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The variable-step deferred correction method constructs high-order schemes by modifying the low-order schemes, which not only relaxes the strict step-ratio constraints related to higher-order schemes, but also more accurately captures multi-scale evolution processes compared to lower-order schemes. It also provides internal error estimators to adjust the step sizes in adaptive time-stepping algorithms. Two third-order deferred correction methods based on the variable-step second-order BDF formula are analyzed for the Cahn–Hilliard model. The stability of the proposed variable-step deferred correction schemes is established under the step-ratio restriction of 0 < r k < 4.864 . By utilizing the discrete orthogonal convolution kernels and some discrete convolution embedding inequalities, the modified energy dissipation law and the L 2 norm error estimate at the discrete levels are established. Numerical experiments validate the effectiveness of our approaches. [ABSTRACT FROM AUTHOR]

An adaptive time-stepping scheme is developed for the Zakharov-Rubenchik system to resolve the multiple time scales accurately and to improve the computational efficiency during long-time simulations. The Crank-Nicolson formula and the Fourier pseudo-spectral method are respectively utilized for the temporal and spatial approximations. The proposed numerical method is proved to preserve the mass and energy conservative laws in the discrete levels exactly so that the magnetic field, the density of mass, and the fluid speed are stable on a general class of nonuniform time meshes. With the aid of the priori estimates derived from the discrete invariance and the newly proved discrete Gronwall inequality on variable time grids, sharp convergence analysis of the fully discrete scheme is established rigorously. Error estimate shows that the suggested adaptive time-stepping method can attain the second-order accuracy in time and the spectral accuracy in space. Extensive numerical experiments coupled with an adaptive time-stepping algorithm are presented to show the effectiveness of our numerical method in capturing the multiple time scale evolution for various velocity cases during the interactions of solitons. [ABSTRACT FROM AUTHOR]

In this article, we formulate and solve the representation problem for diffusion equations: giving a discretization of the Laplace transform of a diffusion equation under a space discretization over a space scale determined by an increment h > 0 , can we construct a continuous in h family of Cauer ladder networks whose constitutive equations match for all h > 0 the discretization. It is proved that for a finite differences discretization over a uniform geometric space scale, the representation problem over fractal Cauer networks is possible if and only if the coefficients of the diffusion are exponential functions in the space variable. Such diffusion equations admit a (Laplace) transfer function with a fractional behavior whose exponent is explicit. This allows us to justify previous works made by Sabatier and co-workers in [15-16] and Oustaloup and co-workers [14]. [ABSTRACT FROM AUTHOR]

The paper concerns the numerical solution for the wave propagation problem in periodic heterogeneous media. The homogenization method is utilized for the solution in the bounded periodic structure with highly oscillating coefficients. The perfectly matched layer (PML) technique is adopted to truncate the unbounded physical domain into a bounded computational domain, and the exponential convergence of Cartesian PML is generalized to the Helmholtz transmission problem in periodic heterogeneous media. An efficient adaptive finite element algorithm based on reliable a posteriori error estimate is extended to solve the homogenized PML problem, and the reliability of the estimator is established. Numerical experiments are included to demonstrate the competitive behavior of the proposed method. [ABSTRACT FROM AUTHOR]

We build an asymptotically compatible energy of the variable-step L2- 1 σ scheme for the time-fractional Allen–Cahn model with the Caputo’s fractional derivative of order α ∈ (0 , 1) , under a weak step-ratio constraint τ k / τ k - 1 ≥ r ⋆ (α) for k ≥ 2 , where τ k is the k-th time-step size and r ⋆ (α) ∈ (0.3865 , 0.4037) for α ∈ (0 , 1) . It provides a positive answer to the open problem in Liao et al. (J Comput Phys 414:109473, 2020), and, to the best of our knowledge, it is the first second-order nonuniform time-stepping scheme to preserve both the maximum bound principle and the energy dissipation law of time-fractional Allen–Cahn model. The compatible discrete energy is constructed via a novel discrete gradient structure of the second-order L2- 1 σ formula by a local-nonlocal splitting technique. It splits the discrete fractional derivative into two parts: one is a local term analogue to the trapezoid rule of the first derivative and the other is a nonlocal summation analogue to the L1 formula of Caputo derivative. Numerical examples with an adaptive time-stepping strategy are provided to show the effectiveness of our scheme and the asymptotic properties of the associated modified energy. [ABSTRACT FROM AUTHOR]

Electrical impedance tomography (EIT) is a non-ionizing imaging modality suitable for bedside pulmonary imaging. Currently, lung imaging with EIT in the ICU is tomographic due to the need for real-time images. However, linearized reconstruction methods have the potential to provide 3-D images in real time at the bedside. In this paper, the 3-D implementation of Calderón's method provided in Shin and Mueller (Inverse Probl 36:124005, 2020) is further developed for a patient-specific domain. Results of reconstructions from simulated data of two realistic human torsos using the Calderón-based method and one step of an iterative Gauss-Newton method are made for four different electrode configurations. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of exponential stabilization of a one-dimensional thermoelastic diffusion problem of type II for isotropic and homogeneous materials with frictional damping and time delay. The system of equations under consideration is a coupling of three hyperbolic equations involving three variables considering frictional damping and time delay on each variable. Using semigroup theory, we prove the well-posedness by the Lumer–Phillips theorem and the exponential stability by exploring the dissipative properties of the linear operator associated with the model through the Gearhart–Huang–Pruss theorem. [ABSTRACT FROM AUTHOR]

In this paper, an effective fully-discrete implicit scheme for solving linear reaction-diffusion equations is constructed by using the variable-time-step two-step backward differentiation formula (VSBDF2) in time combining with the nonconforming finite element methods in space. By introducing a modified energy projection operator, a discrete Laplace operator, the discrete orthogonal convolution kernels, we obtain the optimal and sharp error estimates of order O (h 2 + τ 2) in L 2 -norm and O (h + τ 2) in H 1 -norm under a mild restriction 0 < r k < r max ≈ 4.8645 for the ratio of adjacent time steps r k . Furthermore, with the help of a modified discrete Grönwall inequality and the combination technique of interpolation and projection operators, we achieved the superclose result between the interpolation function I h u and finite element solution u h in H 1 -norm of order O (h 2 + τ 2) , which together with the interpolation postprocessing operator Π 2 h leads to the global superconvergence result about u - Π 2 h u h in H 1 -norm of order O (h 2 + τ 2) . Finally, numerical tests are provided to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

A fully implicit numerical scheme is established for solving the time fractional Swift-Hohenberg equation with a Caputo time derivative of order α ∈ (0 , 1) . The variable-step L1 formula and the finite difference method are employed for the time and the space discretizations, respectively. The unique solvability of the numerical scheme is proved by the Brouwer fixed-point theorem. With the help of the discrete convolution form of L1 formula, the time-stepping scheme is shown to preserve a discrete energy dissipation law which is asymptotically compatible with the classic energy law as α → 1 - . Furthermore, the L ∞ norm boundedness of the discrete solution is obtained. Combining with the global consistency error analysis framework, the L 2 norm convergence order is shown rigorously. Several numerical examples are provided to illustrate the accuracy and the energy dissipation law of the proposed method. In particular, the adaptive time-stepping strategy is utilized to capture the multi-scale time behavior of the time fractional Swift-Hohenberg model efficiently. [ABSTRACT FROM AUTHOR]

In this paper, we study a second-order accurate and linear numerical scheme for the nonlocal Cahn-Hilliard equation. The scheme is established by combining a modified Crank-Nicolson approximation and the Adams-Bashforth extrapolation for the temporal discretization, and by applying the Fourier spectral collocation to the spatial discretization. In addition, two stabilization terms in different forms are added for the sake of the numerical stability. We conduct a complete convergence analysis by using the higher-order consistency estimate for the numerical scheme, combined with the rough error estimate and the refined estimate. By regarding the numerical solution as a small perturbation of the exact solution, we are able to justify the discrete ℓ∞ bound of the numerical solution, as a result of the rough error estimate. Subsequently, the refined error estimate is derived to obtain the optimal rate of convergence, following the established ℓ∞ bound of the numerical solution. Moreover, the energy stability is also rigorously proved with respect to a modified energy. The proposed scheme can be viewed as the generalization of the second-order scheme presented in an earlier work, and the energy stability estimate has greatly improved the corresponding result therein. [ABSTRACT FROM AUTHOR]

In this paper, the convex-splitting BDF2 method with variable time-steps (proposed in Chen et al. SIAM J Numer Anal 57:495–525, 2019) is reconsidered for the Cahn–Hilliard model. We adopt the Fourier pseudo-spectral discretization in space. With the help of the discrete gradient structure of the BDF2 formula and some embedded inequalities, we prove that the scheme preserves a modified energy dissipation law under the updated time-step-ratio restriction 0 < r k = τ k / τ k - 1 < 4.864 . By utilizing the discrete orthogonal convolution kernels, some discrete convolution inequalities and some proof techniques (Lemma 4.6), we tackle the difficulty brought from the Douglas-Dupont regularization stabilized term and prove the robust L 2 norm convergence of the scheme under the same mild time-step-ratio restriction 0 < r k < 4.864. Numerical experiments are carried out to support our theoretical analysis and a time adaptive strategy is applied to accelerate the simulation of the multi-scale characteristics of the Cahn–Hilliard model. [ABSTRACT FROM AUTHOR]

In this paper, we propose a quaternion formulation for the orientation variable in the three-dimensional Kobayashi–Warren model for the dynamics of polycrystals. We obtain existence of solutions to the L 2 -gradient descent flow of the constrained energy functional via several approximating problems. In particular, we use a Ginzburg–Landau-type approach and some extra regularizations. Existence of solutions to the approximating problems is shown by the use of nonlinear semigroups. Coupled with good a priori estimates, this leads to successive passages to the limit up to finally showing existence of solutions to the proposed model. Moreover, we also obtain an invariance principle for the orientation variable. [ABSTRACT FROM AUTHOR]

The generalized Hirota-Satsuma coupled KdV system with fractional-order derivative plays a significant role to simulate the interaction of nearby identical-weight particles in a crystal lattice structure, two long waves interaction with different dispersion relationships, the description of shallow-water wave propagation, ion-acoustic waves, plasma physics and some other fields. In this study, diverse analytical wave solutions in general and standard structures are established of the stated equation by introducing two viable techniques, namely, the generalized Kudryashov scheme and the two variables G'/G,1/G-expansion approach. The solutions are established in terms of elementary functions, specifically trigonometric functions, rational functions, exponential functions, and hyperbolic functions. For definite values of free parameters, the obtained analytical wave solutions transform into solitary wave solutions. The graphical patterns of the wave solutions with there-, two-, and contour plots are depicted magnificently to elucidate the internal structure of the phenomenon. The methods contribute as powerful mathematical tools and appear to be further effective, computerized, and user-friendly to investigate nonlinear fractional equations as well as comprehensive analytical solutions of nonlinear evolution equations in engineering, technology, and mathematical physics.

We analyze here a finite element space semidiscretization of the Allen-Cahn/Cahn-Hilliard system coupled with heat equation and based on the Maxwell-Cattaneo law. We prove that the semidiscrete solution converges weakly to the continuous solution as the discretization parameter tends to 0. We obtain optimal a priori error estimates in energy norm and related norms, assuming enough regularity on the solution. We also show that the semidiscrete solution converges to an equilibrium as time goes to infinity. [ABSTRACT FROM AUTHOR]

The backward differential formulas of order 3-5 (BDF3-5) are applied to simulate the molecular beam epitaxial (MBE) model with slop selection. The fully implicit BDF3-5 schemes are uniquely solvable under an estimated time-step requirement. We prove that the schemes conserve the modified discrete energy dissipation laws by utilizing the discrete gradient structures of BDF3-5 formulas. Furthermore, we present a new convolution inequality to handle the nonlinear term for the error estimate. More importantly, the L 2 norm convergence analysis of the BDF3-5 schemes are proved rigorously without the assumption of Lipschitz boundedness. Numerical examples are shown to verify the efficiency and the accuracy of the developed schemes. [ABSTRACT FROM AUTHOR]

This paper is concerned with the long time dynamics of the semilinear wave equation including localized interior damping term under acoustic boundary conditions. With appropriate assumptions, the existence of a global attractor for the semigroup generated by the considered problem is proved. Moreover, imposing more strict conditions on the nonlinearity in the equation, the regularity and finite dimensionality of the attractor are established. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the stability and convergence analysis of the second order backward differentiation formula (BDF2) scheme with variable time steps for the no-slope-selection (NSS) equation of the epitaxial thin film growth model, with Fourier pseudo-spectral method in physical domain. Under the adjoint time-step ratio condition r k : = τ k / τ k - 1 < 4.864 , the variable-step BDF2 scheme is uniquely solvable and stable in L 2 -norm, also we establish a rigorous error estimate with a novel discrete orthogonal convolution kernels involved in the analysis. Finally, the adaptive time step strategy is used to accelerate the calculation of the steady-state solution, and the theoretical results are verified by numerical examples. In addition, the long time simulation results have indicated a logarithm law for the energy decay, as well as the power laws for growth of the surface roughness and the mound width. [ABSTRACT FROM AUTHOR]

This paper presents the current state of mathematical modelling of the electrochemical behaviour of lithium-ion batteries (LIBs) as they are charged and discharged. It reviews the models developed by Newman and co-workers, both in the cases of dilute and moderately concentrated electrolytes and indicates the modelling assumptions required for their development. Particular attention is paid to the interface conditions imposed between the electrolyte and the active electrode material; necessary conditions are derived for one of these, the Butler–Volmer relation, in order to ensure physically realistic solutions. Insight into the origin of the differences between various models found in the literature is revealed by considering formulations obtained by using different measures of the electric potential. Materials commonly used for electrodes in LIBs are considered and the various mathematical models used to describe lithium transport in them discussed. The problem of upscaling from models of behaviour at the single electrode particle scale to the cell scale is addressed using homogenisation techniques resulting in the pseudo-2D model commonly used to describe charge transport and discharge behaviour in lithium-ion cells. Numerical solution to this model is discussed and illustrative results for a common device are computed. [ABSTRACT FROM AUTHOR]

A fully implicit two-step backward differentiation formula (BDF2) scheme with variable time steps is considered for solving the phase field crystal (PFC) model by combining with pseudo-spectral method in space. We show that the BDF2 scheme inherits a modified energy dissipation law under a mild ratio restriction A1, i.e., 0 < r k : = τ k / τ k - 1 < r max ≈ 4.8645 , which justifies the thermodynamic consistency of the PFC model numerically. In addition, the optimal second-order convergence of the proposed scheme is also established under the ratio restriction A1. The proof involves the tools of DOC and DCC kernels, and some generalized properties of the DOC kernels. As far as we know, the ratio restriction A1 required in our results is mildest so far for the variable time-step BDF2 scheme for calculating PFC model. Numerical examples are provided to demonstrate our theoretical results. [ABSTRACT FROM AUTHOR]

The two-step backward differential formula (BDF2) with unequal time-steps is applied to construct an energy stable convex-splitting scheme for the Cahn–Hilliard model. We focus on the numerical influences of time-step variations by using the recent theoretical framework with the discrete orthogonal convolution kernels. Some novel discrete convolution embedding inequalities with respect to the orthogonal convolution kernels are developed such that a concise L 2 norm error estimate is established at the first time under an updated step-ratio restriction 0 < r k : = τ k / τ k - 1 ≤ r user , where r user can be chosen by the user such that r user < 4.864 . The stabilized convex-splitting BDF2 scheme is shown to be mesh-robustly convergent in the sense that the convergence constant (prefactor) in the error estimate is independent of the adjoint time-step ratios. The suggested method is proved to preserve a modified energy dissipation law at the discrete levels if 0 < r k ≤ r user , such that it is mesh-robustly stable in an energy norm. On the basis of ample tests on random time meshes, a useful adaptive time-stepping strategy is applied to efficiently capture the multi-scale behaviors and to accelerate the long-time simulation approaching the steady state. [ABSTRACT FROM AUTHOR]

This paper proposes a new type of multilevel method for solving the Steklov eigenvalue problem based on Newton’s method. In this iteration method, solving the Steklov eigenvalue problem is replaced by solving a small-scale eigenvalue problem on the coarsest mesh and a sequence of augmented linear problems on refined meshes, derived by Newton step. We prove that this iteration scheme obtains the optimal convergence rate with linear complexity, which improves the overall efficiency of solving the Steklov eigenvalue problem. Moreover, an adaptive iteration scheme for multi eigenvalues based on this new multilevel method is given. Finally, some numerical experiments are provided to illustrate the efficiency of the proposed multilevel scheme. [ABSTRACT FROM AUTHOR]

Boundary element methods produce dense linear systems that can be accelerated via multipole expansions. Solved with Krylov methods, this implies computing the matrix-vector products within each iteration with some error, at an accuracy controlled by the order of the expansion, p. We take advantage of a unique property of Krylov iterations that allows lower accuracy of the matrix-vector products as convergence proceeds, and propose a relaxation strategy based on progressively decreasing p. In extensive numerical tests of the relaxed Krylov iterations, we obtained speed-ups of between 1.5 × and 2.3 × for Laplace problems and between 2.7 × and 3.3 × for Stokes problems. We include an application to Stokes flow around red blood cells, computing with up to 64 cells and problem size up to 131k boundary elements and nearly 400k unknowns. The study was done with an in-house multi-threaded C++ code, on a hexa-core CPU. The code is available on its version-control repository, , and we share reproducibility packages for all results in . [ABSTRACT FROM AUTHOR]

The backward differential formulas of order k (BDF- k ) for 3 ≤ k ≤ 5 are analyzed for the molecular beam epitaxial (MBE) model without slope selection. We show that the fully implicit uniform BDF- k schemes are convex and uniquely solvable under a weak time-step constraint. Then the BDF methods are proved to preserve the modified discrete energy dissipation laws by using the discrete gradient structures of BDF- k (3 ≤ k ≤ 5) formulas. Furthermore, with the help of discrete orthogonal convolution kernels and the corresponding convolution Young inequalities, the L 2 norm stability and convergence analysis of the BDF- k (3 ≤ k ≤ 5) schemes for the MBE model are established by the recent discrete energy technique. To the best of our knowledge, it is the first time to present a unified approach to establish the discrete energy dissipation laws and L 2 norm convergence of non-A-stable BDF- k methods for the MBE model. Numercial simulations are presented to demonstrate the accuracy and efficiency of the proposed schemes. [ABSTRACT FROM AUTHOR]

We consider the initial boundary value problem for the two dimensional equation describing the evolution of the systems with the fast phase transition. Under critical growth and dissipativity conditions on the nonlinearities, we prove the existence of the global attractor. In particular, we establish the existence of the global attractor for the strong solutions of the weakly damped 2D Kirchhoff plate equation involving p-Laplacian operator with the critical exponent p = 5 , and thereby improve the previously obtained results. We also show that the approach of this paper can be applied to the hyperbolic relaxation of the 2D Cahn-Hilliard equation with the critical quartic nonlinearity. [ABSTRACT FROM AUTHOR]

Neumann condition has so many applications in mechanical engineering. This paper suggests a numerical method for solving Neumann problem with discontinuous coefficients. The accuracy of mentioned method will be illustrated with some numerical examples with given exact solutions. [ABSTRACT FROM AUTHOR]