Semi-discrete Equations Research Articles

Semi-discrete hyperbolic equations admitting five dimensional characteristic x-ring

The necessary and sufficient conditions for a hyperbolic semi-discrete equation to have five dimensional characteristic x-ring are derived. For any given chain, the derived conditions are easily verifiable by straightforward calculations.

Boundary Observability of Semi-Discrete Second-Order Integro-Differential Equations Derived from Piecewise Hermite Cubic Orthogonal Spline Collocation Method

In this paper we consider space semi-discretization of some integro-differential equations using the harmonic analysis method. We study the problem of boundary observability, i. e., the problem of whether the initial data of solutions can be estimated uniformly in terms of the boundary observation as the net-spacing \(h\rightarrow 0\). When \(h\rightarrow 0\) these finite-dimensional spaces increase and eventually cover the whole space. We thus recover the well-known observability property of the continuous system as the limit of discrete observability estimates as the mesh size tends to zero. We shall consider the piecewise Hermite cubic orthogonal spline collocation semi-discretization.

Canonical Euler splitting method for nonlinear composite stiff evolution equations

In this paper, a new splitting method, called canonical Euler splitting method (CES), is constructed and studied, which can be used for the efficient numerical solution of general nonlinear composite stiff problems in evolution equations of various type, such as ordinary differential equations (ODEs), semi-discrete unsteady partial differential equations (PDEs) and ordinary or partial Volterra functional differential equations (VFDEs), and can significantly improve the computing speed on the basis of ensuring the computing quality. Stability, consistency and convergence theories of this method are established. A series of numerical experiments are given which check the efficiency of CES method and confirm our theoretical results.

A semi-discrete Kadomtsev-Petviashvili equation and its coupled integrable system

We establish connections between two cascades of integrable systems generated from the continuum limits of the Hirota-Miwa equation and its remarkable nonlinear counterpart under the Miwa transformation, respectively. Among these equations, we are mainly concerned with the semi-discrete bilinear Kadomtsev-Petviashvili (KP) equation which is seldomly studied in literature. We present both of its Casorati and Grammian determinant solutions. Through the Pfaffianization procedure proposed by Hirota and Ohta, we are able to derive the coupled integrable system for the semi-discrete KP equation.

The numerical solution of space-dependent neutron kinetics equations in hexagonal-z geometry using Diagonally Implicit Runge Kutta method

The numerical solution of the space-dependent neutron kinetics equations in hexagonal-z geometry is studied in this paper. The equations are discretized by Diagonally Implicit Runge–Kutta (DIRK) method in the temporal direction. The semi-discretized equations are in the form of the fixed source problems, which can be solved by the function expansion nodal method in the hexagonal-z geometry. The intranodal fluxes in each group are expanded by exponential functions and orthogonal polynomials up to the second order. The two-and three-dimensional benchmark problems are used to verify this method. Compared with the backward Euler method, the numerical results show that the current method is more accurate.

New energy-preserving schemes for Klein–Gordon–Schrödinger equations

In this manuscript, we focus on new conservative numerical methods for Klein–Gordon–Schrödinger equations. By expressing Klein–Gordon–Schrödinger equations in an infinite-dimensional Hamiltonian form, we firstly discretize spatial derivatives by using Sinc collocation method then approximate the associated semi-discrete ordinary differential equations by discrete gradient method. Based on two different discrete gradients, two new energy-preserving schemes are provided, respectively. Furthermore, it is proved that both schemes preserve the discrete charge conservation law as well. Finally, numerical experiments are presented to show the excellent long-time conservation behavior and efficiency of the new energy-preserving schemes.

Stability of Newton TVD Runge–Kutta scheme for one-dimensional Euler equations with adaptive mesh

In this paper, we propose a moving mesh method with a Newton total variation diminishing (TVD) Runge–Kutta scheme for the Euler equations. Our scheme improves time discretization in the moving mesh algorithms. By analyzing the semi-discrete Euler equations with the discrete moving mesh equations as constraints, the stability of the Newton TVD Runge–Kutta scheme is proved. Thus, we can conclude that the proposed algorithm can generate a weak solution to the Euler equations. Finally, numerical examples are presented to verify the theoretical results and demonstrate the accuracy of the proposed scheme.

Rational Solutions for Lattice Potential KdV Equation and Two Semi-discrete Lattice Potential KdV Equations

Abstract By imposing reduction conditions on rational solutions for a system involving the Hirota–Miwa equation, rational solutions for lattice potential KdV equation are constructed. Besides, the rational solutions for two semi-discrete lattice potential KdV equations are also considered. All these rational solutions are in the form of Schur function type.

Elastodynamics of FGM plates by mesh-free method

The paper deals with transient analysis of homogeneous as well as FGM (functionally graded material) thin and/or thick plates subjected to transversal dynamic loading. The Young modulus and mass density are allowed to be continuously variable through the thickness of the plate. The unified formulation is developed for plate bending problems including three various theories such as the classical Kirchhoff–Love theory for bending of thin elastic plates, the 1st and 3rd order shear deformations plate theories. Switching among these three theories is controlled by two key factors. From the derived equations of motion, the coupling between the bending and in-plane deformation modes is discovered in FGM plates with specification of the necessary conditions. For the numerical implementation, the strong formulation is proposed with meshless approximation of spatial variations of field variables. The semi-discretized equations of motion yield a system the ordinary differential equations which can be solved by standard time stepping techniques. The great attention is paid to the numerical study of coupling effects in FGM plates subjected to transversal Heaviside impact loading and/or Heaviside pulse loading. The achieved numerical results are thoroughly discussed and interpreted.

A family of three-stage third order AMF-W-methods for the time integration of advection diffusion reaction PDEs.

In this paper new three-stage W-methods for the time integration of semi-discretized advection diffusion reaction Partial Differential Equations (PDEs) are provided. In particular, two three-parametric families of W-methods of order three are obtained under a realistic assumption regarding the commutator of the exact Jacobian and the approximation of the Jacobian which defines the corresponding W-method. Specific methods are selected by minimizing error coefficients, enlarging stability regions or increasing monotonicity factors, and embedded methods of order two for an adaptive time integration are derived by further assuming first order approximations to the Jacobian. The relevance of the newly proposed methods in connection with the Approximate Matrix Factorization technique is discussed and numerical illustration on practical PDE problems revealing that the new methods are good competitors over existing integrators in the literature is provided.

On Generating Integrable Dynamical Systems in 1+1 and 2+1 Dimensions by Using Semisimple Lie Algebras

Abstract We deduce a set of integrable equations under the framework of zero curvature equations and obtain two sets of integrable soliton equations, which can be reduced to some new integrable equations including the generalised nonlinear Schrödinger (NLS) equation. Under the case where the isospectral functions are one-order polynomials in the parameter λ, we generate a set of rational integrable equations, which are reduced to the loop soliton equation. Under the case where the derivative λ t of the spectral parameter λ is a quadratic algebraic curve in λ, we derive a set of variable-coefficient integrable equations. In addition, we discretise a pair of isospectral problems introduced through the Lie algebra given by us for which a set of new semi-discrete nonlinear equations are available; furthermore, the semi-discrete MKdV equation and the Hirota lattice equation are followed to produce, respectively. Finally, we apply the Lie algebra to introduce a set of operator Lax pairs with an operator, and then through the Tu scheme and the binomial-residue representation method proposed by us, we generate a 2+1-dimensional integrable hierarchy of evolution equations, which reduces to a generalised 2+1-dimensional Davey-Stewartson (DS) equation.

Integrable discretizations and self-adaptive moving mesh method for a coupled short pulse equation

In the present paper, integrable semi-discrete and fully discrete analogues of a coupled short pulse (CSP) equation are constructed. The key to the construction are the bilinear forms and determinant structure of the solutions of the CSP equation. We also construct N-soliton solutions for the semi-discrete and fully discrete analogues of the CSP equations in the form of Casorati determinants. In the continuous limit, we show that the fully discrete CSP equation converges to the semi-discrete CSP equation, then further to the continuous CSP equation. Moreover, the integrable semi-discretization of the CSP equation is used as a self-adaptive moving mesh method for numerical simulations. The numerical results agree with the analytical results very well.

A hybrid Taylor–Galerkin variational multi-scale stabilization method for the level set equation

A stabilized finite element formulation of the level set equation is proposed for the numerical simulation of water droplet dynamics for in-flight ice accretion problems. The variational multi-scale and Taylor–Galerkin approaches are coupled such that the temporal derivative in the weak Galerkin formulation is replaced with a Taylor series expansion improving the temporal accuracy of the scheme. The variational multi-scale approach is then applied to the semi-discrete equation, allowing the stabilization terms to appear naturally. Taylor series expansions up to the fourth order have been studied in terms of accuracy and convergence rates. A second order implicit expansion was found to provide a good trade-off between accuracy and computational cost when compared to a fourth order implicit expansion. Validation is done through a number of benchmark cases considering droplet stretching and high-speed advection. Results indicate good mass conservation characteristics compared to other methods available in the literature.

Reducing spurious oscillations in discontinuous wave propagation simulation using high-order finite elements

In this paper, a modified explicit time integration scheme is proposed for simulating the propagation of discontinuous waves. The spatial domain is discretized by the finite element method. To obtain accurate results, the standard finite element method requires a very fine mesh, which is why the computational effort can be very time consuming. The use of high-order finite element methods–such as the spectral element method based on Lagrange polynomials through Gauss–Lobatto–Legendre points or the iso-geometric analysis using non-uniform rational B-splines–can reduce the enormous computational costs significantly, compared to the standard finite element method. However, explicit time integration schemes such as the central difference method cannot eliminate the spurious oscillation near the front wave. The procedure proposed here is basically similar to the explicit method of Noh and Bathe, but the semi-discrete equation of motion is modified by introducing a damping parameter to suit the high-order FEM analysis of the discontinuous wave propagation. The performance due to this modification is tested for one-dimensional wave propagation problems using both lumped and consistent mass matrices. Then, the idea of a combination of the consistent and row sum lumped mass matrices is evaluated. The proposed method is studied also in two-dimensions by considering the Lamb problem of wave propagation. The results are promising enough to provide a better simulation of the discontinuous wave propagation problem.

A finite element method for the inverse problem of boundary data recovery in an oxygen balance model

Abstract The inverse problem under investigation consists of the boundary data completion in a deoxygenation-reaeration model in stream-waters. The unidimensional model we deal with is based on the one introduced by Streeter and Phelps, augmented by Taylor dispersion terms. The missing boundary condition is the load or/and the flux of the biochemical oxygen demand indicator at the upstream point. The counterpart is the availability of two boundary conditions on the dissolved oxygen tracer at the same point. The major consequence of these non-standard boundary conditions is that dispersion-reaction equations on both oxygen indicators are strongly coupled and the resulting system becomes ill-posed. The main purpose here is a finite element space-discretization of the variational problem. Mixed finite elements turn out to be well fitted and yield a non-symmetric saddle point system. The obtained semi-discrete problem is a differential algebraic equation that needs specific tools for its analysis. Combining analytical calculations and theoretical justifications, we try to elucidate the main properties of this ill-posed dynamical problem and understand its mathematical structure.

Dynamics of a differential-difference integrable (2+1)-dimensional system.

A Kadomtsev-Petviashvili- (KP-) type equation appears in fluid mechanics, plasma physics, and gas dynamics. In this paper, we propose an integrable semidiscrete analog of a coupled (2+1)-dimensional system which is related to the KP equation and the Zakharov equation. N-soliton solutions of the discrete equation are presented. Some interesting examples of soliton resonance related to the two-soliton and three-soliton solutions are investigated. Numerical computations using the integrable semidiscrete equation are performed. It is shown that the integrable semidiscrete equation gives very accurate numerical results in the cases of one-soliton evolution and soliton interactions.

Solutions and Continuum Limits of Two Semi-Discrete Lattice Potential Korteweg-de Vries Equations* *Supported by National Natural Science Foundation of China under Grant No. 11371241 and National Natural Science Foundation of Shanghai under Grant No. 13ZR1416700

We derive bilinear forms and Casoratian solutions for two semi-discrete potential Korteweg-de Vries equations. Their continuum limits go to the counterparts of the continuous potential Korteweg-de Vries equation.

Integrable discrete nonautonomous quad-equations as Bäcklund auto-transformations for known Volterra and Toda type semidiscrete equations

We construct integrable discrete nonautonomous quad-equations as Bäcklund autotransformations for known Volterra and Toda type semidiscrete equations, some of which are also nonautonomous. Additional examples of this kind are found by using transformations of discrete equations which are invertible on their solutions. In this way we obtain integrable examples of different types: discrete analogs of the sine-Gordon equation, the Liouville equation and the dressing chain of Shabat. For Liouville type equations we construct general solutions, using a specific linearization. For sine-Gordon type equations we find generalized symmetries, conservation laws and L—A pairs.

Integrable semi-discretization of a multi-component short pulse equation

In the present paper, we mainly study the integrable semi-discretization of a multi-component short pulse equation. First, we briefly review the bilinear equations for a multi-component short pulse equation proposed by Matsuno [J. Math. Phys. 52, 123702 (2011)] and reaffirm its N-soliton solution in terms of pfaffians. Then by using a Bäcklund transformation of the bilinear equations and defining a discrete hodograph (reciprocal) transformation, an integrable semi-discrete multi-component short pulse equation is constructed. Meanwhile, its N-soliton solution in terms of pfaffians is also proved.

The $$q$$ q -PushASEP: A New Integrable Model for Traffic in $$1+1$$ 1 + 1 Dimension

We introduce a new interacting (stochastic) particle system $$q$$ -PushASEP which interpolates between the $$q$$ -TASEP of Borodin and Corwin (Probab Theory Relat Fields 158(1–2):225–400, 2014; see also Borodin et al., Ann Probab 42(6):2314–2382, 2014; Borodin and Corwin, Int Math Res Not 2:499–537, 2015; O’Connell and Pei, Electron J Probab 18(95):1–25, 2013; Borodin et al., Comput Math, 2013) and the $$q$$ -PushTASEP introduced recently (Borodin and Petrov, Adv Math, 2013). In the $$q$$ -PushASEP, particles can jump to the left or to the right, and there is a certain partially asymmetric pushing mechanism present. This particle system has a nice interpretation as a model of traffic on a one-lane highway. Using the quantum many body system approach, we explicitly compute the expectations of a large family of observables for this system in terms of nested contour integrals. We also discuss relevant Fredholm determinantal formulas for the distribution of the location of each particle, and connections of the model with a certain two-sided version of Macdonald processes and with the semi-discrete stochastic heat equation.