Development Of Efficient Numerical Methods For Several Types Of Wave Propagation PDEs

In the fields of nature,economy and military,the partial differential equa-tions(PDEs)play an important role in describing the development of movement.With the expansion of the depth and breadth for objects studied in physics and other disciplines,its application is more extensive,and it can model a variety of phenomenons,such as thermals,quantum mechnics and fluid dynamics,etc.The wave propagation PDEs are used to show the phenomenon of movement in nature and physics,for example,the wave propagation in acoustic,electromagetic and fluid dynamics.However,for most PDEs,especially for nonlinear equations,it is difficult to obtain exact or analytical solutions by analytical methods.With the rapid development of software technology and computer hardware,numerical simulation has increasingly become an important method for solving PDEs[2,35].In this paper,we mainly study the development of efficient numerical methods for three types of wave propagation PDEs:Schrodinger equation,Maxwell's e-quation coupled to current equation in magnetized plasma and the Regularized Long Wave equation.The Schrodinger equation was proposed by Austrian physicists in 1926 to describe the equations of motion for wave functions in quantun mechanics.It reveals the basic law of matter movement in the microscopic physical world[20],as Newton's second law plays in classical mechnics,it is a very powerful tool in atomic physics dealing with all non-relativistic problems,and it is widely used in atoms,molecules,fluid physics,condensed physics,nulcar physics,etc.The Schrodinger equation,commonly known as a soliton wave equation,is a partial differential equation that describes the development of physics systems.Non-linear Schrodinger equation[117]also has many successful applications in non-linear physics[22,74,75,120].The nonlinear wave problem in nonlinear optics,Bose-Einstein condensation(BEC),plasma physcis physics,fluid mechanics,self-focusing in laser pulse,thermal pulse propagation in crystals and other systems all can be solved by nonlinear Schrodinger equation[4,20,26,48,70,99,122].Based the above reasons,it is an important task to establish a numerical method that can effectively solve the Schrodinger equation.There are many investigations on numerical solutions for the initial-boundary value problems of the Schrodinger equations,such as the discontinuous Galerkin methods([73,140]),the finite ele-ment methods([10,51,62]),spectral methods([13,76]),meshless methods([37])and split-step methods([36,98]).Recently,the study of the compact finite dif-ference schemes for Schrodinger equations has been of great interest.However,the ordinary Crank-Nicolson.scheme can bring tremendous computations when solving the practical problem of Schrodinger equation in high dimensional or large domain.Therefore,it is very important to propose a conservative compact split-step finite difference scheme to solve the high dimensional Schrodinger equation.As we know,many physical problems are often defined on unbounded domain in the practical application.However,the unboundedness of the computation-al domain presents a great numerical challenge,since any general domain-based method,such as finite difference or finite element,only can deal with a system of finite degrees,and it is impossible when the computational domain is unbounded.In order to solve numerically such a differential equation defined on an unbound-ed domain,one has to consider a finite subdomain and to impose an artificial boundary condition[8,15,40,90,110,157].There are some difficulties in con-structing the discrete artificial conditions,especially for nonlinear equation.The discrete artificial boundary conditions often destroy the stability of the whole fi-nite difference scheme,for example,the unconditional stability of the underlying Crank-Nicolson method for the Schrodinger equation is destroyed[129].Further-more,some numerical reflections at the artificial boundaries may appear.For the nonlinear Schrodinger equation with unbounded domain,it is significant to construct a numerical scheme that is unconditionally stable and does not cause any turbulence at the boundaries.Plasma physics that began to develop in the 1930s is a discipline to study the interaction between plasmas and electromagnetic fields or other matter forms.The interaction between plasmas and electromagnetic waves,i.e.,the transmis-sion,reflection and attenuation of electromagnetic wave in plasma,has always been a practical fields.In recent years,the application of plasma technology in plasma antennas,plasma waveguides,stealthy aircraft and "agile mirror" radars has been paid more and more attention by military powers.It's principle is to achieve the new performance to meet the continuous industrial and military needs by utilizing the interaction,between electromagnetic waves and plasmas,and replacing with the plasma material or improving the original work equip-ment[55,56,72,130,147].The interaction between electromagnetic waves and plasmas dates back to the launch of Soviet Union artificial satellites in the 1957s,and this is the first time that human have obtained the information about the interactions between them[126].Since then,the research on their interaction has been developing continuously.From the original theory of propagation of electromagnetic wave in plasma,it has been widely used in practical and deeper theoretical researches,such as surface wave plasma,plasma stealth,the interac-tion between plasmas and Terahertz band waves and plasma antennas,etc.Magnetized plasma can be modeled using the non-stationary Maxwell's equa-tions coupled with a linear current model.A number of explicit finite-difference time-domain(FDTD)methods have been developed for analyzing EM waves radi-ation,for example,the recursive convolution FDTD method[69,88],the piecewise linear current density recursive convolution(PLCDRC)FDTD method[85,86],the electric field and current-density convolution(JEC)explicit FDTD method[138,139],etc.In fact,as early as the 1966s,Yee[144]proposed the time-domain finite difference method to solve classical Maxwell's equations.However,based on the explicit Yee's scheme of staggered grid and central difference,there are some drawbacks that can not be neglected.The explicit schemes are conditional sta-ble,because the time-step size is limited by the CFL condition.Such a limitation maktes the explicit methods less practical for applications where fine geometric details and a high quality factor are required[127].To overcome the limitation of the CFL condition,some researchers proposed unconditionally stable algorith-m to solve finite difference time-domain magneto-plasmon models.For instance,one-step leapfrog alternative directional implicit FDTD method was developed by[31,136],[65]proposed a three-step 1D-FDTD method with local unconditional stability.It is important to develop numerical methods preserving the discrete energy in magnetized plasma to model and simulate the practical electromagnetic problems in magnitized plasma much more efficiently and effectively.Thus,it has been an important and difficult task to develop energy conservative FDT-D schemes for computing electromagnetic waves propagation in an anisotropic magnetized plasma.Regularized Long Wave equation is one of the most important partial differ-ential equations in involving time variation.It was first proposed by Peregrine[102],and used to investigate the one-way propagation of long wave in fluids of nonlinear dispersive media.The regularized long wave equation plays a very im-portant role in physics and engineering,such as solitons,nonliear shear waves in shallow water,ion and magnetohydrodynamic wave in plasma,longitudinal dispersion waves in elastic rods[17,18].The nonlinearity of the regularized long wave equation makes it difficult to solve its analytical solution,so many experts have done a lot of researches in solving numerical solution by using dif-ferent methods.In recent decades,many scholars have proposed some methods to discuss the numerical solution,for instance,finite element method,meshless method,multi-symplectic method,spline method,pseudo-spectral method,finite difference method,and so on[23,27,33,38,49,54,57,71,92-94,101,107,114].Due to the existence of nonlinear terms in the regularized long wave equation,some scholars have constructed the finite difference scheme of second-order ac-curacy in time and space[101].However,it is difficult to analyze the high-order energy-conserved and compact finite difference scheme for the regularized long wave equation.As Li and Vu-Quoc[79]stated that "In some areas,the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation".In the practical calculations,better algorithms should maintain some inherent physical properties of the original prob-lems as much as possible.Therefore,constructing an energy-conserved scheme is the key to ensure accurate numerical solutions,which retains important physical properties,especially for long time wave propagation.It has been recognized that developing the efficient numerical schemes to preserve their physical invariance is significantly important in computations of electromagnetic wave equations,reg-ularized long wave equations,and Schrodinger equation[29,41,96],etc.In this paper,we investigate the efficient numerical solutions for three kinds of wave propagation PDEs.Based on the background of the problem and the inherent structure of the equation,a series of effective and stable finite difference algo-rithms are proposed,and the conservation of discrete energy,numerical stability,convergence property and error estimate are theoretically analyzed.In addition,we test the numerical experiments and numerical simulation for nonlinear wave propagation.The dissertation is divided into six chapters.The outline is as follows.In Chapter 1,we consider the two-dimensional linear Schrodinger equations with Dirichlet boundary condition.Papers[81,131]proposed the compact ADI schemes for solving linear Schrodiger equations.However,these schemes do not p-reserve the physical conversation laws.By combining the splitting technique with the high-order compact operators,we propose a new conservative splitting high-order compact finite difference scheme,which preserves the physical invariance of charge and energy.Following the splitting process,we divide the equations into x-and y-directional splitting equations respectively,in each time interval.The proposed splitting scheme contains three steps in order to obtain second-order ac-curacy in time step size.Then we impose a fourth-order compact finite difference operator on both sides of each splitting equation to ensure of high-order accuracy in spatial step size.At each step,it leads to the tridiagonal and symmetric sys-tems of equations,which can be easily solved by Thomas's algorithm.We strictly prove that the scheme is unconditionally stable and satisfies the charge and ener-gy conservations.The fourth-order compact finite difference operators only use three mesh points along the x-direction and the y-direction respectively.While they have fourth-order accuracy in spatial step size,the fourth-order compact differences will not go out of the domain when acting at the near boundary mesh points,which ensures without loss of charge near boundary.we prove that the scheme is of fourth order accuracy in space step size and of second order accuracy in time step size in the discrete L2-norm.The scheme can be easily extended to three-dimensional or high-dimensional Schrodinger equations.Numerical exper-iments are given to show the performance of the scheme and numerical results confirm our theoretical analysis.In addition,the scheme stimulated perfectly the physical movements of transient Gaussian distributions.In Chapter 2,we analyze the two-dimensional nonlinear Schrodinger equa-tions(NLSE).The NLSE describes and predicts the propagation of important nonlinear waves and nonlinear effects such as vortices and solitons.Paper[119]proposed an approach for steady-state higher-order compact difference methods to time-dependent problems.Paper[24]described an easy-to-implement two-step high-order compact scheme for the Laplacian operator and its implementation in-to an explicit finite-difference scheme for simulating the NLSE.However,these schemes do not preserve the physical conservation laws.Thus,it is important and difficult to develop the conservative compact split-step finite difference schemes for solving high dimensional nonlinear Schrodinger equations,which ensure to p-reserve the conserved property.In this chapter,combining the operator splitting technique,we propose a new conservative split-step compact difference scheme for solving the two-dimensional nonlinear Schrodinger equations.The significan-t feature is that the scheme is conservative and unconditionally stable while it improves the accuracy by introducing a compact operator for discretization of space and does not increase the computational cost.At each time step,we split the equation into a linear part and nonlinear part.For the nonlinear part,we can solve it exactly.For the linear part,we use a space splitting technique to solve it.We prove the proposed scheme to be charge conserved and uncondition-ally stable,and we further prove that the scheme is of fourth order accuracy in space step size and of second order accuracy in time step size in the discrete L2-norm.Numerical experiments are given to show the performance of the scheme and numerical results confirm our theoretical analysis.In addition,we study the physical propagation of nonlinear waves with different focusing parameters ?.In Chapter 3,we consider the nonlinear Schrodinger equations with un-bounded domain condition.It is of great practical significance to study the physical problems of unbounded areas.For the linear Schrodinger equations on unbounded domain,papers[60,123,124]proposed and analyzed the finite difference schemes for the one-dimensional problem.The artificial boundary con-ditions of the finite computational domain were derived by the Laplace method,and papers[123,124]proved the proposed finite difference schemes to be uncon-ditionally stable and obtained error estimates for the linear Schrodinger equations with artificial boundary conditions.For the nonlinear Schrodinger equations on unbounded domain,papers[7,153,156]developed the corresponding artificial boundary conditions,however,there was no convergence analysis of the numer-ical methods for nonlinear Schodinger equations on unbounded domains in the papers[7,9,21,34,39,141,151-153,156].In this chapter,we construct the artificial boundary conditions for the finite difference discretization of nonlinear Schrodinger equations on unbounded domain,and theoretically analyze the pro-posed finite scheme.The whole-space problem is cut,into three subproblems,the interior problem on the interval x?(xL,xR),the left and right exterior problems.The analytic artificial boundary conditions are obtained by solving the two exte-rior problem with Laplace transform,and our discretized boundary conditions are exact in spatial direction.Therefore,by introducing the two artificial boundaries,the original Schrodinger equation with the unbounded domain is truncated to the problem of the initial-boundary value with the bounded region.We introduce a auxiliary variable and propose the coupled finite difference scheme for the non-linear Schrodinger equations with artificial boundary conditions.The significant feature of the proposed finite difference scheme is that a extrapolation operator is introduced to solve the nonlinear term while the scheme also keeps uncondi-tionally stable and does not induce any oscillations at the artificial boundaries.We prove that the overall scheme is unconditionally stable and convergent for the nonlinear Schrodinger equations with artificial boundary conditions.Our nunerical examples show that there is uo numerical reflection at the artificial boundaries.We also investigate the impact of discontinuous potential functions on the evolution of waves.The collision of solitons is simulated,and all solitons can recovery their shapes and move by their specific speed afterwards despite a strongly nonlinear interaction.In Chapter 4,we investigate the electromagnetic waves propagation through an anisotropic magnetized plasma slab.The plasma exhibits anisotropic behavior in the presence of an external magnetic field.When the electromagnetic(EM)waves propagate in the magnetized plasma,the plasma not only attenuates the energy of the incident wave but also changes the propagation direction.Thus plasma material has wide applications in satellite communications,space weath-er hazards,remote sensing,geophysics and the radar cross section control for various objects or scatters[109,138,139].A number of explicit finite-difference time-domain(FDTD)methods[69,85,88,116,139]have been developed for an-alyzing EM waves propagation in plasma.The all methods mentioned above are conditionally stable because the time-step size is limited by the CFL condition.One-step leapfrog alternative directional implicit FDTD method was developed by[31,136].Hosseini et al.[65]proposed a three-step 1D-FDTD method with lo-cal unconditional stability.Although the methods provide unconditional stability,numerical solutions obtained by these methods do not satisfy energy conservation laws.In this chapter,we focus on the development of energy conservative FDTD methods for the propagation of EM waves in anisotropic plasma.Due to the inter-action of the EM waves and the magnetized plasma,it is an important and difficult task to develop energy conservative FDTD schemes for computing electromagnet-ic waves propagation in an anisotropic magnetized plasma with varying plasma and cyclotron frequencies.We first give the FDTD scheme I(namely,FDTD-I),which is an implicit FDTD scheme,to the problem but it only preserves energies for the problem of constant plasma and cyclotron frequencies.We then propose to improve FDTD-I to obtain a new energy conservative FDTD scheme(namely,FDTD-EC)for the the propagation of EM waves in anisotropic plasma with vary-ing plasma and cyclotron frequencies.We prove theoretically that the FDTD-EC scheme satisfies two energy conservation relations in the discrete norm sense and hence it leads to unconditional stability.In addition,we prove the convergence of the numerical solution by the energy method and the solution accurate to order two in time and space.We numerically compare the performance of both meth-ods to that of the conventional FDTD method.Numerical results are consistent with our theoretical findings.We further apply both methods to simulate the electromagnetic waves propagation in anisotropic magnetized plasma excited by point source.Our results illustrate that the magnetized plasma can efficiently absorb EM waves and change the propagation direction.The characteristics of absorption and anisotropy mainly depend on several parameters such as the EM waves source frequency,plasma frequency(determined by electron density):and cyclotron frequency(determined by external magnetic fields).The findings can provide us useful insight to understand the impact of the plasma and cyclotron frequencies on EM wave propagation in plasma and to design an optimal plasma material for specific applications.In Chapter 5,we develop an efficient numerical method to simulate the EM wave propagation in magnetized plasma.Generally,the traditional FDT-D method is limited by CFL conditions,and it is conditional stable.For the classical Maxwell's equation,many scholars proposed the unconditional stable and splitting time-domain finite difference method to overcome the limitation of CFL.But for Maxwell's equations coupled to current equations in magnetized plasma models,there are few unconditionally stable schemes.[64,136]proposed alternative directional implicit FDTD methods.[65,66]proposed a three-step unconditionally stable FDTD method.Although the methods provide uncon-ditional stability,numerical solutions obtained by these methods do not satisfy energy conservation laws.In this chapter,we propose a new energy-conserved splitting finite difference time domain scheme(namely,EC-S-FDTD)to study the electromagnetic waves propagation through an anisotropic magnetized plas-ma slab.For varying plasma and cyclotron frequencies,it is difficult to construct an energy conservative scheme for non-stationary Maxwell's equations coupled with a linear current model with density fluctuations.In addition,the split of the scheme may break the properties of energy conservation.The proposed s-plitting scheme contains three steps in order to obtain second-order accuracy in time step size.The first step and the third step include five equations,and these five equations can be rewritten into a symmetric tridiagonal matrice and then effectively solved according to the Thomas' method.The second step consists of four equations.To satisfy the energy-conserved,the cross product terms in the current equations are all put into the second step.The combination of current density Jx and Jy is easy to solve.We theoretically prove that the EC-S-FDTD scheme satisfies two energy conservation relations in the discrete norm sense and hence it leads to unconditional stability.By the energy method,we prove the convergence of the splitting scheme is second order in time and space.Numerical examples validate our theoretical analysis,and we simulate the wave propagation in the magnetized plasma.In Chapter 6,we study the nonlinear regularized long wave equation.Since there is a nonlincar term included in the RLW equation,it is difficult to find its an-alytical solution.Therefore,there have been various investigations on numerical solutions of the initial and boundary condition for the RLW equation.Zheng et al.[155]presented a finite difference method using the Richardson extrapolation technique.They get fourth-order convergence in space by five points.Akbari in[5]presented a new compact finite difference method for solving the generalized long wave equation.But the above methods don't satisfy energy conservative laws.In this chapter,we propose two conservative and fourth-order compact difference schemes for analyzing the numerical solution of the regularized long wave equation.The first scheme is two-level and nonlinear implicit.The sec-ond scheme is three-level and linear implicit.The first nonlinear scheme makes the computation relatively time-consuming,while the second linearized scheme makes the computation time-saving.We present two fourth-order compact finite difference operators Lx and Mx,which only use three mesh points along the x-direction.The existence of nonlinear terms and the use of compact operators increase the difficulty of proving conservation and convergence.Conservations of the discrete mass and energy,and unique solvability of the numerical solutions are proved.Convergence and unconditional stability are also derived without any restrictions on the grid ratios by using discrete energy method.The optimal error estimates in norm ? · ? and ?·?? are of fourth-order and second-order accuracy for the spatial and temporal step sizes,respectively.Numerical examples support the theoretical analysis.Finally,we simulate the collision and separation for two solitary waves or three solitary waves.In addition,we simulate the wave propa-gation under Maxwellian initial conditions for different parameters.