Semi-discrete Equations Research Articles

Auto-Bäcklund transformations for a matrix partial differential equation

We derive auto-Bäcklund transformations, analogous to those of the matrix second Painlevé equation, for a matrix partial differential equation. We also then use these auto-Bäcklund transformations to derive matrix equations involving shifts in a discrete variable, a process analogous to the use of the auto-Bäcklund transformations of the matrix second Painlevé equation to derive a discrete matrix first Painlevé equation. The equations thus derived then include amongst other examples a semidiscrete matrix equation which can be considered to be an extension of this discrete matrix first Painlevé equation. The application of this technique to the auto-Bäcklund transformations of the scalar case of our partial differential equation has not been considered before, and so the results obtained here in this scalar case are also new. Other equations obtained here using this technique include a scalar semidiscrete equation which arises in the case of the second Painlevé equation, and which does not seem to have been thus derived previously.

Linear second order in time energy stable schemes for hydrodynamic models of binary mixtures based on a spatially pseudospectral approximation

We develop two linear, second order energy stable schemes for solving the governing system of partial differential equations of a hydrodynamic phase field model of binary fluid mixtures. We first apply the Fourier pseudo-spectral approximation to the partial differential equations in space to obtain a semi-discrete, time-dependent, ordinary differential and algebraic equation (DAE) system, which preserves the energy dissipation law at the semi-discrete level. Then, we discretize the DAE system by the Crank-Nicolson (CN) and the second-order backward differentiation/extrapolation (BDF/EP) method in time, respectively, to obtain two fully discrete systems. We show that the CN method preserves the energy dissipation law while the BDF/EP method does not preserve it exactly but respects the energy dissipation property of the hydrodynamic model. The two new fully discrete schemes are linear, unconditional stable, second order accurate in time and high order in space, and uniquely solvable as linear systems. Numerical examples are presented to show the convergence property as well as the efficiency and accuracy of the new schemes in simulating mixing dynamics of binary polymeric solutions.

A new high-order scheme based on numerical dispersion analysis of the wave phase velocity for semidiscrete wave equations

In the numerical computation of wave equations, numerical dispersion is a persistent problem arising from inadequate discretization of the continuous wave equation. To thoroughly understand the mechanism of numerical dispersion, we separately analyze the numerical dispersion relations of time-stepping and spatial discretization schemes by Fourier analysis. The relevant results show that the numerical dispersion errors of time-marching schemes depend on the time step length or the Courant number, whereas the numerical dispersion errors of spatial discretization schemes are determined by the error between the eigenvalues of the numerical spatial differential operator and the continuous spatial differential operator. We also find that the much better numerical dispersion accuracy of the stereo-modeling discrete (SMD)-type operator can be attributed to the inclusion of diversified basis functions for its eigenvalue. Based on these findings, we combine the optimal four-stage symplectic partitioned Runge-Kutta and eighth-order SMD as a new fully discrete scheme. The subsequent analysis of its normalized phase velocity is consistent with the numerical dispersion analysis in semidiscrete forms. This is followed by an acoustic wave simulation in a homogeneous model and corresponding computational efficiency comparison. The results show that the new scheme is much more accurate and antidispersive on a coarse grid. In the final two numerical experiments, we use the new scheme to model the acoustic wave propagation in a three-layer model and the Marmousi model. The convolutional perfectly matched layer is applied to eliminate artificial boundary reflections. Our semidiscrete numerical dispersion analysis provides an efficient tool to quantitatively evaluate the time-stepping and spatial discretization schemes. It can facilitate the development of more accurate fully discrete numerical schemes for solving seismic wave equations.

Darboux Transformation for a Semidiscrete Short-Pulse Equation

We define a Darboux transformation in terms of a quasideterminant Darboux matrix on the solutions of a semidiscrete short-pulse equation. We also give a quasideterminant formula for N-loop soliton solutions and obtain a general expression for the multiloop solution expressed in terms of quasideterminants. Using quasideterminants properties, we find explicit solutions and as an example compute one- and two-loop soliton solutions in explicit form.

On soliton solutions of multi-component semi-discrete short pulse equation

The short pulse (SP) equation is an integrable equation. Multi-component generalizations of the SP equation are important for describing the polarization or anisotropic effects in optical fibers. An integrable semi-discretization of multi-component SP equation via Lax pair and Darboux transformation (DT) has been presented. We derive a Lax pair representation for the multi-component semi-discrete short pulse (sdSP) equation in the form of a block matrices by generalizing the 2 × 2 Lax pair matrices to the case of . A DT is studied for the multi-component sdSP equation and is used to compute soliton solutions of the system. Further, by expanding quasideterminants, we compute cuspon-soliton, smooth-soliton and loop-soliton solutions of the complex sdSP equation.

On the discretization of Laine equations

We consider the discretization of Darboux integrable equations. For each of the integrals of a Laine equation we constructed either a semi-discrete equation which has that integral as an n-integral, or we proved that such an equation does not exist. It is also shown that all constructed semi-discrete equations are Darboux integrable.

Convergence of exponential attractors for a time semi-discrete reaction-diffusion equation

We consider a time semi-discretization of a generalized Allen–Cahn equation with time-step parameter $$\tau $$ . For every $$\tau $$ , we build an exponential attractor $$\mathcal {M}_\tau $$ of the discrete-in-time dynamical system. We prove that $$\mathcal {M}_\tau $$ converges to an exponential attractor $$\mathcal {M}_0$$ of the continuous-in-time dynamical system for the symmetric Hausdorff distance as $$\tau $$ tends to 0. We also provide an explicit estimate of this distance and we prove that the fractal dimension of $$\mathcal {M}_\tau $$ is bounded by a constant independent of $$\tau $$ . Our construction is based on the result of Efendiev, Miranville and Zelik concerning the continuity of exponential attractors under perturbation of the underlying semi-group. Their result has been applied in many situations, but here, for the first time, the perturbation is a discretization. Our method is applicable to a large class of dissipative problems.

Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian

We study the equations\begin{document}$\begin{align}\partial_t u(t, n) = L u(t, n) + f(u(t, n), n); \partial_t u(t, n) = iL u(t, n) + f(u(t, n), n)\end{align}$\end{document}and\begin{document}$\begin{align}\partial_{tt} u(t, n) =Lu(t, n) + f(u(t, n), n), \end{align}$\end{document}where $n∈ \mathbb{Z}$, $t∈ (0, ∞)$, and $L$ is taken to be either the discrete Laplacian operator $Δ_\mathrm{d} f(n)=f(n+1)-2f(n)+f(n-1)$, or its fractional powers $-(-Δ_{\mathrm{d}})^{σ}$, $0<σ<1$. We combine operator theory techniques with the properties of the Bessel functions to develop a theory of analytic semigroups and cosine operators generated by $Δ_\mathrm{d}$ and $-(-Δ_\mathrm{d})^{σ}$. Such theory is then applied to prove existence and uniqueness of almost periodic solutions to the above-mentioned equations. Moreover, we show a new characterization of well-posedness on periodic Hölder spaces for linear heat equations involving discrete and fractional discrete Laplacians. The results obtained are applied to Nagumo and Fisher-KPP models with a discrete Laplacian. Further extensions to the multidimensional setting $\mathbb{Z}^N$ are also accomplished.

Insensitizing controls for a semilinear parabolic equation: A numerical approach

In this paper, we study the insensitizing control problem in the discrete setting of finite-differences. We prove the existence of a control that insensitizes the norm of the observed solution of a 1-D semi discrete parabolic equation. We derive a (relaxed) observability estimate that yields a controllability result for the cascade system arising in the insensitizing control formulation. Moreover, we deal with the problem of computing numerical approximations of insensitizing controls for the heat equation by using the Hilbert Uniqueness Method (HUM). We present various numerical illustrations.

Accurate artificial boundary conditions for the semi-discretized linear Schrödinger and heat equations on rectangular domains

The aim of this paper is to design some accurate artificial boundary conditions for the semi-discretized linear Schrödinger and heat equations in rectangular domains. The Laplace transform in time and discrete Fourier transform in space are applied to get Green’s functions of the semi-discretized equations in unbounded domains with single-source. An algorithm is given to compute these Green’s functions accurately through some recurrence relations. Furthermore, the finite-difference method is used to discretize the reduced problem with accurate boundary conditions. Numerical simulations are presented to illustrate the accuracy of our method in the case of the linear Schrödinger and heat equations. It is shown that the reflection at the corners is correctly eliminated.

N-Bright-Dark Soliton Solution to a Semi-Discrete Vector Nonlinear Schrödinger Equation

In this paper, a general bright-dark soliton solution in the form of Pfaffian is constructed for an integrable semi-discrete vector NLS equation via Hirota's bilinear method. One- and two-bright-dark soliton solutions are explicitly presented for two-component semi-discrete NLS equation; two-bright-one-dark, and one-bright-two-dark soliton solutions are also given explicitly for three-component semi-discrete NLS equation. The asymptotic behavior is analysed for two-soliton solutions.

The modified semi-discrete two-dimensional Toda lattice with self-consistent sources

In this paper, we derive the Grammian determinant solutions to the modified semi-discrete two-dimensional Toda lattice equation, and then construct the semi-discrete two-dimensional Toda lattice equation with self-consistent sources via source generation procedure. The algebraic structure of the resulting coupled modified differential–difference equation is clarified by presenting its Grammian determinant solutions and Casorati determinant solutions. As an application of the Grammian determinant and Casorati determinant solution, the explicit one-soliton and two-soliton solution of the modified semi-discrete two-dimensional Toda lattice equation with self-consistent sources are given. We also construct another form of the modified semi-discrete two-dimensional Toda lattice equation with self-consistent sources which is the Backlund transformation for the semi-discrete two-dimensional Toda lattice equation with self-consistent sources.

Localized Linear Systems in Sequential Implicit Simulation of Two-Phase Flow and Transport

Summary Implicit-reservoir-simulation models offer improved robustness compared with semi-implicit or explicit alternatives. The implicit treatment gives rise to a large nonlinear algebraic system of equations that must be solved at each timestep. Newton-like iterative methods are often used to solve these nonlinear systems. At each nonlinear iteration, large and sparse linear systems must be solved to obtain the Newton update vector. It is observed that these computed Newton updates are often sparse, even though the sum of the Newton updates over a converged iteration may not be. Sparsity in the Newton update suggests the presence of a spatially localized propagation of corrections along the nonlinear iteration sequence. Substantial computational savings may be realized by restricting the linear-solution process to obtain only the nonzero update elements. This requires an a priori identification of the set of nonzero update elements. To preserve the convergence behavior of the original Newton-like process, it is necessary to avoid missing any nonzero element in the identification procedure. This ensures that the localized and full linear computations result in the same solution. As a first step toward the development of such a localization method for general fully implicit simulation, the focus is on sequential implicit methods for general two-phase flow. Theoretically conservative, a priori estimates of the anticipated Newton-update sparsity pattern are derived. The key to the derivation of these estimates is in forming and solving simplified forms of infinite-dimensional Newton iteration for the semidiscrete residual equations. Upon projection onto the discrete mesh, the analytical estimates produce a conservative indication on the update's sparsity pattern. The algorithm is applied to several large-scale computational examples, and more than a 10-fold reduction in simulation time is attained. The results of the localized and full simulations are identical, as is the nonlinear convergence behavior.

On a class of Darboux-integrable semidiscrete equations

We consider a classification problem for Darboux-integrable hyperbolic semidiscrete equations. In particular, we obtain a complete description for a special class of equations admitting four-dimensional characteristic x-rings and two-dimensional characteristic n-rings. For all described equations, the corresponding x- and n-integrals are constructed.

Strong Convergence Analysis of the Stochastic Exponential Rosenbrock Scheme for the Finite Element Discretization of Semilinear SPDEs Driven by Multiplicative and Additive Noise

In this paper, we consider the numerical approximation of a general second order semilinear stochastic spartial differential equation (SPDE) driven by multiplicative and additive noise. Our main interest is on such SPDEs where the nonlinear part is stronger than the linear part also called stochastic reactive dominated transport equations. Most numerical techniques, including current stochastic exponential integrators lose their good stability properties on such equations. Using finite element for space discretization, we propose a new scheme appropriated on such equations, called stochastic exponential Rosenbrock scheme based on local linearization at every time step of the semi-discrete equation obtained after space discretization. We consider noise with finite trace and give a strong convergence proof of the new scheme toward the exact solution in the root-mean-square $$L^2$$ norm. Numerical experiments to sustain theoretical results are provided.

Multisoliton solutions of integrable discrete and semi-discrete principal chiral equations

Using a quasideterminant Darboux transformation matrix, we construct soliton solutions of nonlinear integrable discrete and semi-discrete principal chiral equations (PCEs). A Darboux transformation is defined for the matrix solutions of the discrete PCE in terms of matrix solutions to the Lax pair. The solutions are expressed in terms of quasideterminants. By taking continuum limit of one independent discrete variable, we also compute quasideterminant solutions of semi-discrete PCEs. Explicit expressions of one and two soliton solutions of the discrete PCE are obtained from a seed solution by using properties of quasideterminants. It has been shown that the soliton solutions of the discrete system reduce to those of semi-discrete and usual continuum PCEs by applying appropriate limits.

Analysis of time integration methods for the compressible two-fluid model for pipe flow simulations

In this paper we analyse different time integration methods for the two-fluid model and propose the BDF2 method as the preferred choice to simulate transient compressible multiphase flow in pipelines. Compared to the prevailing Backward Euler method, the BDF2 scheme has a significantly better accuracy (second order) while retaining the important property of unconditional linear stability (A-stability). In addition, it is capable of damping unresolved frequencies such as acoustic waves present in the compressible model (L-stability), opposite to the commonly used Crank–Nicolson method. The stability properties of the two-fluid model and of several discretizations in space and time have been investigated by eigenvalue analysis of the continuous equations, of the semi-discrete equations, and of the fully discrete equations. A method for performing an automatic von Neumann stability analysis is proposed that obtains the growth rate of the discretization methods without requiring symbolic manipulations and that can be applied without detailed knowledge of the source code.The strong performance of BDF2 is illustrated via several test cases related to the Kelvin–Helmholtz instability. A novel concept called Discrete Flow Pattern Map (DFPM) is introduced which describes the effective well-posed unstable flow regime as determined by the discretization method. Backward Euler introduces so much numerical diffusion that the theoretically well-posed unstable regime becomes numerically stable (at practical grid and timestep resolution). BDF2 accurately identifies the stability boundary, and reveals that in the nonlinear regime ill-posedness can occur when starting from well-posed unstable solutions. The well-posed unstable regime obtained in nonlinear simulations is therefore in practice much smaller than the theoretical one, which might severely limit the application of the two-fluid model for simulating the transition from stratified flow to slug flow. This should be taken very seriously into account when interpreting results from any slug-capturing simulations.

On the nonlinear dynamics of shell structures: Combining a mixed finite element formulation and a robust integration scheme

In this work, we present an approach to analyze the nonlinear dynamics of shell structures, which combines a mixed finite element formulation and a robust integration scheme. The structure is spatially discretized with extensible-director-based solid-degenerate shells. The semi-discrete equations are temporally discretized with a momentum-preserving, energy-preserving/decaying method, which allows to mitigate the effects due to unresolved high-frequency content. Additionally, kinematic constraints are employed to render structural junctions. Finally, the method, which can be used to analyze blades of wind turbines or wings of airplanes effectively, is tested and its capabilities are illustrated by means of examples.

Spectral variational integrators for semi-discrete Hamiltonian wave equations

Abstract In this paper, we present a highly accurate Hamiltonian structure-preserving numerical method for simulating Hamiltonian wave equations. This method is obtained by applying spectral variational integrators (SVI) to the system of Hamiltonian ODEs which are derived from the spatial semi-discretization of the Hamiltonian PDE. The spatial variable is discretized by using high-order symmetric finite-differences. An efficient implementation of SVI for high-dimensional systems of ODEs is presented.

On a new semi-discrete integrable combination of Burgers and Sharma-Tasso-Olver equation.

In this paper, a new semi-discrete integrable combination of Burgers and Sharma-Tasso-Olver equation is investigated. The underlying integrable structures like the Lax pair, the infinite number of conservation laws, the Darboux-Bäcklund transformation, and the solutions are presented in the explicit form. The theory of the semi-discrete equation including integrable properties yields the corresponding theory of the continuous counterpart in the continuous limit. Finally, numerical experiments are provided to demonstrate the effectiveness of the developed integrable semi-discretization algorithms.