Numerical Investigations on the Resonance Errors of Multiscale Discontinuous Galerkin Methods for One-Dimensional Stationary Schrödinger Equation

In this paper,numerical experiments are carried out to investigate the impact of penalty parameters in the numerical traces on the resonance errors of high-order multiscale discontinuous Galerkin(DG)methods(Dong et al.in J Sci Comput 66:321–345,2016;Dong and Wang in J Comput Appl Math 380:1–11,2020)for a one-dimensional stationary Schrödinger equation.Previous work showed that penalty parameters were required to be positive in error analysis,but the methods with zero penalty parameters worked fine in numerical simulations on coarse meshes.In this work,by performing extensive numerical experiments,we discover that zero penalty parameters lead to resonance errors in the multiscale DG methods,and taking positive penalty parameters can effectively reduce resonance errors and make the matrix in the global linear system have better condition numbers.

One-dimensional consolidation of double-layered soil with non-Darcian flow described by exponent and threshold gradient

Based on non-Darcian flow law described by exponent and threshold gradient within a double-layered soil, the classic theory of one-dimensional consolidation of double-layered soil was modified to consider the change of vertical total stress with depth and time together. Because of the complexity of governing equations, the numerical solutions were obtained in detail by finite difference method. Then, the numerical solutions were compared with the analytical solutions in condition that non-Darcian flow law was degenerated to Dary's law, and the comparison results show that numerical solutions are reliable. Finally, consolidation behavior of double-layered soil with different parameters was analyzed, and the results show that the consolidation rate of double-layered soil decreases with increasing the value of exponent and threshold of non-Darcian flow, and the exponent and threshold gradient of the first soil layer greatly influence the consolidation rate of double-layered soil. The larger the ratio of the equivalent water head of external load to the total thickness of double-layered soil, the larger the rate of the consolidation, and the similitude relationship in classical consolidation theory of double-layered soil is not satisfied. The other consolidation behavior of double-layered soil with non-Darcian flow is the same as that with Darcy's law.

One-dimensional nonlinear consolidation analysis of soil with continuous drainage boundary

Following the assumptions proposed by MESRI and ROKHSAR,the one-dimensional nonlinear consolidation problem of soil under constant loading is studied by introducing continuous drainage boundary.The numerical solution is derived by using finite difference method and its correctness is assessed by comparing with existing analytical and numerical solutions.Based on the present solution,the effects of interface parameters,stress ratios(i.e.,final effective stress over initial effective stress,N_(σ))and the ratio c_(c)/c_(k)of compression index to permeability index on the consolidation behavior of soil are studied in detail.The results show that,the characteristics of one-dimensional nonlinear consolidation of soil are not only related to c_(c)/c_(k)and N_(σ),but also related to boundary conditions.In the engineering practice,the soil drainage rate of consolidation process can be designed by adjusting the values of interface parameters.

Nonlinear analytical solution for one-dimensional consolidation of soft soil under cyclic loading

This paper presents an analytical solution for one-dimensional consolidation of soft soil under some common types of cyclic loading such as trapezoidal cyclic loading, based on the assumptions proposed by Davis and Raymond （1965） that the decrease in permeability is proportional to the decrease in compressibility during the consolidation process of the soil and that the distribution of initial effective stress is constant with depth the solution obtained, some diagrams are prepared and the It is verified by the existing analytical solutions in special cases. Using telex ant consolidation behavior is investigated.

Analytical solution to one-dimensional consolidation in unsaturated soils

This paper presents an analytical solution of the one-dimensional consolidation in unsaturated soil with a finite thickness under vertical loading and confinements in the lateral directions. The boundary contains the top surface permeable to water and air and the bottom impermeable to water and air. The analytical solution is for Fredlund＇s one-dimensional consolidation equation in unsaturated soils. The transfer relationship between the state vectors at top surface and any depth is obtained by using the Laplace transform and Cayley-Hamilton mathematical methods to the governing equations of water and air, Darcy＇s law and Fick＇s law. Excess pore-air pressure, excess pore-water pressure and settlement in the Laplace-transformed domain are obtained by using the Laplace transform with the initial conditions and boundary conditions. By performing inverse Laplace transforms, the analytical solutions are obtained in the time domain. A typical example illustrates the consolidation characteristics of unsaturated soil from an- alytical results. Finally, comparisons between the analytical solutions and results of the finite difference method indicate that the analytical solution is correct.

Study on one-dimensional consolidation of soil under cyclic loading and with varied compressibility

This paper presents a semi-analytical method to solve one dimensional consolidation problem by taking consideration of varied compressibility of soil under cyclic loading. In the method, soil stratum is divided equally into n layers while load and consolidation time are also divided into small parts and time intervals accordingly. The problem of one-dimensional consolidation of soil stratum under cyclic loading can then be dealt with at each time interval as one-dimensional linear consolidation of multi-layered soils under constant loading. The compression or rebounding of each soil layer can be judged by the effective stress of the layer. When the effective stress is larger than that in the last time interval, the soil layer is compressed, and when it is smaller, the soil layer rebounds. Thus, appropriate compressibility can be chosen and the consolidation of the layered system can be analyzed by the available analytical linear consolidation theory. Based on the semi-analytical method, a computer program was developed and the behavior of one-dimensional consolidation of soil with varied compressibility under cyclic loading was investigated, and compared with the available consolidation theory which takes no consideration of varied compressibility of soil under cyclic loading. The results showed that by taking the variable compressibility into account, the rate of consolidation of soil was greater than the one predicted by conventional consolidation theory.

General semi-analytical solutions to one-dimensional consolidation for unsaturated soils

This paper presents general semi-analytical solutions to Fredlund and Hasan＇s one-dimensional （1D） consolidation equations for unsaturated soils subject to different initial conditions, homogeneous boundaries and time-dependent loadings. Two variables are introduced to transform the two-coupled governing equations of pore-water and poreair pressures into an equivalent set of partial differential equations （PDEs）, which are solved with the Laplace transform method. The pore-water and pore-air pressures and settlement are obtained in the Laplace transform domMn. The Crump＇s method is used to perform inverse Laplace transform to obtain the solutions in the time domain. The present solutions are more general in practical applications and show good agreement with the previous solutions in the literature.

STABILITY OF VISCOUS SHOCK WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

We study the large-time behavior toward viscous shock waves to the Cauchy problem of the one-dimensional compressible isentropic Navier-Stokes equations with density- dependent viscosity. The nonlinear stability of the viscous shock waves is shown for certain class of large initial perturbation with integral zero which can allow the initial density to have large oscillation. Our analysis relies upon the technique developed by Kanel~ and the continuation argument.

Exact Solution of Terzaghi’s Consolidation Equation and Extension to Two/Three-Dimensional Cases

The differential equation by Terzaghi and Fr?hlich, better known as Terzaghi’s one-dimensional consolidation equation, simulates the visco-elastic behaviour of soils depending on the loads applied as it happens, for example, when foundations are laid and start carrying the weight of the structure. Its application is traditionally based on Taylor’s solution that approximates experimental results by introducing non-dimensional variables that, however, contradict the actual behaviour of soils. The proposal of this research is an exact solution consisting in a non-linear equation that can be considered correct as it meets both mathematical and experimental requirements. The solution proposed is extended to include differential equations relating to two/three dimensional consolidation by adopting a transversally isotropic model more consistent with the inner structure of soils.

ONE-DIMENSIONAL CONSOLIDATION OF LAYERED AND VISCO-ELASTIC SOILS UNDER ARBITRARY LOADING

Based on the layered visco-elastic soil model, according to the Terzaghi's one dimensional consolidation theory, by the method of Laplace transform and matrix transfer technique, the problems about the consolidation of layered and saturated visco-elastic soils under arbitrary loading were solved. Through deductions, the general solution, in the terms of layer thickness, the modulus and the coefficients of permeability and Laplacian transform's parameters was obtained. The strain and deformation of the layered and saturated visco-elastic soils under arbitrary loading can be calculated by Laplace inversion. According to the results of several numerical examples, the consolidation of visco-elastic soils logs behind that of elastic soils. The development of effective stress and the displacement is vibrant process under cyclic loading. Finally, an engineering case is studied and the results prove that the methods are very effective.

Semi-analytical solution to one-dimensional consolidation in unsaturated soils

In this paper, a series of semi-analytical solutions to one-dimensional consolidation in unsaturated soils are obtained. The air governing equation by Fredlund for unsaturated soils consolidation is simplified. By applying the Laplace transform and the Cayley-Hamilton theorem to the simplified governing equations of water and air, Darcy＇s law, and Fick＇s law, the transfer function between the state vectors at top and at any depth is then constructed. Finally, by the boundary conditions, the excess pore-water pressure, the excess pore-air pressure, and the soil settlement are obtained under several kinds of boundary conditions with the large-area uniform instantaneous loading. By the Crump method, the inverse Laplace transform is performed, and the semi-analytical solutions to the excess pore-water pressure, the excess pore-air pressure, and the soils settlement are obtained in the time domain. In the case of one surface which is permeable to air and water, comparisons between the semi-analytical solutions and the analytical solutions indicate that the semi-analytical solutions are correct. In the case of one surface which is permeable to air but impermeable to water, comparisons between the semi-analytical solutions and the results of the finite difference method are made, indicating that the semi-analytical solution is also correct.

One-dimensional dynamic equations of a piezoelectric semiconductor beam with a rectangular cross section and their application in static and dynamic characteristic analysis

Within the framework of continuum mechanics, the double power series ex- pansion technique is proposed, and a series of reduced one-dimensional （1D） equations for a piezoelectric semiconductor beam are obtained. These derived equations are universal, in which extension, flexure, and shear deformations are all included, and can be degen- erated to a number of special cases, e.g., extensional motion, coupled extensional and flexural motion with shear deformations, and elementary flexural motion without shear deformations. As a typical application, the extensional motion of a ZnO beam is analyzed sequentially. It is revealed that semi-conduction has a great effect on the performance of the piezoelectric semiconductor beam, including static deformations and dynamic be- haviors. A larger initial carrier density will evidently lead to a lower resonant frequency and a smaller displacement response, which is a little similar to the dissipative effect. Both the derived approximate equations and the corresponding qualitative analysis are general and widely applicable, which can clearly interpret the inner physical mechanism of the semiconductor in the piezoelectrics and provide theoretical guidance for further experimental design.

Analysis of one-dimensional consolidation of soft soils with non-Darcian flow caused by non-Newtonian liquid

Based on non-Darcian flow caused by non-Newtonian liquid, the theory of one-dimensional (1D) consolidation was modified to consider variation in the total vertical stress with depth and time. The finite difference method (FDM) was adopted to obtain numerical solutions for excess pore water pressure and average degree of consolidation. When non-Darcian flow is degenerated into Darcian flow, a comparison between numerical solutions and analytical solutions was made to verify reliability of finite difference solutions. Finally, taking into account the ramp time-dependent loading, consolidation behaviors with non-Darcian flow under various parameters were analyzed. Thus, a comprehensive analysis of 1D consolidation combined with non-Darcian flow caused by non-Newtonian liquid was conducted in this paper.

NONLINEAR STABILITY OF VISCOUS SHOCK WAVES FOR ONE-DIMENSIONAL NONISENTROPIC COMPRESSIBLE NAVIER–STOKES EQUATIONS WITH A CLASS OF LARGE INITIAL PERTURBATION

We study the nonlinear stability of viscous shock waves for the Cauchy problem of one-dimensional nonisentropic compressible Navier–Stokes equations for a viscous and heat conducting ideal polytropic gas. The viscous shock waves are shown to be time asymptotically stable under large initial perturbation with no restriction on the range of the adiabatic exponent provided that the strengths of the viscous shock waves are assumed to be sufficiently small.The proofs are based on the nonlinear energy estimates and the crucial step is to obtain the positive lower and upper bounds of the density and the temperature which are uniformly in time and space.

Analytical solution to one-dimensional consolidation in unsaturated soils under sinusoidal cyclic loading

An analytical solution was presented to the unsaturated soil with a finite thickness under confinement in the lateral direction and sinusoidal cyclic loading in the vertical direction based on Fredlund's one-dimensional consolidation equation for unsaturated soil. The transfer relationship between the state vectors at the top surface and any depth was gained by applying the Laplace transform and Cayley-Hamilton mathematical methods to the governing equations of water and air, Darcy's law and Fick's law. The excess pore-air and pore-water pressures and settlement in the Laplace-transformed domain were obtained by using the Laplace transform with the initial and boundary conditions. The analytical solutions of the excess pore-air and pore-water pressures at any depth and settlement were obtained in the time domain by performing the inverse Laplace transforms. A typical example illustrates the consolidation characteristics of unsaturated soil under sinusoidal loading from analytical results. Finally, comparisons between the analytical solutions and results of the numerical method indicate that the analytical solution is correct.

ONE-DIMENSIONAL CONSOLIDATION OF LAYERED SOILS WITH IMPEDED BOUNDARIES UNDER TIME-DEPENDENT LOADINGS

On the basis of Terzaghi's one-dimensional consolidation theory, the variation of effective stress ratio in layered saturated soils with impeded boundaries under time-dependent loading was studied. By the method of Laplace transform, the solution was presented. Influences of different kinds of cyclic loadings and impeded boundaries conditions were discussed. Through numerical inversion of Laplace transform, useful illustrations were given considering several common time-dependent loadings. Pervious or impervious boundary condition is just the special case of the problem considered here. Compared with average index method,the results from the method illustrated are more accurate.

Potential Symmetries, One-Dimensional Optimal System and Invariant Solutions of the Coupled Burgers’ Equations

In this paper, we discuss one-dimensional optimal system and the invariant solutions of Coupled Burgers’ equations. By using Wu-differential characteristic set algorithm with the aid of Mathematica software, the classical symmetries of the Coupled Burgers’ equations are calculated, and the one-dimensional optimal system of Lie algebra is constructed. And we obtain the invariant solution of the Coupled Burgers’ equations corresponding to one element in one dimensional optimal system by using the invariant method. The results generalize the exact solutions of the Coupled Burgers’ equations.

LOCAL ONE-DIMENSIONAL ASE-I SCHEME FOR 2D DIFFUSION EQUATION

A local alternating segment explicit - implicit method for the solution of 2D diffusion equations is presented in this paper .The method is unconditionally stable and has the obvious property of parallelism. Some numerical experiments show the method is not only simple but also more accurate.

Dynamical Soliton Wave Structures of One-Dimensional Lie Subalgebras via Group-Invariant Solutions of a Higher-Dimensional Soliton Equation with Various Applications in Ocean Physics and Mechatronics Engineering

Having realized various significant roles that higher-dimensional nonlinear partial differ-ential equations(NLPDEs)play in engineering,we analytically investigate in this paper,a higher-dimensional soliton equation,with applications particularly in ocean physics and mechatronics(electrical electronics and mechanical)engineering.Infinitesimal generators of Lie point symmetries of the equation are computed using Lie group analysis of differen-tial equations.In addition,we construct commutation as well as Lie adjoint representation tables for the nine-dimensional Lie algebra achieved.Further,a one-dimensional optimal system of Lie subalgebras is also presented for the soliton equation.This consequently enables us to generate abundant group-invariant solutions through the reduction of the understudy equation into various ordinary differential equations(ODEs).On solving the achieved nonlinear differential equations,we secure various solitonic solutions.In conse-quence,these solutions containing diverse mathematical functions furnish copious shapes of dynamical wave structures,ranging from periodic,kink and kink-shaped nanopteron,soliton(bright and dark)to breather waves with extensive wave collisions depicted.We physically interpreted the resulting soliton solutions by imploring graphical depictions in three dimensions,two dimensions and density plots.Moreover,the gained group-invariant solutions involved several arbitrary functions,thus exhibiting rich physical structures.We also implore the power series technique to solve part of the complicated differential equa-tions and give valid comments on their results.Later,we outline some applications of our results in ocean physics and mechatronics engineering.

Alternating Group Explicit Iterative Methods for One-Dimensional Advection-Diffusion Equation

The finite difference method such as alternating group iterative methods is useful in numerical method for evolutionary equations and this is the standard approach taken in this paper. Alternating group explicit (AGE) iterative methods for one-dimensional convection diffusion equations problems are given. The stability and convergence are analyzed by the linear method. Numerical results of the model problem are taken. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show that the behavior of the method with emphasis on treatment of boundary conditions is valuable.