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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]

The matrix-valued Allen-Cahn (MAC) equation was first introduced as a model problem of finding the stationary points of an energy for orthogonal matrix-valued functions and has attracted much attention in recent years. It is well known that the MAC equation satisfies the maximum bound principle (MBP) with respect to either the matrix 2-norm or the Frobenius norm, which plays a key role in understanding the physical meaning and the wellposedness of the model. To preserve this property, we extend the explicit integrating factor Runge-Kutta (IFRK) method to the MAC equation. Moreover, we construct a new three-stage third-order and a new four-stage fourth-order IFRK schemes based on the classical Runge-Kutta schemes. Under a reasonable time-step constraint, we prove the MBP preservation of the IFRK method with respect to the matrix 2-norm, based on which, we further establish their optimal error estimates in the matrix 2-norm. Although numerical results indicate that the IFRK method preserves the MBP with respect to the Frobenius norm, a detailed analysis shows that it is hard to prove this preservation by using the same approach for the case of 2-norm. Several numerical experiments are carried out to test the convergence of the IFRK schemes and to verify the MBP preservation with respect to the matrix 2-norm and the Frobenius norm, respectively. Energy stability is also observed, which clearly indicates the orthogonality of the stationary solution of the MAC equation. In addition, we simulate the coarsening dynamics to verify the motion law of the interface for different initial conditions. [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]

This paper proposes a framework for constructing high-order linearly implicit exponential integrators that conserve a quadratic invariant. This is then applied to the scalar auxiliary variable (SAV) approach. Quadratic invariants are significant objects that are present in various physical equations and also in computationally efficient conservative schemes for general invariants. For instance, the SAV approach converts the invariant into a quadratic form by introducing scalar auxiliary variables, which have been intensively studied in recent years. In this vein, Sato et al. (Appl. Numer. Math. 187, 71-88 2023) proposed high-order linearly implicit schemes that conserve a quadratic invariant. In this study, it is shown that their method can be effectively merged with the Lawson transformation, a technique commonly utilized in the construction of exponential integrators. It is also demonstrated that combining the constructed exponential integrators and the SAV approach yields schemes that are computationally less expensive. Specifically, the main part of the computational cost is the product of several matrix exponentials and vectors, which are parallelizable. Moreover, we conduct some mathematical analyses on the proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new framework for deriving partitioned implicit-exponential integrators for stiff systems of ordinary differential equations and construct several time integrators of this type. The new approach is suited for solving systems of equations where the forcing term is comprised of several additive nonlinear terms. We analyze the stability, convergence, and efficiency of the new integrators and compare their performance with existing schemes for such systems using several numerical examples. We also propose a novel approach to visualizing the linear stability of the partitioned schemes, which provides a more intuitive way to understand and compare the stability properties of various schemes. Our new integrators are A-stable, second-order methods that require only one call to the linear system solver and one exponential-like matrix function evaluation per time step. [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]

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]

In this paper, we study a new splitting method for the semi-linear system of ordinary differential equation, where the linear part is stiff. Firstly, the stiff part is split into two parts. The first stiff part, that is called the stiff-cutter and expected to be easily inverted, is implicitly treated. The second stiff part and the remaining nonlinear part are explicitly treated. Therefore, such stiff-cut method can be fast implemented and save the CPU time. Theoretically, we rigorously prove that the proposed method is unconditionally stable and convergent, if the stiff-cutter is chosen to be well-matched in the stiff part. As an application, we apply our method to solve a spatial-fractional reaction-diffusion equation and give a way for how to choose a suitable stiff-cutter. Finally, numerical experiments are carried out to illustrate the accuracy and efficiency of the proposed stiff-cut method. [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]

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 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 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]

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]

We establish a class of uniformly accurate nested Picard iterative integrator (NPI) Fourier pseudospectral methods for the nonlinear Klein-Gordon-Schrödinger equation (KGS) in the nonrelativistic regime, involving a dimensionless parameter ε ≪ 1 inversely proportional to the speed of light. Actually, the solution propagates waves in time with O(ε2) wavelength when 0 < ε ≪ 1, which brings significant difficulty in designing accurate and efficient numerical schemes. The NPI method is designed by separating the oscillatory part from the non-oscillatory part, and integrating the former exactly. Based on the Picard iteration, the NPI method can be applied to derive arbitrary higher-order methods in time with optimal and uniform accuracy (w.r.t. ε ∈ (0,1]), and the corresponding error estimates are rigorously established. In addition, the practical implementation of the second-order NPI method via Fourier pseupospectral discretization is clearly demonstrated, with extensions to the third-order NPI. Some numerical examples are provided to support our theoretical results and show the accuracy and efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

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]

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]

Due to the seminal works of Hochbruck and Ostermann (Appl Numer Math 53(2–4):323–339, 2005, Acta Numer 19:209–286, 2010) exponential splittings are well established numerical methods utilizing operator semigroup theory for the treatment of semilinear evolution equations whose principal linear part involves a sectorial operator with angle greater than π 2 (meaning essentially the holomorphy of the underlying semigroup). The present paper contributes to this subject by relaxing the sectoriality condition, but in turn requiring that the semigroup operators act consistently on an interpolation couple (or on a scale of Banach spaces). Our conditions (on the semigroup and on the semilinearity) are inspired by the approach of Kato (Math Z 187(4):471–480, 1984) to the local solvability of the Navier–Stokes equation, where the L p - L r -smoothing of the Stokes semigroup was fundamental. The present abstract operator theoretic result is applicable for this latter problem (as was already the result of Hochbruck and Ostermann), or more generally in the setting of Hochbruck and Ostermann (2005), but also allows the consideration of examples, such as non-analytic Ornstein–Uhlenbeck semigroups or the Navier–Stokes flow around rotating bodies. [ABSTRACT FROM AUTHOR]

Structure-preserving algorithms and algorithms with uniform error bound have constituted two interesting classes of numerical methods. In this paper, we blend these two kinds of methods for solving nonlinear systems with highly oscillatory solution, and the blended algorithms inherit and respect the advantage of each method. Two kinds of algorithms are presented to preserve the symplecticity and energy of the Hamiltonian systems, respectively. Long time energy conservation is analysed for symplectic algorithms and the proposed algorithms are shown to have uniform error bound in the position for the highly oscillatory structure. Moreover, some methods with uniform error bound in the position and in the velocity are derived and analysed. Two numerical experiments are carried out to support all the theoretical results established in this paper by showing the performance of the blended algorithms. [ABSTRACT FROM AUTHOR]

The aim of this work is the numerical homogenization of a parabolic problem with several time and spatial scales using the heterogeneous multiscale method. We replace the actual cell problem with an alternate one, using Dirichlet boundary and initial values instead of periodic boundary and time conditions. Further, we give a detailed a priori error analysis of the fully discretized, i.e., in space and time for both the macroscopic and the cell problem, method. Numerical experiments illustrate the theoretical convergence rates. [ABSTRACT FROM AUTHOR]

Solving the nonlinear Dirac equation in the nonrelativistic limit regime numerically is difficult, because the solution oscillates in time with frequency of O ε - 2 , where 0 < ε ≪ 1 is inversely proportional to the speed of light. Yongyong Cai and Yan Wang have shown, however, that such solutions can be approximated up to an error of O ε 2 by solving the semi-nonrelativistic limit system, which is a non-oscillatory problem. For this system, we construct a two-step method, called the explicit exponential midpoint rule, and prove second-order convergence of the semi-discretization in time. Furthermore, we construct a benchmark method based on standard techniques and compare the efficiency of both methods. Numerical experiments show that the new integrator reduces the computational costs per time step to 40% and within a given runtime improves the accuracy significantly. [ABSTRACT FROM AUTHOR]

Recently, a so-called direct method has been developed for solving 2D Maxwell’s equations in Kerr-type nonlinear media. This method is free of iteration error and more efficient than classical iterative method. We investigate this method from a theoretical point of view by proving the stability of 3D Maxwell’s equations. Numerical results have been achieved to verify the second-order convergence rate in both time and space. [ABSTRACT FROM AUTHOR]

Many chemical systems exhibit a range of patterns, a noticeable and interesting class of numerical patterns that arise in autocatalytic reactions which changes with increasing spatial domains. In this paper, autocatalytic spatiotemporal patterns were demonstrated using the system of chemical species modeled with the time-fractional Caputo derivatives of subdiffusive orders. It is not new that a spectral algorithm with its entire nature is more accurate when compared with a finite-difference scheme to solve a range of integer and non-integer order time partial differential equations. This is because the Fourier spectral techniques have the upper hand on high-order spectral accuracy, and are computationally efficient. Hence, it is regarded as the best approach to existing lower-order methods for integrating the second-order partial derivatives in space. This motivates the present study to explore the usefulness of Fourier spectral methods in resolving and obtaining complex Turing patterns arising from nonlinear fractional autocatalytic reaction-diffusion problems in high dimensions. The autocatalysis model was examines for linear stability in an attempt to obtain the correct choice of parameters that are likely to lead to the formation of new complex turing-like patterns. Numerical experiments in the 2D lead to a striking range of patterns arising from catalytic reactions of fractional-order labyrinthine pattern-like structures. Analysis of pattern formation was also extended to 3D dynamics to obtain a new set of patterns like a star-, cyclic-, diamond-like, and the emergence of apple-shaped structures, which are greatly influenced by either the choice parameters that are involved or that of the initial conditions. [ABSTRACT FROM AUTHOR]

In this study, we introduce a numerical method to reconstruct, from posterior measurements of the solution, the initial condition in the one-dimensional heat conduction problem. The method uses a wavelet-based Galerkin method for the spatial discretization and spectral decomposition of the basis stiffness matrix to get the numerical solution without time discretization as in classical approaches. In fact, according to the proposed method, setting an error bound and properly selecting the eigenvalues of the discretization system, we are able to reconstruct the initial conditions without exceeding the required error. The applicability and computational efficiency of the methods are investigated by solving some numerical examples with toy solutions and the experimental results confirm its accuracy. [ABSTRACT FROM AUTHOR]

In this work, we consider the numerical solutions of a dispersion-managed nonlinear Schrödinger equation (DM-NLS) and a nonlinearity-managed NLS equation (NM-NLS). The two equations arise from the soliton managements in optics and matter waves, and they involve temporal discontinuous coefficients with possible frequent jumps and stiffness which cause numerical difficulties. We analyze to see the order reduction problems of some popular traditional methods, and then we propose a class of exponential-type dispersion-map integrators for both DM-NLS and NM-NLS. The proposed methods are explicit, efficient under Fourier pseudospectral discretizations and second order accurate in time regardless the jumps/jump-period in the dispersion map. The extension to the fast & strong management regime of DM-NLS is made with uniform accuracy. [ABSTRACT FROM AUTHOR]

In this manuscript, we present a common tensor framework which can be used to generalize one-dimensional numerical tasks to arbitrary dimension d by means of tensor product formulas. This is useful, for example, in the context of multivariate interpolation, multidimensional function approximation using pseudospectral expansions and solution of stiff differential equations on tensor product domains. The key point to obtain an efficient-to-implement BLAS formulation consists in the suitable usage of the μ-mode product (also known as tensor-matrix product or mode-n product) and related operations, such as the Tucker operator. Their MathWorks MATLAB®/GNU Octave implementations are discussed in the paper, and collected in the package KronPACK. We present numerical results on experiments up to dimension six from different fields of numerical analysis, which show the effectiveness of the approach. [ABSTRACT FROM AUTHOR]

In this paper, we construct and analyze a new dynamical low-rank integrator for second-order matrix differential equations. The method is based on a combination of the projector-splitting integrator introduced in Lubich and Oseledets (BIT 54(1):171–188, 2014. ) and a Strang splitting. We also present a variant of the new integrator which is tailored to semilinear second-order problems. [ABSTRACT FROM AUTHOR]

In this paper, we construct and analyze a new dynamical low-rank integrator for second-order matrix differential equations. The method is based on a combination of the projector-splitting integrator introduced in Lubich and Oseledets (BIT 54(1):171–188, 2014. ) and a Strang splitting. We also present a variant of the new integrator which is tailored to semilinear second-order problems. [ABSTRACT FROM AUTHOR]

Dynamical low-rank integrators for matrix differential equations recently attracted a lot of attention and have proven to be very efficient in various applications. In this paper, we propose a novel strategy for choosing the rank of the projector-splitting integrator of Lubich and Oseledets adaptively. It is based on a combination of error estimators for the local time-discretization error and for the low-rank error with the aim to balance both. This ensures that the convergence of the underlying time integrator is preserved. The adaptive algorithm works for projector-splitting integrator methods for first-order matrix differential equations and also for dynamical low-rank integrators for second-order equations, which use the projector-splitting integrator method in its substeps. Numerical experiments illustrate the performance of the new integrators. [ABSTRACT FROM AUTHOR]

In this paper, one-stage explicit trigonometric integrators for solving quasilinear wave equations are formulated and studied. For solving wave equations, we first introduce trigonometric integrators as the semidiscretization in time and then consider a spectral Galerkin method for the discretization in space. We show that one-stage explicit trigonometric integrators in time have second-order convergence and the result is also true for the fully discrete scheme without requiring any CFL-type coupling of the discretization parameters. The results are proved by using energy techniques, which are widely applied in the numerical analysis of methods for partial differential equations. [ABSTRACT FROM AUTHOR]

Exponential integrators, due to their robust stability properties, have been considered as reliable schemes for numerical solutions of stiff systems. In this paper, we propose generalized exponential time differencing (GETD) schemes for nonlinear fractional differential equations of order α ∈ (0,1). First, we improve the suboptimal performance of the multistep GETD schemes. Using graded mesh, uniform optimal convergence rates under no additional smoothness requirements are obtained. Second, we develop and analyze novel second-order and third-order accurate predictor-corrector type GETD schemes. Using linear stability analysis and numerical illustrations, we demonstrate that the newly introduced schemes have better stability properties than the multistep GETD schemes. Partial fraction decompositions of global Padé approximations for Mittag-Leffler function are used for efficient implementation. Numerical examples involving nonlinear scalar equations and stiff systems are provided to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

For the Dirac equation with potentials characterized by a small parameter ε ∈ (0,1], the numerical methods for long-time dynamics have received more and more attention. Recently, two exponential wave integrator Fourier pseudo-spectral (EWIFP) methods for the Dirac equation have been proposed (Feng et al., Appl. Numer. Math. 172, 50–66, 2022) which are uniformly accurate about ε and perform well over the classical methods. However, the EWIFP methods cannot preserve the mass and energy, which are important structural features of the Dirac equation from the perspective of geometric numerical integration. In addition, the EWIFP methods are not time symmetric or only are conditionally stable under specific stability condition which implies CFL condition restrictions on the grid ratio. In this work, we propose a structure-preserving EWIFP (SPEWIFP) method. The proposed method is proved to be time symmetric, stable only under the condition τ ≲ 1 , and preserves the discrete energy and modified mass. Without any CFL condition restrictions on the grid ratio, we carry out a rigourously error analysis and give uniform error bounds of the method at O (h m 0 + ε 1 − β τ 2) up to the time at O(1/εβ) with β ∈ [0,1], mesh size h, time step τ, and an integer m0 determined by the regularity conditions. In general, the Dirac equation with small potentials can be converted to an oscillatory Dirac equation with wavelength at O(εβ) in time which includes the case of simultaneous massless and nonrelativistic regime. It is easy to extend the error bounds and structure-preservation properties to the oscillatory Dirac equation. Numerical experiments support our error bounds and structure-preservation properties. [ABSTRACT FROM AUTHOR]

This paper is concerned with Krylov subspace methods based on the tensor t-product for computing certain quantities associated with generalized third-order tensor functions. We use the tensor t-product and define the tensor global Golub-Kahan bidiagonalization process for approximating tensor functions. Pairs of Gauss and Gauss-Radau quadrature rules are applied to determine the desired quantities with error bounds. An application to the computation of the tensor nuclear norm is presented and illustrates the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

The solution of linear non-autonomous ordinary differential equation systems (also known as the time-ordered exponential) is a computationally challenging problem arising in a variety of applications. In this work, we present and study a new framework for the computation of bilinear forms involving the time-ordered exponential. Such a framework is based on an extension of the non-Hermitian Lanczos algorithm to 4-mode tensors. Detailed results concerning its theoretical properties are presented. Moreover, computational results performed on real-world problems confirm the effectiveness of our approach. [ABSTRACT FROM AUTHOR]