2. 5-D RESISTIVITY TOMOGRAPHY USINGBOUNDARY INTEGRAL EQUATIONS

DC Resistivity Tomography is a non-linear inversion problem. So far there are mainly two kinds of inversion methods, based on the finite-element method and alpha centers method. In this paper, the disadvantages of these two kinds of methods were analysed,and a new method of forward modeling and inversion (Tomography) based on boundary integral equations was proposed. This scheme successfuly overcomes the difficulties of the two formarly methods. It isn’t necessary to use the linearization approximation and calculate the Jacobi matrix. Numerical modeling results given in this paper showed that the computation speed of our method is fast, and there is not any special requirement for initial model, and satisfying results of tomography can be obtained in the case of great contrast of conductivity. So it has wide applications.

Abundant solutions of Wick-type stochastic fractional 2D KdV equations

A modified fractional sub-equation method is applied to Wick-type stochastic fractional two-dimensional （2D） KdV equations. With the help of a Hermit transform, we obtain a new set of exact stochastic solutions to Wick-type stochastic fractional 2D KdV equations in the white noise space. These solutions include exponential decay wave solutions, soliton wave solutions, and periodic wave solutions. Two examples are explicitly given to illustrate our approach.

THE NAVIER-STOKES EQUATIONS IN STREAM LAYER AND ON STREAM SURFACE AND A DIMENSION SPLIT METHODS

In this paper,we proposal stream surface and stream layer.By using classical tensor calculus,we derive 3-D Navier-Stokes Equations(NSE)in the stream layer under semigeodesic coordinate system,Navier-Stokes equation on the stream surface and 2-D Navier-Stokes equations on a two dimensional manifold. After introducing stream function on the stream surface,a nonlinear initial-boundary value problem satisfies by stream function is obtained,existence and uniqueness of its solution are proven.Based this theory we proposal a new method called"dimension split method"to solve 3D NSE.

A correction of the 2D KdV equation of Djordjevic & Redekopp in exponentially stratified fluid

To further study the nal solitons on the continental fission laws of initial intershelf/slope, we rederive and correct the 2D KdV equation of Djordjevic ＆ Redekopp for exponentially stratified fluid （or ocean） and with twodimensional topography. Through a combination of theoretical study and numerical experiments, we show that solitons in the odd vertical modes can fission. However, because of the corrections, the fission conditions are different from those of Djordjevic ＆ Redekopp. The even modes cannot fission unless the initial internal solitons propagate from shallow sea to deep sea. This conclusion is entirely opposite to that of Djordjevic ＆ Redekopp.

Direct Reduction and Exact Solutions for Generalized Variable Coefficients 2D KdV Equation under Some Integrability Conditions

Based on the closed connections among the homogeneous balance （HB） method and Clarkson-KruSkal （CK） method, we study the similarity reductions of the generalized variable coefficients 2D KdV equation. In the meantime it is shown that this leads to a direct reduction in the form of ordinary differential equation under some integrability conditions between the variable coefficients. Two different cases have been discussed, the search for solutions of those ordinary differential equations yielded many exact travelling and solitonic wave solutions in the form of hyperbolic and trigonometric functions under some constraints between the variable coefficients.

Asymptotic behavior of 2D generalized stochastic Ginzburg-Landau equation with additive noise

The 2D generalized stochastic Ginzburg-Landau equation with additive noise is considered. The compactness of the random dynamical system is established with a priori estimate method, showing that the random dynamical system possesses a random attractor in H^1 0.

Diagonal form fast multipole boundary element method for 2D acoustic problems based on Burton-Miller boundary integral equation formulation and its applications

This paper describes formulation and implementation of the fast multipole boundary element method （FMBEM） for 2D acoustic problems. The kernel function expansion theory is summarized, and four building blocks of the FMBEM are described in details. They are moment calculation, moment to moment translation, moment to local translation, and local to local translation. A data structure for the quad-tree construction is proposed which can facilitate implementation. An analytical moment expression is derived, which is more accurate, stable, and efficient than direct numerical computation. Numerical examples are presented to demonstrate the accuracy and efficiency of the FMBEM, and radiation of a 2D vibration rail mode is simulated using the FMBEM.

Coupled thermo-hydro-mechanical simulation of CO2 enhanced gas recovery with an extended equation of state module for TOUGH2MP-FLAC3D

As one of the most important ways to reduce the greenhouse gas emission,carbon dioxide（CO2）enhanced gas recovery（CO2-EGR） is attractive since the gas recovery can be enhanced simultaneously with CO2sequestration.Based on the existing equation of state（EOS） module of TOUGH2 MP,extEOS7C is developed to calculate the phase partition of H2O-CO2-CH4-NaCl mixtures accurately with consideration of dissolved NaCI and brine properties at high pressure and temperature conditions.Verifications show that it can be applied up to the pressure of 100 MPa and temperature of 150℃.The module was implemented in the linked simulator TOUGH2MP-FLAC3 D for the coupled hydro-mechanical simulations.A simplified three-dimensional（3D）1/4 model（2.2 km×1 km×1 km） which consists of the whole reservoir,caprock and baserock was generated based on the geological conditions of a gas field in the North German Basin.The simulation results show that,under an injection rate of 200,000 t/yr and production rate of 200,000 sm3/d,CO2breakthrough occurred in the case with the initial reservoir pressure of 5 MPa but did not occur in the case of 42 MPa.Under low pressure conditions,the pressure driven horizontal transport is the dominant process;while under high pressure conditions,the density driven vertical flow is dominant.Under the considered conditions,the CO2-EGR caused only small pressure changes.The largest pore pressure increase（2 MPa） and uplift（7 mm） occurred at the caprock bottom induced by only CO2injection.The caprock had still the primary stress state and its integrity was not affected.The formation water salinity and temperature variations of ±20℃ had small influences on the CO2-EGR process.In order to slow down the breakthrough,it is suggested that CO2-EGR should be carried out before the reservoir pressure drops below the critical pressure of CO2.

Two-dimensional equations for thin-films of ionic conductors

A theoretical model is developed for predicting both conduction and diffusion in thin-film ionic conductors or cables. With the linearized Poisson-Nernst-Planck(PNP)theory, the two-dimensional(2D) equations for thin ionic conductor films are obtained from the three-dimensional(3D) equations by power series expansions in the film thickness coordinate, retaining the lower-order equations. The thin-film equations for ionic conductors are combined with similar equations for one thin dielectric film to derive the 2D equations of thin sandwich films composed of a dielectric layer and two ionic conductor layers. A sandwich film in the literature, as an ionic cable, is analyzed as an example of the equations obtained in this paper. The numerical results show the effect of diffusion in addition to the conduction treated in the literature. The obtained theoretical model including both conduction and diffusion phenomena can be used to investigate the performance of ionic-conductor devices with any frequency.

A novel(2+1)-dimensional integrable KdV equation with peculiar solution structures

The celebrated(1+1)-dimensional Korteweg de-Vries(KdV)equation and its(2+1)-dimensional extension,the Kadomtsev-Petviashvili(KP)equation,are two of the most important models in physical science.The KP hierarchy is explicitly written out by means of the linearized operator of the KP equation.A novel(2+1)-dimensional KdV extension,the cKP3-4 equation,is obtained by combining the third member(KP3,the usual KP equation)and the fourth member(KP4)of the KP hierarchy.The integrability of the cKP3-4 equation is guaranteed by the existence of the Lax pair and dual Lax pair.The cKP3-4 system can be bilinearized by using Hirota's bilinear operators after introducing an additional auxiliary variable.Exact solutions of the cKP3-4 equation possess some peculiar and interesting properties which are not valid for the KP3 and KP4 equations.For instance,the soliton molecules and the missing D'Alembert type solutions(the arbitrary travelling waves moving in one direction with a fixed model dependent velocity)including periodic kink molecules,periodic kink-antikink molecules,few-cycle solitons,and envelope solitons exist for the cKP3-4 equation but not for the separated KP3 equation and the KP4 equation.

Shock Formation for 2D Isentropic Euler Equations with Self-similar Variables

The author studies the 2D isentropic Euler equations with the ideal gas law.He exhibits a set of smooth initial data that give rise to shock formation at a single point near the planar symmetry.These solutions to the 2D isentropic Euler equations are associated with non-zero vorticity at the shock and have uniform-in-time-1/3-Holder bound.Moreover,these point shocks are of self-similar type and share the same profile,which is a solution to the 2D self-similar Burgers equation.The proof of the solutions,following the 3D construction of Buckmaster,Shkoller and Vicol(in 2023),is based on the stable 2D self-similar Burgers profile and the modulation method.

Numerical three-dimensional modeling of earthen dam piping failure

A physically-based numerical three-dimensional earthen dam piping failure model is developed for homogeneous and zoned soil dams.This model is an erosion model,coupled with force/moment equilibrium analyses.Orifice flow and two-dimensional(2D)shallow water equations(SWE)are solved to simulate dam break flows at different breaching stages.Erosion rates of different soils with different construction compaction efforts are calculated using corresponding erosion formulae.The dam's real shape,soil properties,and surrounding area are programmed.Large outer 2D-SWE grids are used to control upstream and downstream hydraulic conditions and control the boundary conditions of orifice flow,and inner 2D-SWE flow is used to scour soil and perform force/moment equilibrium analyses.This model is validated using the European Commission IMPACT(Investigation of Extreme Flood Processes and Uncertainty)Test#5 in Norway,Teton Dam failure in Idaho,USA,and Quail Creek Dike failure in Utah,USA.All calculated peak outflows are within 10%errors of observed values.Simulation results show that,for a V-shaped dam like Teton Dam,a piping breach location at the abutment tends to result in a smaller peak breach outflow than the piping breach location at the dam's center;and if Teton Dam had broken from its center for internal erosion,a peak outflow of 117851 m'/s,which is 81%larger than the peak outflow of 65120 m3/s released from its right abutment,would have been released from Teton Dam.A lower piping inlet elevation tends to cause a faster/earlier piping breach than a higher piping inlet elevation.

The Hugoniot equation of state of the fluid He+D_2 mixtures

The fluid variational theory and effective one-component model have been used to calculate the Hugoniot equation of state （EOS） of fluid He, D2, and He+D2 mixtures with different He:D2 compositions under high pressures and temperatures. An examination of the confidence of above computation is performed by comparing experiment and calculation, in which the similar calculation procedure used for He+D2 is adopted, of He and D2 each, since no experimental data are available to conduct this kind of comparison. Good agreement in both comparisons is found. This fact may be looked as if an indirect positive verification of calculation procedure used here at least in the pressure and temperature domain covered by the experimental data of He and D2 used for comparison, numerically nearly up to 35 GPa and 105K.

Deformed two-dimensional rogue waves in the (2+1)-dimensional Korteweg–de Vries equation

Within the(2+1)-dimensional Korteweg–de Vries equation framework,new bilinear B¨acklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation.By introducing an arbitrary functionφ(y),a family of deformed soliton and deformed breather solutions are presented with the improved Hirota’s bilinear method.By choosing the appropriate parameters,their interesting dynamic behaviors are shown in three-dimensional plots.Furthermore,novel rational solutions are generated by taking the limit of the obtained solitons.Additionally,twodimensional(2D)rogue waves(localized in both space and time)on the soliton plane are presented,we refer to them as deformed 2D rogue waves.The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane,and its evolution process is analyzed in detail.The deformed 2D rogue wave solutions are constructed successfully,which are closely related to the arbitrary functionφ(y).This new idea is also applicable to other nonlinear systems.

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.

Analysis on Sixth-Order Compact Approximations with Richardson Extrapolation for 2D Poisson Equation

By using Richardson extrapolation and fourth-order compact finite difference scheme on different scale grids, a sixth-order solution is computed on the coarse grid. Other three techniques are applied to obtain a sixth-order solution on the fine grid, and thus give out three kinds of Richardson extrapolation-based sixth order compact computation methods. By carefully analyzing the truncation errors respectively on 2D Poisson equation, we compare the accuracy of these three sixth order methods theoretically. Numerical results for two test problems are discussed.

Painlevé property, local and nonlocal symmetries, and symmetry reductions for a (2+1)-dimensional integrable KdV equation

The Painlevé property for a(2+1)-dimensional Korteweg–de Vries(KdV) extension, the combined KP3(Kadomtsev–Petviashvili) and KP4(cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé expansion is used to find the Schwartz form, the Bäcklund/Levi transformations, and the residual nonlocal symmetry. The residual symmetry is localized to find its finite Bäcklund transformation. The local point symmetries of the model constitute a centerless Kac–Moody–Virasoro algebra. The local point symmetries are used to find the related group-invariant reductions including a new Lax integrable model with a fourth-order spectral problem. The finite transformation theorem or the Lie point symmetry group is obtained by using a direct method.

A Local Discontinuous Galerkin Method for Two-Dimensional Time Fractional Diffusion Equations

For two-dimensional(2D)time fractional diffusion equations,we construct a numerical method based on a local discontinuous Galerkin(LDG)method in space and a finite differ-ence scheme in time.We investigate the numerical stability and convergence of the method for both rectangular and triangular meshes and show that the method is unconditionally stable.Numerical results indicate the effectiveness and accuracy of the method and con-firm the analysis.

An Extended Subequation Rational Expansion Method and Solutions of (2+1)-Dimensional Cubic Nonlinear Schr(?)dinger Equation

An extended subequation rational expansion method is presented and used to construct some exact,analyt-ical solutions of the (2+1)-dimensional cubic nonlinear Schrdinger equation.From our results,many known solutionsof the (2+1)-dimensional cubic nonlinear Schrdinger equation can be recovered by means of some suitable selections ofthe arbitrary functions and arbitrary constants.With computer simulation,the properties of new non-travelling waveand coefficient function's soliton-like solutions,and elliptic solutions are demonstrated by some plots.

Time Dependent Wave Propagation Modeling Using Finite Difference Scheme of 2D Wave Equation Based on Absorbing and Reflecting Boundaries

Boundary procedure is an important phenomenon in numerical simulation. To reduce or eliminate the spurious reflections significantly which is occurred in boundary is a challenging and vital approach. The appropriate artificial numerical boundaries can be applied to eliminate the effect of unnecessary spurious reflections in case of the numerical simulations of wave propagation phenomena problems. Typically, to reduce the artificial reflections, the absorbing boundary conditions are necessary. In this paper, we overview and investigate the appropriate typical absorbing boundary conditions and analyzed the boundary effect of two dimensional wave equation numerically. Reflections over the wide-ranging incident angles are complicated to eliminate, but the absorbing boundary conditions that we have applied are computationally cost efficient, easy to apply and able to reduce reflections significantly. For numerical solution, finite difference method is applied to develop numerical scheme using 2D wave equation. Using the developed numerical scheme, we obtain the numerical solution of the governing equation as an initial boundary value problem and realize the qualitative behavior of the solution in infinite space. The finite difference numerical scheme has been investigated by developing MATLAB programming language code. Numerical results have been discussed and analyzed with presenting different qualitative behavior of the numerical scheme. The accuracy and efficiency of the numerical scheme has been illustrated. The stability analysis was discussed and verified stability condition. Using the numerical scheme and absorbing boundary conditions, the boundary effects and absorption of spurious reflection of boundary have been demonstrated.