conditions, boundary conditions or even spatiotemporal forcing. Importantly, VisualPDE is freely available, open source and highly customisable. We give several examples in teaching, research and knowledge exchange, providing high-level discussions of how it may be employed in different settings. This includes designing web-based course materials structured around interactive simulations, or easily crafting specific simulations that can be shared with students or collaborators via a simple URL. We envisage VisualPDE becoming an invaluable resource for teaching and research in mathematical biology and beyond. We also hope that it inspires other efforts to make mathematics more interactive and accessible. [ABSTRACT FROM AUTHOR]

In this paper, two semidiscrete low regularity exponential-type integrators are proposed and analyzed for the "good" Boussinesq equation, including a first-order integrator and a second-order one. Compared to the existing numerical methods, the convergence rate can be achieved under weaker regularity assumptions on the exact solution. Specifically, the first-order integrator is convergent linearly in $H^r$ for solutions in $H^{r+1}$ if $r>1/2$ , i.e., the boundedness of one additional derivative of the solution is required to achieve the first-order convergence. When $r>7/6$ , we can even prove linear convergence in $H^r$ for solutions in $H^{r+2/3}$. What's more, half-order convergence is established in $H^{r}(r>3/2)$ for any solutions in $H^r$ , i.e., no additional smoothness assumptions are needed. For the second-order integrator, the quadratic convergence in $H^{r}$ $(r>1/2)$ (or $L^2$) is demonstrated, when the solutions belong to $H^{r+2}$ (or $H^{9/4}$). Numerical examples illustrating the convergence analysis are included. A comparison with other methods demonstrates the superiority of the newly proposed exponential-type integrators for rough data. [ABSTRACT FROM AUTHOR]

We derive a numerical method, based on operator splitting, to abstract parabolic semilinear boundary coupled systems. The method decouples the linear components that describe the coupling and the dynamics in the abstract bulk- and surface-spaces, and treats the nonlinear terms similarly to an exponential integrator. The convergence proof is based on estimates for a recursive formulation of the error, using the parabolic smoothing property of analytic semigroups, and a careful comparison of the exact and approximate flows. This analysis also requires a deep understanding of the effects of the Dirichlet operator (the abstract version of the harmonic extension operator), which is essential for the stable coupling in our method. Numerical experiments, including problems with dynamic boundary conditions, reporting on convergence rates are presented. [ABSTRACT FROM AUTHOR]

A new harmonic analysis technique using the Littlewood–Paley dyadic decomposition is developed for constructing low-regularity integrators for the one-dimensional cubic nonlinear Schrödinger equation in a bounded domain under Neumann boundary condition, when the frequency analysis based on the Fourier series cannot be used. In particular, a low-regularity integrator is constructively designed through the consistency analysis by the Littlewood–Paley decomposition of the solution, in order to have almost first-order convergence (up to a logarithmic factor) in the $L^{2}$ norm for $H^{1}$ initial data. A spectral method in space, using fast Fourier transforms with $\mathcal{O}(N\ln N)$ operations at every time level, is constructed without requiring any Courant-Friedrichs-Lewy (CFL) condition, where $N$ is the degrees of freedom in the spatial discretization. The proposed fully discrete method is proved to have an $L^{2}$ -norm error bound of $\mathcal{O}(\tau [\ln (1/\tau)]^{2}+ N^{-1})$ for $H^{1}$ initial data, where $\tau $ is the time-step size. [ABSTRACT FROM AUTHOR]

A flexible extended Krylov subspace method (F -EKSM) is considered for numerical approximation of the action of a matrix function f (A) to a vector b, where the function f is of Markov type. F -EKSM has the same framework as the extended Krylov subspace method (EKSM), replacing the zero pole in EKSM with a properly chosen fixed nonzero pole. For symmetric positive definite matrices, the optimal fixed pole is derived for F -EKSM to achieve the lowest possible upper bound on the asymptotic convergence factor, which is lower than that of EKSM. The analysis is based on properties of Faber polynomials of A and (I − A / s) − 1 . For large and sparse matrices that can be handled efficiently by LU factorizations, numerical experiments show that F -EKSM and a variant of RKSM based on a small number of fixed poles outperform EKSM in both storage and runtime, and usually have advantages over adaptive RKSM in runtime. [ABSTRACT FROM AUTHOR]

The convective Allen–Cahn (CAC) equation has been widely used for simulating multiphase flows of incompressible fluids, which contains an extra convective term but still maintains the same maximum bound principle (MBP) as the classic Allen–Cahn equation. Based on the operator splitting approach, we propose a second-order semi-Lagrangian exponential time differencing method for solving the CAC equation, that preserves the discrete MBP unconditionally. In our scheme, the AC equation part is first spatially discretized via the central finite difference scheme, then it is efficiently solved by using the exponential time differencing method with FFT-based fast implementation. The transport equation part is computed by combining the semi-Lagrangian approach with a cut-off post-processing within the finite difference framework. MBP stability and convergence analysis of our fully discretized scheme are presented. In particular, we conduct an improved error estimation for the semi-Lagrangian method with variable velocity, so that the error of our scheme is not spoiled by the reciprocal of the time step size. Extensive numerical tests in two and three dimensions are also carried out to validate the theoretical results and demonstrate the performance of our scheme. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new scheme for the integration of the periodic nonlinear Schrödinger equation and rigorously prove convergence rates at low regularity. The new integrator has decisive advantages over standard schemes at low regularity. In particular, it is able to handle initial data in Hs for 0 < s ≤ 1. The key feature of the integrator is its ability to distinguish between low and medium frequencies in the solution and to treat them differently in the discretization. This new approach requires a well-balanced filtering procedure which is carried out in Fourier space. The convergence analysis of the proposed scheme is based on discrete (in time) Bourgain space estimates which we introduce in this paper. A numerical experiment illustrates the superiority of the new integrator over standard schemes for rough initial data. [ABSTRACT FROM AUTHOR]

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This paper considers an effective numerical collocation method for numerical solution of the KdV–Kawahara equation. This numerical method relies on a finite element formulation and spline interpolation with a trigonometric quintic B-spline basis. First, the KdV–Kawahara equation is reduced to a coupled equation via an auxiliary variable of the form . The collocation method is then applied to the coupled equation together with the forward difference and the Crank–Nicholson formula. This results in a system of algebraic equations in terms of time variables with the trigonometric quintic B-spline basis. For determination of the error between the numerical and exact solutions, the error norms and are calculated. The results are illustrated by two numerical examples with their graphical representation and comparison with other methods. [ABSTRACT FROM AUTHOR]

We apply the ultraspherical spectral method to solving time-dependent PDEs by proposing two approaches to discretization based on the method of lines and show that these approaches produce approximately same results. We analyze the stability, the error, and the computational cost of the proposed method. In addition, we show how adaptivity can be incorporated to offer adequate spatial resolution efficiently. Both linear and nonlinear problems are considered. We also explore time integration using exponential integrators with the ultraspherical spatial discretization. Comparisons with the Chebyshev pseudospectral method are given along the discussion and they show that the ultraspherical spectral method is a competitive candidate for the spatial discretization of time-dependent PDEs. [ABSTRACT FROM AUTHOR]

The tensor t-function, a formalism that generalizes the well-known concept of matrix functions to third-order tensors, is introduced in Lund (Numer Linear Algebra Appl 27(3):e2288). In this work, we investigate properties of the Fréchet derivative of the tensor t-function and derive algorithms for its efficient numerical computation. Applications in condition number estimation and nuclear norm minimization are explored. Numerical experiments implemented by the t-Frechet toolbox hosted at https://gitlab.com/katlund/t-frechet illustrate properties of the t-function Fréchet derivative, as well as the efficiency and accuracy of the proposed algorithms. [ABSTRACT FROM AUTHOR]

We present an implementation of the multiconfiguration time-dependent Hartree–Fock method based on the adaptive finite element method for molecules under intense laser pulses. For efficient simulations, orbital functions are propagated by a stable propagator using the short iterative Arnoldi scheme and our implementation is parallelized for distributed memory computing. This is demonstrated by simulating high-harmonic generation from a water molecule and achieves a simulation of multielectron dynamics with overwhelmingly less computational time, compared to our previous work [Sawada, R. et al. Phys. Rev. A, 2016, 93, 023434]. [ABSTRACT FROM AUTHOR]

We reviewed the time-dependent multiconfiguration self-consistent-field (TD-MCSCF) method and the time-dependent optimized coupled-cluster (TD-OCC) method for first-principles simulations of high-field phenomena such as tunneling ionization and high-order harmonic generation in atoms and molecules irradiated by a strong laser field. These methods provide a flexible and systematically improvable description of the multielectron dynamics by expressing the all-electron wavefunction by configuration interaction expansion or coupled-cluster expansion, using time-dependent one-electron orbital functions. The time-dependent variational principle plays a key role in deriving these methods while satisfying gauge invariance and Ehrenfest theorem. The real-time/real-space implementation with an absorbing boundary condition enables the simulation of high-field processes involving multiple excitation and ionization. We present a detailed, comprehensive discussion of such features of the TD-MCSCF and TD-OCC methods. [ABSTRACT FROM AUTHOR]

A large class of differential Riccati equations (DREs) satisfy positivity property in the sense that the time-dependent solution preserves for any time its symmetric and positive semidefinite structure. This positivity property plays a crucial role in understanding the wellposedness of the DRE, and whether it could be inherited in the discrete level is a significant issue in numerical simulations. In this paper, we study positivity preserving time integration schemes by means of exponential integrators. The proposed exponential Euler and exponential midpoint schemes are linear and proven to be positivity preserving and unconditionally stable. Sharp error estimates of the schemes are also obtained. Numerical experiments are carried out to illustrate the performance of the proposed integrators. [ABSTRACT FROM AUTHOR]

Integral representations of solutions of linear multiplicatively perturbed differential equations are obtained, the diffusion part of which is bilinear on the state vector and the vector of independent Wiener processes. Equations of such class serve as models of stochastic systems with control functioning under conditions of parametric uncertainty or undesirable influence of external disturbances. The concepts and analytical apparatus of the theory of Lie algebras are used to find integral representations and fundamental matrices of the equations. [ABSTRACT FROM AUTHOR]

We consider Kerr frequency combs in a dual-pumped microresonator as time-periodic and spatially 2 π -periodic traveling wave solutions of a variant of the Lugiato–Lefever equation, which is a damped, detuned and driven nonlinear Schrödinger equation given by i a τ = (ζ - i) a - d a xx - | a | 2 a + i f 0 + i f 1 e i (k 1 x - ν 1 τ) . The main new feature of the problem is the specific form of the source term f 0 + f 1 e i (k 1 x - ν 1 τ) which describes the simultaneous pumping of two different modes with mode indices k 0 = 0 and k 1 ∈ N . We prove existence and uniqueness theorems for these traveling waves based on a-priori bounds and fixed point theorems. Moreover, by using the implicit function theorem and bifurcation theory, we show how non-degenerate solutions from the 1-mode case, i.e., f 1 = 0 , can be continued into the range f 1 ≠ 0 . Our analytical findings apply both for anomalous ( d > 0 ) and normal ( d < 0 ) dispersion, and they are illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]

We study linear time-dependent Maxwell equations on a cuboid consisting of two homogeneous subcuboids. At the interface, we allow for nonzero surface charge density and surface current. This model is a first step towards a detailed mathematical analysis of the interaction of single-layer materials with electromagnetic fields. The main results of this paper provide several wellposedness and regularity statements for the solutions of the Maxwell system. To prove the statements, we employ extension arguments using interpolation theory, as well as semigroup theory and regularity theory for elliptic transmission problems. [ABSTRACT FROM AUTHOR]

We describe a proof‐of‐concept development and application of a phase‐averaging technique to the nonlinear rotating shallow‐water equations on the sphere, discretised using compatible finite‐element methods. Phase averaging consists of averaging the nonlinearity over phase shifts in the exponential of the linear wave operator. Phase averaging aims to capture the slow dynamics in a solution that is smoother in time (in transformed variables), so that larger timesteps may be taken. We overcome the two key technical challenges that stand in the way of studying the phase averaging and advancing its implementation: (1) we have developed a stable matrix exponential specific to finite elements and (2) we have developed a parallel finite averaging procedure. Following recent studies, we consider finite‐width phase‐averaging windows, since the equations have a finite timescale separation. In our numerical implementation, the averaging integral is replaced by a Riemann sum, where each term can be evaluated in parallel. This creates an opportunity for parallelism in the timestepping method, which we use here to compute our solutions. Here, we focus on the stability and accuracy of the numerical solution. We confirm that there is an optimal averaging window, in agreement with theory. Critically, we observe that the combined time discretisation and averaging error is much smaller than the time discretisation error in a semi‐implicit method applied to the same spatial discretisation. An evaluation of the parallel aspects will follow in later work. [ABSTRACT FROM AUTHOR]