A linearized compact ADI numerical method for the two-dimensional nonlinear delayed Schrödinger equation

ABSTRACTAs we know, second-order backward differentiation formula (BDF2) is L-stable scheme, which can damp unwanted finite oscillations, while Crank-Nicolson (C-N) method is only A-stable scheme and usually causes numerical oscillations. Thus, BDF2 may be more popular than C-N. However, up to now, most of studies focus on the error analysis of the compact alternating direction implicit (ADI) method on the basis of C-N method for temporal discretization in -norm. This paper is devoted to the convergence analysis of a compact multi-step ADI method used to solve a two-dimensional (2D) parabolic equation in -, - and -norms. Besides, the existence of the ADI solution is also analyzed in detail. A class of Richardson extrapolation algorithms are established to improve computational efficiency. By introducing a transformation, our algorithms are easily generalized to the solution of convection-diffusion problems with constant coefficients. Numerical results confirm the performance of our algorithms and support theoretical analysis.

This study was suggested by previous work on the simulation of evolution equations with scale-dependent processes, e.g., wave-propagation or heat-transfer, that are modeled by wave equations or heat equations. Here, we study both parabolic and hyperbolic equations. We focus on ADI (alternating direction implicit) methods and LOD (locally one-dimensional) methods, which are standard splitting methods of lower order, e.g. second-order. Our aim is to develop higher-order ADI methods, which are performed by Richardson extrapolation, Crank-Nicolson methods and higher-order LOD methods, based on locally higher-order methods. We discuss the new theoretical results of the stability and consistency of the ADI methods. The main idea is to apply a higher- order time discretization and combine it with the ADI methods. We also discuss the dis- cretization and splitting methods for first-order and second-order evolution equations. The stability analysis is given for the ADI method for first-order time derivatives and for the LOD (locally one-dimensional) methods for second-order time derivatives. The higher-order methods are unconditionally stable. Some numerical experiments verify our results. AMS subject classifications: 35J60, 35J65, 65M99, 65N12, 65Z05, 74S10, 76R50, 80A20, 80M25

An alternating direction implicit (ADI) numerical method was used to simulate transient solute movement resulting from a step‐function increase of solute concentration in water applied to a shallow sand tank system with a subsurface drain. Steady water flow in the saturated portion of the tank was simulated using an iterative alternating direction implicit (IADI) numerical method. The saturated‐unsaturated condition in the actual sand tank was approximated by a smaller rectangular flow region of saturation in the model simulation. Isopleths of hydraulic head and solute concentration within the soil demonstrated a decrease in the rate of infiltration from the point vertically above the drain to the point vertically above the midpoint between the drains. Consequently, solute transport was much faster in the surface soil overlying the drain than in the soil near the mid‐point between drains. When cumulative effluent volume was expressed as the number of pore volumes, breakthrough curves for solute concentration in the drainage water revealed early arrival of solute compared to a one‐dimensional system. In spite of the different boundary conditions imposed at the soil surface in the tank and in the model, reasonable agreement was obtained between simulated breakthrough curves and experimental data from the sand tank.

In this paper, a novel compact alternating direction implicit (ADI) scheme is proposed for solving the time-fractional subdiffusion equation in two space dimensions. The established scheme is based on the modified L1 method in time and the compact finite difference method in space. The unique solvability, unconditionally stability and convergence of the scheme are proved. The derived compact ADI scheme is coincident with the one for 2D integer order parabolic equation when the , where is the order of the Riemann–Liouville derivative operator. In addition, the novel ADI scheme is used to solve the 2D modified fractional diffusion equation, and the corresponding stability and convergence results are also given. Numerical results are provided to verify the theoretical analysis.

Scattering theory for the wave equation in two-dimensional space, perturbed by a finite function of a radial variable, integrable everywhere except, perhaps, the origin of coordinates, is considered from the point of view of the LaxPhillips scheme. The compression operator, related to the corresponding scattering problem, is considered. It is shown that this compression has one-dimensional defect subspaces, and its characteristic operator-function is a meromorphic function, whose zeros and poles coincide, respectively, with the corresponding values of a dissipative operator and its adjoint. The solution of the inverse scattering problem is obtained by reducing it to the inverse problem with two spectra for the singular self-adjoint Sturm-Liouville operator.

In this article, we develop an exponential high order compact alternating direction implicit (EHOC ADI) method for solving three dimensional (3D) unsteady convection–diffusion equations. The method, which requires only a regular seven‐point 3D stencil similar to that in the standard second‐order methods, is second order accurate in time and fourth‐order accurate in space and unconditionally stable. The resulting EHOC ADI scheme in each alternating direction implicit (ADI) solution step corresponding to a strictly diagonally dominant matrix equation can be solved by the application of the one‐dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. Numerical experiments for three test problems are carried out to demonstrate the performance of the present method and to compare it with the classical Douglas–Gunn ADI method and the Karaa's high‐order compact ADI method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

On the complete integrability of a nonlinear system of partial differential equations in two-dimensional space

In this paper, an analytical solution for the stress fields in the close neighbourhoods of radiused notches in thick orthotropic plates under shear loading and twisting is provided. In the first step, the equations of the three-dimensional theory of elasticity are successfully reduced to two uncoupled equations in two-dimensional space. Later, the 3D stress field solution for orthotropic plates with radiused notches is presented and its degree of accuracy is discussed by comparing theoretical results and numerical data from 3D FE analyses. The solution proposed can be satisfactorily used to characterise the stress field in plates made with polymeric composite materials, such as fibre-reinforced polymers and natural composites.

This paper compares the finite difference and finite volume methods for solving time-fractional diffusion equations. These methods are widely known for diffusion equations with integer order, but their effectiveness for time-fractional diffusion equations has not been sufficiently studied. The definition of the Grunwald-Letnikov fractional derivative is used to approximate the equation. An explicit difference scheme for the finite difference method is obtained and a stability condition for the fractional time order difference scheme is derived, which is also a generalisation for parabolic and hyperbolic type equations, which was previously unknown for schemes with a fractional time order. An explicit discrete form for solving subdiffusion equations in two-dimensional space with fractional time order by the finite volume method is presented. Numerical results show that the finite difference method demonstrates high accuracy, while the finite volume method is better suited for complex geometries. These findings provide insights for future developments in anomalous diffusion modeling.

The authors consider a simple transport equation in one-dimensional space and the linearized shallow water equations in two-dimensional space, and describe and implement a multilevel finite-volume discretization in the context of the utilization of the incremental unknowns. The numerical stability of the method is proved in both cases.

In this paper a general integral formula for the fundamental solution of a linear partial difference equation with constant coefficients is investigated. For equations in twodimensional space, especially for the discretization of the Helmholtz equation, special integral formulae are given and the convergence of corresponding integrals is proved. The limits of the fundamental solution at infinity are calculated in several cases.

In this chapter we will show that to each ℤ-graded Lie algebra there is associated a whole series of systems of equations in two-dimensional space which are completely integrable in the sense of the Goursat problem with initial values determined on characteristics. The explicit form of the nonlinear systems arising in this way depends on the structure of the Lie algebra and its grading. The integration of the system is closely related to the representation theory of the Lie algebra and the corresponding group.

Basis functions associated with the two-component hyperbolic Dirac equation were obtained. The expansion via the basis functions separates time and spatial variables in a linearized Davey–Stewartson I equation. It is possible to solve this equation by the Fourier method.

Classical finite element method (FEM) has been applied to solve some fractional differential equations, but its scheme has too many degrees of freedom. In this paper, a low-dimensional FEM, whose number of basis functions is reduced by the theory of proper orthogonal decomposition (POD) technique, is proposed for the time fractional diffusion equation in two-dimensional space. The presented method has the properties of low dimensions and high accuracy so that the amount of computation is decreased and the calculation time is saved. Moreover, error estimation of the method is obtained. Numerical example is given to illustrate the feasibility and validity of the low-dimensional FEM in comparison with traditional FEM for the time fractional differential equations.

Two higher-order compact finite difference approaches, the Hermitian and the Lax-Wendroff are examined by applying them to the viscous Burgers' equation. The difference equations obtained by the two methods were integrated in time through the third-order Strong-Stability-Preserving Runge-Kutta scheme. Absolute errors are computed by using an exact solution. The results are also compared with a second order central difference scheme. The Hermitian approach is far easier to implement. On uniform grids the Lax-Wendroff scheme produces smaller errors during the initial stages, but both methods are equally good for larger durations of integration. The convergence rate of the Hermitian scheme is slightly higher than the Lax-Wendroff scheme although both are of fourth order. Both schemes are unstable beyond a certain step size in time and space. When numerical boundary conditions are imposed, second-order conditions produce the best results whereas linear extrapolation proved to be the worst. It was also observed that large domains were required to implement the numerical boundary conditions properly. There was no detrimental effect on the accuracy of the results obtained through either of the two schemes when the size of the domain was greatly increased. Both schemes showed remarkable improvement in accuracy when clustered grids were employed. However much smaller time steps are required for stable solutions.