Semi-discrete Systems Research Articles

Implicit-Explicit schemes for decoupling multicontinuum problems in porous media

In this work, we present an efficient way to decouple the multicontinuum problems. To construct decoupled schemes, we propose Implicit-Explicit time approximation in general form and study them for the fine-scale and coarse-scale space approximations. We use a finite-volume method for fine-scale approximation, and the nonlocal multicontinuum (NLMC) method is used to construct a coarse-scale approximation. The NLMC method is a multiscale method for developing an accurate and physically meaningful coarse-scale model based on defining the macroscale variables. The multiscale basis functions are constructed in local domains by solving constraint energy minimization problems and projecting the system to the coarse grid. The resulting basis functions have exponential decay properties and lead to the accurate approximation on a coarse grid. We construct a fully Implicit time approximation for semi-discrete systems arising after fine-scale and coarse-scale space approximations. We investigate the stability of the two and three-level schemes for fully Implicit and Implicit-Explicit time approximations schemes for multicontinuum problems in fractured porous media. We show that combining the decoupling technique with multiscale approximation leads to developing an accurate and efficient solver for multicontinuum problems.

Lagrangian multiform structure of discrete and semi-discrete KP systems

A variational structure for the potential AKP system is established using the novel formalism of a Lagrangian multiforms. The structure comprises not only the fully discrete equation on the 3D lattice, but also its semi-discrete variants including several differential-difference equations asssociated with, and compatible with, the partial difference equation. To this end, an overview is given of the various (discrete and semi-discrete) variants of the KP system, and their associated Lax representations, including a novel `generating PDE' for the KP hierarchy. The exterior derivative of the Lagrangian 3-form for the lattice potential KP equation is shown to exhibit a double-zero structure, which implies the corresponding generalised Euler-Lagrange equations. Alongside the 3-form structures, we develop a variational formulation of the corresponding Lax systems via the square eigenfunction representation arising from the relevant direct linearization scheme.

Direct/split invariant-preserving Fourier pseudo-spectral methods for the rotation-two-component Camassa–Holm system with peakon solitons

The Fourier pseudo-spectral method is well suited to solve PDEs under the periodic boundary condition due to its high-order accuracy and easy-to-implement feature. In this paper, we explore as well as comparatively study four classes of Fourier pseudo-spectral schemes for solving the rotation-two-component Camassa–Holm system which possibly owns peakon solitons. Via exploiting inherent structural properties of the system, we reformulate it into two kinds of different equivalent forms and then apply the Fourier pseudo-spectral method to derive two spatial semi-discrete systems, both of which are proved to preserve the corresponding invariants including mass, momentum and energy. Subsequently, we construct two linearly implicit schemes based on Strang splitting technique and two nonlinear schemes, respectively, for both semi-discrete systems. Owing to the different equivalent forms in the structure, one of the nonlinear schemes preserves discrete mass and momentum, while the other one is shown to preserve all three invariants. Numerical results under the situation of smooth/nonsmooth initial values are provided for distinct types of solutions to test the accuracy in long time simulation and to verify the capacity of predicting water wave propagation, as well as advantages in preserving these invariants. For instance, the present schemes are shown to be at least 14 significant digits, improving upon 10 from ones in previous references.

Development and Analysis of Novel Integrable Nonlinear Dynamical Systems on Quasi-One-Dimensional Lattices. Two-Component Nonlinear System with the On-Site and Spatially Distributed Inertial Mass Parameters

The main principles of developing the evolutionary nonlinear integrable systems on quasi-onedimensional lattices are formulated in clear mathematical and physical terms discarding the whimsical mathematical formulations and computer-addicted presentations. These basic principles are substantiated by the actual development of novel semi-discrete integrable nonlinear system, whose auxiliary spectral and evolutionary operators are given by 4 × 4 square matrices. The procedure of reduction from the prototype nonlinear integrable system with twelve field functions to the physically meaningful nonlinear integrable system with four field functions is described in details prompted by our previous cumulative experience. The obtained ultimate semi-discrete nonlinear integrable system comprises the two subsystems of essentially distinct physical origins. Thus, the first subsystem is the subsystem of the Toda type. It is characterized by the on-site (spatially local) mass parameter and the positively defined elasticity coefficient. In contrast, the second subsystem is characterized by the spatially distributed mass parameters and the negatively defined elasticity coefficient responsible for the low-amplitude instability. We believe our scrupulous consideration of all main steps in developing the semidiscrete nonlinear integrable systems will be useful for the researchers unfamiliar with the numerous stumbling blocks inevitable in such an interesting and prospective scientific field as the theory of semi-discrete nonlinear integrable systems.

A variant of the discrete gradient method for the solution of the semilinear wave equation under different boundary conditions

In this paper, energy-preserving schemes based upon the discrete gradient are used to numerically solve the semilinear wave equation under periodic boundary conditions, Dirichlet boundary conditions and Neumann boundary conditions. Both the integral form and the evolutionary behaviour of the energy depend on the associated boundary conditions. The wave equations are semi-discretized in different manners for different boundary conditions such that the Hamiltonian functions of the resulting semi-discrete Hamiltonian systems are the discrete analogue of the original energy. Furthermore, the Hamiltonian functions of the resulting semi-discrete systems have similar evolutionary behaviours as the original energy. It is noted that the semi-discrete system exhibits an oscillatory structure. Subsequently, the semi-discrete approximation of the energy evolution as well as the oscillatory structure is passed to the fully-discrete problem by applying extended discrete gradient formula in the temporal direction. Numerical experiments are carried out to show the effectiveness of the new schemes.

The exponential stabilization of a heat-wave coupled system and its approximation

In this paper, we consider exponential stabilization for a heat-wave coupled system under boundary control and collocated observation. It is known that the system is polynomially stable without control. Two kinds of feedback strategies namely static negative proportional feedback and dynamic feedback are designed. By the Lyapunov function direct method, the exponential stabilities of the closed-loop systems under different two feedbacks are firstly verified. Secondly, the H∞ robustness is further analyzed and the related sufficient conditions are developed. Furthermore, the closed-loop systems are discretized into semi-discrete systems by a new finite difference method, and the uniform exponential stabilities of the discrete systems are established by the approach paralleling to the continuous counterpart.

Semi-discrete Lagrangian 2-forms and the Toda hierarchy

We present a variational theory of integrable differential-difference equations (semi-discrete integrable systems). This is an extension of the ideas known by the names ‘Lagrangian multiforms’ and ‘Pluri-Lagrangian systems’, which have previously been established in both the fully discrete and fully continuous cases. The main feature of these ideas is to capture a hierarchy of commuting equations in a single variational principle. Semi-discrete Lagrangian multiforms provide a new way to relate differential-difference equations and partial differential equations. We discuss this relation in the context of the Toda lattice, which is part of an integrable hierarchy of differential-difference equations, each of which involves a derivative with respect to a continuous variable and a number of lattice shifts. We use the theory of semi-discrete Lagrangian multiforms to derive PDEs in the continuous variables of the Toda hierarchy, which hold as a consequence of the differential-difference equations, but do not involve any lattice shifts. As a second example, we briefly discuss the semi-discrete potential Korteweg-de Vries equation, which is related to the Volterra lattice.

Davydov–Kyslukha model as the starting point in the development of integrable multi-component nonlinear dynamical systems on quasi-one-dimensional lattices

The Davydov–Kyslukha nonlinear exciton-phonon model on a regular one-dimensional lattice is asserted to be the driving force for the development of integrable multi-component nonlinear dynamical systems encompassing excitonic, vibrational and orientational degrees of freedom. The two most representative quasi-one-dimensional integrable multi-component nonlinear systems inspired by the Davydov–Kyslukha model are presented explicitly in their concise Hamiltonian forms. The new six-subsystem integrable nonlinear model on a regular quasi-one-dimensional lattice is proposed and its derivation based upon the appropriate zero-curvature representation is presented. The constructive aspect of the famous Davydov motto is illustrated by the examples of semi-discrete integrable nonlinear dynamical systems canonicalizeable via the proper point transformations to the physically motivated field variables.

Branched-dispersion generalizations of Lotka–Volterra and Ablowitz–Ladik nonlinear integrable systems revisited from the intersite coupling standpoint

The scrupulous inspection of actual inter-site coupling in generalized Lotka–Volterra and Ablowitz–Ladik nonlinear integrable systems with branched dispersion shows that the true essence, typifying each of them, is manifested as the collection of several basic physically independent semi-discrete (differential-difference) nonlinear integrable systems settled along the respective autonomous helical-like infinite chains.

Efficient computation of Jacobian matrices for entropy stable summation-by-parts schemes

Entropy stable schemes replicate an entropy inequality at the semi-discrete level. These schemes rely on an algebraic summation-by-parts (SBP) structure and a technique referred to as flux differencing. We provide simple and efficient formulas for Jacobian matrices for the semi-discrete systems of ODEs produced by entropy stable discretizations. These formulas are derived based on the structure of flux differencing and derivatives of flux functions, which can be computed using automatic differentiation (AD). Numerical results demonstrate the efficiency and utility of these Jacobian formulas, which are then used in the context of two-derivative explicit time-stepping schemes and implicit time-stepping.

Highly accurate, linear, and unconditionally energy stable large time-stepping schemes for the Functionalized Cahn–Hilliard gradient flow equation

This paper aims to develop a class of large time-stepping schemes for solving the Functionalized Cahn–Hilliard (FCH) gradient flow equation. By using the Invariant Energy Quadratization (IEQ) approach, we successfully construct several highly efficient linear schemes that allow large time step sizes to be used in computations. All the developed schemes with a fixed time step size are proved to be unconditionally uniquely solvable and unconditionally energy stable in theory. Moreover, we show that the obtained semi-discrete systems can be solved efficiently by using the preconditioned Krylov subspace methods, since all the linear operators of these systems are proved to be symmetric and positive definite. Finally, several numerical experiments are carried out to verify the accuracy, efficiency and energy stability of the developed IEQ schemes. We observe that our IEQ schemes are more accurate than the Scalar Auxiliary Variable (SAV) schemes. This can be attributed to the fact that the IEQ method can preserve the operator structures of the original equation as much as possible.

Two efficient spectral methods for the nonlinear fractional wave equation in unbounded domain

This paper is concerned with two fast and efficient numerical methods to solve the multidimensional nonlinear fractional wave equation in unbounded domain. For the spatial discretization, a spectral-Galerkin method is adopted by using the Fourier-Like Mapped Chebyshev function approximation, which makes fractional Laplacian diagonalized. Then, for the temporal discretization, the second order accurate time-splitting method is proposed and it not only avoids the matrix exponential computation in view of the diagonalized structure of semi-discrete systems, but also results in an explicit and time symmetric scheme; Then, an exponential scalar auxiliary variable (E-SAV) method is developed for solving fractional wave equation, which preserves the original energy conservation and makes nonlinear term to be solved explicitly to obtain a linear system at each time step. In addition, the two schemes both can be implemented by the fast Fourier transform (FFT) related to Chebyshev polynomials. Numerical experiments are provided to test the numerical accuracy and the efficiency in long time simulations.

Integrable discretizations for classical Boussinesq system

In this paper, we propose and study integrable discrete systems related to the classical Boussinesq system. Based on elementary and binary Darboux transformations and associated Bäcklund transformations, both full-discrete systems and semi-discrete systems are constructed. The discrete systems obtained from elementary Darboux transformation are shown to be the discrete systems of relativistic Toda lattice type appeared in the work of Suris (1997) and the ones from binary Darboux transformations are two-component extensions of the lattice potential KdV equation and Kac–van Moerbeke equation. For these discrete systems, their different continuum limits, various interesting reductions and Darboux–Bäcklund transformations are considered. Some solutions such as discrete resonant solitons are also presented.

Integrals and characteristic Lie rings of semi-discrete systems of equations

Integrals and characteristic Lie rings of semi-discrete systems of equations

High-order structure-preserving algorithms for the multi-dimensional fractional nonlinear Schrödinger equation based on the SAV approach

In the paper, we aim to develop a class of high-order structure-preserving algorithms, which are built upon the idea of the newly introduced scalar auxiliary variable approach, for the multi-dimensional space fractional nonlinear Schrödinger equation. First, we reformulate the equation as an infinite-dimension canonical Hamiltonian system, and obtain an equivalent system with a modified energy conservation law by using the scalar auxiliary variable approach. Then, the new system is discretized by Gauss collocation methods to arrive at semi-discrete conservative systems. Subsequently, the Fourier pseudo-spectral method is applied for semi-discrete systems to obtain high-order fully-discrete schemes, which can both preserve the mass and the modified energy exactly in discrete scene. Finally, numerical experiments are provided to demonstrate the conservation and accuracy of the proposed schemes.

The treatment of the Neumann boundary conditions for a new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes

Recently we have developed a new 3-D numerical approach for the time dependent wave and heat equations as well as for the time independent Poisson equation on irregular domains with the Dirichlet boundary conditions. Its extension to the Neumann boundary conditions that was a big issue due to the presence of normal derivatives along irregular boundaries is considered in this paper. Trivial Cartesian meshes and simple 27-point uniform and nonuniform stencil equations are used with the new approach for 3-D irregular domains. The Neumann boundary conditions are introduced as the known right-hand side into the stencil and do not change the width of the stencil. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique. Very small distances (0.1h−10−9h where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not worsen the accuracy of the new technique. At similar 27-point stencils, the accuracy of the new approach is much higher than that for the linear finite elements. The numerical results for irregular domains show that at the same number of degrees of freedom, the new approach is even much more accurate than the high-order (up to the fourth order) tetrahedral finite elements with much wider stencils. The wave and heat equations can be uniformly treated with the new approach. The order of the time derivative in these equations does not affect the coefficients of the stencil equations of the semi-discrete systems. The new approach can be directly applied to other partial differential equations.

On the discretization of Darboux Integrable Systems

We study the discretization of Darboux integrable systems. The discretization is done using x-, y-integrals of the considered continuous systems. New examples of semi-discrete Darboux integrable systems are obtained.

A new domain‐based implicit‐explicit time stepping scheme based on the class of exponential integrators called sEPIRK

AbstractReliable simulations of flows in real applications involve the task of discretizing both space and time in an accurate and efficient way. To cope with the large semidiscrete systems resulting from a space discretization on appropriate grids, which often include comparatively few very small cells near solid walls for boundary layer resolution, an implicit‐explicit time stepping scheme can be the most efficient variant. We present such a type of scheme utilizing recent exponential integrators called sEPIRK and show its computational advantages opposed to IMEX‐Runge‐Kutta schemes.

Nonlinear integrable systems containing the canonical subsystems of distinct physical origins

Four new semi-discrete nonlinear integrable systems relevant for physical applications are suggested. Each system contains two coupled subsystems of distinct physical origin. These integrable hybridizations are (1) the Toda-like subsystem coupled with the induced-trapping subsystem of PT-symmetric excitations, (2) the subsystem of Frenkel-like excitons coupled with the subsystem of nontrivial vibrations, (3) two coupled self-trapping subsystems, (4) the Toda-like subsystem coupled with the self-trapping subsystem akin to the charged particle with electromagnetic field. Each hybrid system admits the clear Hamiltonian representation characterized by two pairs of canonical field variables with the standard Poisson structure. The main general local conserved densities, adaptable to any integrable system under consideration, are found explicitly.

A new 3-D numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes

A new 3-D numerical approach for the time dependent wave and heat equations as well as for the time independent Laplace equation on irregular domains with the Dirichlet boundary conditions has been developed. Trivial Cartesian meshes and simple 27-point uniform and nonuniform stencil equations are used for 3-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique. Very small distances (0.1h−10−9h where h is the grid size) between the grid points of a Cartesian mesh and the boundary do not worsen the accuracy of the new technique. At similar 27-point stencils, the accuracy of the new approach is two orders higher than that for the linear finite elements. The numerical results for irregular domains show that at the same number of degrees of freedom, the new approach is even much more accurate than the high-order (up to the fifth order) tetrahedral finite elements with much wider stencils. The wave and heat equations can be uniformly treated with the new approach. The order of the time derivative in these equations does not affect the coefficients of the stencil equations of the semi-discrete systems. The new approach can be directly applied to other partial differential equations.