Semi-discrete Equations Research Articles

Numerical simulation of a continuum model for bi-directional pedestrian flow

An algorithm for an extended reactive dynamic user equilibrium model of pedestrian counterflow as a continuum is developed. It is based on a cell-centered high-resolution finite volume scheme with a fast sweeping method for an Eikonal-type equation on an orthogonal grid. A high-order total variation diminishing Runge–Kutta method is adopted for the time integration of semi-discrete equations. The numerical results demonstrate the rationality of the model and efficiency of the algorithm. Some crowd pedestrian flow phenomena, such as dynamic lane formation in bi-directional flow, are observed which are helpful for a global comprehension of pedestrian dynamics. Also, the model can be utilized with different potential applications.

Discrete transparent boundary conditions for parabolic equations

There are simple algorithms for constructing transparent boundary conditions (TBCs) for a partial discretization of the basic parabolic equation that is known as a “semi-discrete” parabolic equation. This equation and some of these algorithms are reviewed. Solutions of a semi-discrete parabolic equation in a long rectangular strip subject to TBCs at the long edges of the strip are then considered. These solutions can be computed accurately and efficiently with a pseudospectral method that is based on expansions in Chebyshev polynomials. It is beneficial to combine this method with a conventional split-step FFT solution of a parabolic equation subject to Neumann boundary conditions at the long edges of the strip. This hybrid approach will be called the “decomposition method.” It is demonstrated in a computation of radiation modes from the termination of a truncated nonlinear internal gravity wave duct in a shallow water area.

A nearly analytic symplectically partitioned Runge-Kutta method for 2-D seismic wave equations

SUMMARY In this paper, we develop a new nearly analytic symplectically partitioned Runge–Kutta (NSPRK) method for numerically solving elastic wave equations. In this method, we first transform the elastic wave equations into a Hamiltonian system, and use the nearly analytic discrete operator to approximate the high-order spatial differential operators, and then we employ the partitioned second-order symplectic Runge–Kutta method to numerically solve the resulted semi-discrete Hamiltonian ordinary differential equations (ODEs). We investigate in great detail on the properties of the NSPRK method that includes the stability condition for the P–SV wave in a 2-D homogeneous isotropic medium, the computational efficiency, and the numerical dispersion relation for the 2-D acoustic case. Meanwhile, we apply the NSPRK to simulate the elastic wave propagating in several multilayer models with both strong velocity contrasts and fluctuating interfaces. Both theoretical analysis and numerical results show that the NSPRK can effectively suppress the numerical dispersion resulted from the discretization of the wave equations, and more importantly, it preserves the symplecticity structure for long-time computation. In addition, numerical experiments demonstrate that the NSPRK is effective to combine the split perfectly matched layer boundary conditions to take care of the reflections from the artificial boundaries.

Reseach on Secondary Cooling Dynamic Control Model in Continuous Casting of Slabs

one improved finite volume method is proposed, expatiating on the basic idea and integrating the partial derivative equation for the secondary spary water dynamic cooling control model. The semi-discrete equation for derivative of time is obtained.Combinating with an introducted slab continuous casting production line in some steel mill, according to the Q235 actual parameters, what is obtained is surface temperature distribution, the changing of surface temperature and the water flow rate because of the varying of casting speed. The results of simulations performed using the mathematical model are validated against the measured values and a good agreement is observed, Verifing the validity of the model.The model occupies less computer memory, solving simple

Convergence of an Inverse Problem for a 1-D Discrete Wave Equation

It is by now well known that one can recover a potential in the wave equation from the knowledge of the initial waves, the boundary data, and the flux on a part of the boundary satisfying the Gamma-conditions of J.-L. Lions. We are interested in proving that when trying to fit the discrete flux, given by discrete approximations of the wave equation, with the continuous flux, one recovers, at the limit, the potential of the continuous model. In order to do that, we shall develop a Lax-type argument, usually used for convergence results of numerical schemes, which states that consistency and uniform stability imply convergence. In our case, the most difficult part of the analysis is the one corresponding to the uniform stability that we shall prove using new uniform discrete Carleman estimates, where uniform means with respect to the discretization parameter. We shall then deduce a convergence result for the discrete inverse problems. Our analysis will be restricted to the $1$-d (one-dimensional) case for space semidiscrete wave equations discretized on a uniform mesh using a finite-difference approach.

Implicit peer methods for large stiff ODE systems

Implicit two-step peer methods are introduced for the solution of large stiff systems. Although these methods compute s-stage approximations in each time step one-by-one like diagonally-implicit Runge-Kutta methods the order of all stages is the same due to the two-step structure. The nonlinear stage equations are solved by an inexact Newton method using the Krylov solver FOM (Arnoldi’s method). The methods are zero-stable for arbitrary step size sequences. We construct different methods having order p=s in the multi-implicit case and order p=s−1 in the singly-implicit case with arbitrary step sizes and s≤5. Numerical tests in Matlab for several semi-discretized partial differential equations show the efficiency of the methods compared to other Krylov codes.

Fast approximations of bond option prices under CKLS models

A new computational method for approximating prices of zero-coupon bonds and bond option prices under general Chan–Karolyi–Longstaff–Schwartz models is proposed. The pricing partial differential equations are discretized using second-order finite difference approximations and an exponential time integration scheme combined with best rational approximations based on the Carathéodory–Fejér procedure is employed for solving the resulting semi-discrete equations. The algorithm has a linear computational complexity and provides accurate bond and European bond option prices. We give several numerical results which illustrate the computational efficiency of the algorithm and uniform second-order convergence rates for the computed bond and bond option prices.

Finite dimensional global attractor for a semi-discrete nonlinear Schrödinger equation with a point defect

We consider a semi-discrete in time Crank-Nicolson scheme to discretize a weakly-damped forced nonlinear Schrödinger equation with a delta-function impurity in one space dimension. We prove that such a semi-discrete equation provides a discrete infinite dimensional dynamical system in H 1 ( R ) that possesses a global attractor in H 1 ( R ) . We show also that this global attractor is actually a compact set of H 3 2 - ε ( R ) and has a finite fractal dimension.

Dispersion and transient analyses of Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration for thin beam and plate structures

A dispersion analysis is carried out to study the dynamic behavior of the Hermite reproducing kernel (HRK) Galerkin meshfree formulation for thin beam and plate problems. The HRK approximation utilizes both the nodal deflectional and rotational variables to construct the meshfree approximation of the deflection field within the reproducing kernel framework. The discrete Galerkin formulation is fulfilled with the method of sub-domain stabilized conforming integration. In the dispersion analysis following the HRK Galerkin meshfree semi-discretization, both the deflectional and rotational nodal variables are expressed by harmonic functions and then substituted into the semi-discretized equation to yield the characteristic equation. Subsequently the numerical frequency and phase speed can be obtained. The transient analysis with full-discretization is performed by using the central difference time integration scheme. The results of dispersion analysis of thin beams and plates show that compared to the conventional Gauss integration-based meshfree formulation, the proposed method has more favorable dispersion performance. Thereafter the superior performance of the present method is also further demonstrated by several transient analysis examples.

Multi-stage solvers optimized for damping and propagation

Explicit multi-stage solvers are routinely used to solve the semi-discretized equations that arise in Computational Fluid Dynamics (CFD) problems. Often they are used in combination with multi-grid methods. In that case, the role of the multi-stage solver is to efficiently reduce the high frequency modes on the current grid and is called a smoother. In the past, when optimizing the coefficients of the scheme, only the damping characteristics of the smoother were taken into account and the interaction with the remainder of the multi-grid cycle was neglected. Recently it had been found that coefficients that result in less damping, but allow for a higher Courant–Friedrichs–Lewy (CFL) number are often superior to schemes that try to optimize damping alone. While this is certainly true for multi-stage schemes used as a stand-alone solver, we investigate in this paper if using higher CFL numbers also yields better results in a multi-grid setting. We compare the results with a previous study we conducted and where a more accurate model of the multi-grid cycle was used to optimize the various parameters of the solver. We show that the use of the more accurate model results in better coefficients and that in a multi-grid setting propagation is of little importance. We also look into the gains to be made when we allow the parameters to be different for the pre- and post-smoother and show that even better coefficients can be found in this way.

Analyses of Laminated Cantilever Composite Beams by Model Order Reduction Techniques

In this study, a laminated cantilever composite beam is studied in frequency domain by using some popular model order reduction techniques. In the analyses, component mode synthesis (CMS) and quasi-static mode synthesis (QSM1, QSM2) methods are applied to semi-discrete finite element equations to obtain reduced order models for structural analyses. The performance of the model order reduction methods is compared. In addition, fiber orientation, stacking sequence, ply thickness, and its location effects on the number of modes of the methods and on the natural frequencies are investigated. Results reveal that the performance of the QSM2 method is found to be better than the others. It is observed that the fiber orientation, stacking sequence, ply thickness, and its location parameters affect the natural frequency values and the number of modes of the methods.

Transient heat conduction analysis using the MLPG method and modified precise time step integration method

The meshless local Petrov–Galerkin (MLPG) method in conjunction with the modified precise time step integration method in the time domain is proposed for transient heat conduction analysis in this paper. The MLPG method is often referred to as a truly meshless method because it requires no elements or background cells for either field interpolation or background integration. Local weak forms are developed using weighted residual method locally from the partial differential equation of transient heat conduction. In order to simplify the treatment of essential boundary conditions, the natural neighbour interpolation (NNI) is employed for the construction of trial functions. Moreover, the three-node triangular FEM shape functions are taken as test functions to reduce the order of integrands involved in domain integrals. The semi-discrete heat conduction equation is solved numerically with modified precise time step integration method in the time domain. The availability and accuracy of the present method for transient heat conduction analysis are tested through numerical examples.

Component splitting for semi-discrete Maxwell equations

A time-integration scheme for semi-discrete linear Maxwell equations is proposed. Special for this scheme is that it employs component splitting. The idea of component splitting is to advance the greater part of the components of the semi-discrete system explicitly in time and the remaining part implicitly. The aim is to avoid severe step size restrictions caused by grid-induced stiffness emanating from locally refined space grids. The proposed scheme is a blend of an existing second-order composition scheme which treats wave terms explicitly and the second-order implicit trapezoidal rule. The new blended scheme retains the composition property enabling higher-order composition.

High accuracy wave simulation – Revised derivation, numerical analysis and testing of a nearly analytic integration discrete method for solving acoustic wave equation

The nearly analytic integration discrete (NAID) method for solving the two-dimensional acoustic wave equation has been fully mathematically revised, analyzed and tested. The NAID method is an alternative numerical modeling method for generating synthetic seismograms. The acoustic wave equation is first transformed into a system of first-order ordinary differential equations (ODEs) with respect to time variable t, and then directly integrated at a small time interval of [tn,tn+1] to obtain semi-discrete ordinary differential equations. The integral kernel is expanded into a truncated Taylor series, to which the integration operator is explicitly applied. The high-order temporal derivatives involved in the integral kernel are replaced by high-order spatial derivatives, which then are approximately calculated as a weighted linear combination of the displacement, the particle-velocity, and their spatial gradients. In this article, we investigate the theoretical properties of the revised NAID method, including the discrete error and the stability criteria. Numerical results for constant and layered velocity models show that, comparing to the Lax–Wendroff correction (LWC) scheme and the staggered-grid finite difference method, the NAID method can effectively suppress the numerical dispersion and source-noises caused by the discretization of the acoustic wave equation with too-coarse spatial grids or when models have strong velocity contrasts between adjacent layers. The proposed NAID method has been applied in computing the acoustic wavefields for two heterogeneous models – the corner edge model and the Marmousi model. Promising numerical results illustrate that the NAID method provides an encouraging tool for large-scale and complex wave simulation and inversion problems based on the acoustic equation.

A Modified Semi-Discrete BKP-Type Equation with Self-Consistent Sources

In this paper, a modified semi-discrete BKP-type equation [ t -1 -flow of the BKP hierarchy (KP hierarchy of the B-type)] is firstly proposed and its pfaffian solutions are obtained. Then, utilizing the source generation procedure, this modified semi-discrete BKP-type equation with self-consistent sources (BKP-type ESCS) is presented and its pfaffian solutions are derived. This modified semi-discrete BKP-type ESCS obtained is shown to constitute a bilinear Backlund transformation for the semi-discrete BKP-type ESCS originally given by Wang et al. in 2007. This means that the commutativity of the source generation procedure (SGP) and Backlund transformations (BT) is valid for the semi-discrete BKP-type equation.

On the Numerical Discretization in Space and Time: Part 2 - Total Energy Framework and Satisfaction of Conservation Properties of Discretized Equations of Motion for Linear Dynamics Problems

For the modeling and simulation of computational dynamics applications with particular reference to holonomic-sceleronomic systems towards which this paper is restricted, it is important to ensure that the space and time numerical discretizations enable proper designs such that the fully discretized system also inherits the same physical attributes (such as energy, linear, and angular momentum conservation properties) as in the continuous case for conservative dynamic systems. Of the various computational frameworks (Newtonian, Lagrangian, Hamiltonian, and Total Energy) that we presented in Part 1, here in Part 2 of the exposition we particularly describe and advocate the so-called Total Energy computational framework for elastodynamics applications. This is because it is a self-contained framework in a natural setting which is computationally attractive. Consequently, it naturally enables the design of structure preserving time integration algorithms, and also inherits the advantages of readily ensuring proof of satisfaction of the conservation laws for the space discretized system. In contrast to Part 1, here in Part 2 of the paper we altogether discard the developments starting from the continuous representations to derive the semi-discretized equations. Instead, we describe yet another route directly starting from the supposition of energy conservation for the space discretized system. In particular, via the Total Energy framework, simply for illustration we describe the design of computational algorithms that are energy-momentum conserving within the scope of the particular class of LMS methods involving a single system of equations with a single solve within a single step representation, since these are the most popular in commercial codes. We demonstrate conserving algorithms by design for linear dynamic systems, which is a necessary first step and basis that is required for consequent extensions of the parent linear dynamics algorithms to general nonlinear dynamic applications, which we hope to disseminate in the near future.

A Split-Step Algorithm for Effectively Suppressing the Numerical Dispersion for 3D Seismic Propagation Modeling

In this article, we propose a 3D split-step algorithm (SSA) for seismic- wave simulation. We first transform the wave equations in 3D anisotropic media into a system of first-order partial differential equations with respect to time t. Then we use the multidimensional high-order interpolation method to approximate the high-order spatial derivatives, so that we obtain a system of semidiscrete ordinary differential equations (ODEs). Finally, the third-order implicit Adams method and truncated dif- ferentiator series method are applied to solve the semidiscrete ODEs. We provide the theoretical study on the properties of the 3D SSA, such as stability criteria, theoretical error, numerical error, numerical dispersion, and computational efficiency. We also compare some seismic modeling results of this method against those of some high-order finite-difference schemes. Theoretical analysis and numerical tests show that the 3D SSA is third-order accurate in time and fourth-order accurate in space. However, its computational costs and memory requirements are much less than those of the fourth-order Lax-Wendroff correction method and the fourth-order staggered- grid method. Using a multilayer elastic model with large velocity contrasts and free surface, we compare the result of the 3D SSA with that of the discrete-wavenumber method. We also present the synthetic seismograms in the 3D three-layer isotropic medium, the wave-field snapshots in the 3D two-layer medium, and the 3D trans- versely isotropic medium with a vertical symmetry axis. All these promising numer- ical results illustrate that the 3D SSA can suppress effectively the numerical dispersion caused by discretizing the wave equations when too few sampling points per mini- mum wavelength are used or when models have large velocity contrasts between adjacent layers, further resulting in both increasing the computational efficiency and saving the storage space when big grids are used.

Matrix semidiscrete Ablowitz–Ladik equation hierarchy and a matrix discrete second Painlevé equation

In this article, we first show that the semidiscrete Ablowitz–Ladik equation hierarchy can be extended to the matrix case by considering a matrix semidiscrete Ablowitz–Ladik isospectral linear problem. We then extend this linear problem to the nonisospectral case, and a new coupled matrix semidiscrete integrable equation is constructed. It is shown that the continuum limit of this new matrix semidiscrete integrable equation is the coupled matrix variable-coefficient modified Korteweg-de Vries equation and that a matrix discrete second Painlevé equation can be obtained as a stationary reduction in this new matrix semidiscrete equation.

Localized solutions for the finite difference semi-discretization of the wave equation

We study the propagation properties of the solutions of the finite difference space semi-discrete wave equation on a uniform grid of the whole Euclidean space. We provide a construction of high frequency wave packets that propagate along the corresponding bi-characteristic rays of Geometric Optics with a group velocity arbitrarily close to zero. Our analysis is motivated by control theoretical issues. In particular, the continuous wave equation has the so-called observability property: for a sufficiently large time, the total energy of its solutions can be estimated in terms of the energy concentrated in the exterior of a compact set. This fails to be true, uniformly on the mesh-size parameter, for the semi-discrete schemes and the observability constant blows-up at an arbitrarily large polynomial order. Our contribution consists in providing a rigorous derivation of those wave packets and in analyzing their behavior near that ray, by taking into account the subtle added dispersive effects that the numerical scheme introduces.

Nonlinear Schrödinger Equations and Their Spectral Semi-Discretizations Over Long Times

Cubic Schrodinger equations with small initial data (or small nonlinearity) and their spectral semi-discretizations in space are analyzed. It is shown that along both the solution of the nonlinear Schrodinger equation as well as the solution of the semi-discretized equation the actions of the linear Schrodinger equation are approximately conserved over long times. This also allows us to show approximate conservation of energy and momentum along the solution of the semi-discretized equation over long times. These results are obtained by analyzing a modulated Fourier expansion in time. They are valid in arbitrary spatial dimension.