Efficient High-order Numerical Methods And Theoretical Analysis For Several Types Of Partial Differential Equations

Partial differential equations(PDEs)have been applied widely in the fields of engineering technology and natural science,such as fluid mechanics,acoustics,electromagnetics,quantum mechanics,physics and so on[9,13,45,62,96,115,118,129].With the development of science and technology,all kinds of PDEs based on practical problems have been put forward,and many new methods for solving PDEs have been developed during the study.Due to the initial-boundary value conditions,nonlinearity,complexity and variability of practical problem,for most partial differential equations,there are no exact solutions or it is difficult to obtain exact solutions by analytical method.Hence,numerical simulation is an important method to solve PDEs.The high accuracy numerical method has been widely studied and applied because of its advantages of high computational efficiency,small numerical dissipation and so on.In addition,whether the numer-ical method can keep the structure and property of each physical quantity in the equation is very important.So based on the intrinsic property and structure of the equation,efficient high-order numerical methods and its theoretical analyses are studied for several kinds of PDEs in this paper.Wave equation is a kind of important PDE which used to describe various wave phenomena in nature and widely applied in the fields of fluid mechanics,acoustics,optics,electromagnetics[19,42,48,76,81,107].The acoustic wave equations are very important which describe the propagations of waves in me-dia and commonly found in the geophysics,geoscience,petroleum engineering,telecommunication,medical science and so on[34,49,73].Nonlinear wave equa-tions,such as sine-Gordon equation and Klein-Gordon equation,are commonly used to simulate the behavior of elementary particles,the propagation of dislo-cations in crystals,the interaction of solitons in a collisionless plasma,the nucle-ation and growth phenomenon of nuclei and so forth[40,44,103,123,126].The nonlinear Schrodinger equation often used to models dispersion and nonlinearity in the physical system.It generally arises from many other branches of science and technology,such as quantum mechanics,plasma physics and the dynamics of Bose-Einstein condensate and so on[1,59,68,72,79,119].There have been many studies on numerical methods for solving various kinds of wave propaga-tion equations,such as,finite element method[8.51,61.67,74],finite difference method[18,29,37.43,48,68,69,103],spectral method[10,106,122,132,136].However,many methods or have low-order accuracy in time or do not preserve energy.The averaged vector field(AVF)method was first proposed in[98.110]to solve Hamiltonian systems for time integration,where the Hamiltonian systems are derived from the original PDEs after spatial discretization.The AVF method can preserve the Hamiltonian energy automatically and requires only knowledge of the vector field itself,moreover,the AVF method can achieve high-order ac-curacy in time.Hence,a large number of works have been devoted to solve vari-ous equations using the AVF technique in time discretization incorporating with different space discretization methods.For instance.Kortewcg de Vries(KdV)equation[26.38.76],Cahn-Hilliard equation[75],nonlinear Schrodinger equation[4,82].However,in the existing works,convergence analysis of the fully-discrete scheme is lacking and most of them have low-order accuracy in time.We pro-pose time high accuracy energy conservative schemes based on AVF method for solving several types of wave equations and study its convergence analyses of the fully-discret.e schemes.Convection-diffusion equation is a kind of basic mathematical physics equa-tion,which describes the transport process of mass,energy,heat and some re-action diffusion processes,such as in heat and mass transfer,oil reservoir sim-ulation,groundwater modelling,atmospheric pollution,aerodynamics,biological population,and so on(see,for example,[13,16,39,53,111,112,114,116,118],etc).Standard finite difference methods and finite element methods solving advection-dominated diffusion equations present non-physical oscillations or nu-merical dispersion.This kind of equation has strong hyperbolic property.Char-acteristic method for solving convection-dominated diffusion equation can reduce non-physical oscillation or excessive numerical dissipation essentially,and has no stability constraints required on the time step.However,most of the existing characteristic methods are only of time first-order accuracy.Hence,we pro-pose a time second-order characteristic finite element method to solve nonlinear advection-diffusion problems and study its error theoretical analysis.Energy conservative law is of great importance in various wave propagation equations,it is very important to develop the numerical schemes preserving ener-gy to simulate wave propagation,which often yield physically reasonable results and numerical stability.We propose energy conservative high-order numerical schemes for several types of wave equations,and strict prove the energy con-servation and convergence.Numerical experiments are carried out to confirm theoretical results,and simulate the physical property of wave propagation.For the nonlinear advection-dominated diffusion equation,we use the physical prop-erty of equation to propose the second-order characteristic finite element method and study its error theoretical analysis.Numerical examples verify theoretical result and efficiency of numerical scheme.The dissertation is divided into six chapters,the main research contents and results are as follows:In chapter 1,we consider two-dimensional variable coefficient acoustic wave equations.The acoustic wave equations with variable coefficients describe the propagations of waves in media and have been widely used in the fields of geo-science,medical imaging,seismic exploration and so on.The compact finite difference(CFD)method have the advantages like simplicity and high-order ac-curacy and have been proposed to solve the acoustic wave equations[20,21,41,43,88,89,93].However,most of the methods above do not preserve energy.AVF method can keep energy conservation and achieve high-order accuracy in time.Recently,the AVF methods have been studied to the wave-type PDEs in[26,33,38,76].However,there is no research on theoretical analysis of conver-gence of the AVF compact difference schemes for PDEs.Hence we develop two energy-preserving AVF compact finite difference methods for two-dimensional variable coefficient acoustic wave equations and give its convergence analyses in this chapter.We first,derive an infinite-dimensional Hamiltonian system for the variable coefficient acoustic wave equation in two dimensions.Then fourth-order compact difference operator is used for spatial discretization that yields a semi-discrcete finite dimensional Hamiltonian system.For obtaining time high-order accuracy and preserving energy.AVF technique is applied for the time integra-tion to the system,which leads to the fully-discrete energy conservative time high-order AVF(2)and AVF(4)compact difference schemes.We prove that the semi-discrete scheme satisfies the discrete energy conservation and give its error estimate.We.prove that the fully-discrete AVF(2)and AVF(4)compact differ-ence scheme preserve energy and obtain optimal error estimate.Numerical tests are given to show that the proposed methods have high-order accuracy in both time and space and satisfy energy conservations.Finally,we compute the acous-tic wave equation in the layer medium and show the physical feature of the wave propagation through the layer medium.In chapter 2.we consider the variable coefficient nonlinear wave equations.We study three nonlinear wave equations,that is,sine-Gordon equation,Klein-Gordon equation and wave equation with exponential nonlinearity.The various nonlinearities in equations have more powerful capacity to describe the prop-agation of waves in many physical applications than their linear analogues do[45,115,129].Besides,the theoretical analysis of numerical scheme of nonlinear problems will be more complex and difficult,and there is no analysis about conver-gence of the AVF methods for nonlinear PDEs.Therefore,it is more challenging and significant to study nonlinear wave equations with variable coefficients.We derive an infinite-dimensional Hamiltonian system for variable coeficient nonlin-ear wave equations.Fourth-order compact difference operator is used for space discretization,and the second-order and fourth-order AVF techniques are applied for the time integration to the system,which leads to two fully-discrete schemes AVF(2)-CFD and AVF(4)-CFD.We prove that the fully-discrete schemes satisfy discrete energy conservations.The nonlinear term only satisfies local Lipschitz continuous condition,hence it is more difficult to give the convergence analysis of the numerical scheme.According to the property of energy conservation and discrete Sobolev inequality,we prove that the numerical solution is bounded un-der infinite norm.For AVF(2)-CFD and AVF(4)-CFD scheme,the existence of numerical solution is strictly proved with the help of fixed point theorem.We provide a detailed convergence analysis of schemes and prove its optimal error es-timate.Numerical experiments are given to verify theoretical results and simulate the nonlinear wave propagation in the layered medium.In chapter 3,we consider the two-dimensional variable coefficient nonlin-ear wave equations.We study time high-order AVF compact finite difference schemes for sine-Gordon nonlinear wave equations and Klein-Gordon nonlinear wave equations.The theoretical analysis of numerical method for one-dimensional nonlinear problems can not be extend to high-dimensional problems directly,also its theoretical analysis is more difficult.We first transferred nonlinear wave equa-tions into infinite-dimensional Hamiltonian system using the theory of functional derivative.Then compact difference operator is applied for space discretization and AVF methods are applied for time discretization,respectively,which obtain time second-order and time fourth-order fully-discrete schemes EPAVF(2)-CFD and EPAVF(4)-CFD.We first prove that two schemes satisfy the discrete energy conservations.In the theoretical analysis of numerical scheme,the boundedness of the numerical solution in Lp norm is derived by the property of energy conser-vation and discrete Sobolev inequality.For EPAVF(2)-CFD and EPAVF(4)-CFD scheme,its solvability is proved with the help of fixed point theorem,we give the rigorous convergence analysis by norm inequalities and prove that the op-timal error estimate of proposed scheme is fourth-order accuracy both in time and space.Numerical tests show high accuracy of numerical errors in both time and space,which confirm the energy conservation and convergence of proposed schemes.Finally,numerical experiments are also taken to simulate two types of nonlinear waves in the layer medium.In chapter 4,we consider the nonlinear space-fractional wave equations.It is very important to develop the numerical scheme preserving energy to simu-late nonlinear fractional wave equation,but there is only few work on this field and the existing research work about energy conservative scheme has low-order accuracy in time[94,95,125].[58]propose a scheme by AVF method that only has second-order accuracy in time and there were no unique solvability and con-vergence analyses of the numerical scheme.Hence we develop time fourth-order energy-preserving AVF finite difference method for the nonlinear fractional wave equations and give strict theoretical analysis.Based on the infinite-dimensional Hamiltonian system of nonlinear space-fractional wave equation,we applying fourth-order weighted and shifted difference operator to discrete space-fractional derivative,and fourth-order AVF method to discretize time variable to obtain fully-discrete scheme.We prove that proposed scheme satisfies discrete energy conservation and is unique solvable,we analyze the convergence of numerical scheme and prove that it has fourth-order accuracy in both time and space.Fi-nally,numerical experiments are performed to verify the high-order accuracy and energy conservation of the proposed scheme with different nonlinear term and different fractional order.In chapter 5,we consider the nonlinear Schrodinger(NLS)equations.[4,26,82]apply AVF method to solve NLS equations,however,there ware no conver-gence analyses of their schemes.We first apply the fourth-order compact differ-ence operator and second-order AVF method for space and time discretization respectively,which yield the fully-discrete scheme.We prove that the developed scheme satisfies discrete energy conservation law.Then with the help of fixed point theorem,the numerical solution is existed.We prove that our developed scheme has second-order accuracy in time and fourth-order accuracy in space,which is unconditional for the time step and space size.Finally,numerical tests for defocusing nonlinear Schrodinger problems are given to show the conservation of energy and accuracy of numerical scheme.We give the motion change of dark and bright solitons corresponding to the defocusing and focusing Schrodinger equation,so that the defocusing Schrodinger problem can be better studied.In chapter 6,wo consider the nonlinear advection-diffusion equations.The nonlinear function of population dynamics results from the interaction of birth rate,death rate and the environment factor,which can lead to the density change of population in the global dynamics[16,39,118].Studying such problems has been playing important role and has an actual meaning in spread of dis-eases,population prediction and so on.Standard finite difference methods and fi-nite element methods solving advection-dominated problems present non-physical oscillations or numerical dissipations.Characteristic method can take advantage of the physics characteristics of the convection-diffusion equations,significantly reduce the truncation errors in time and eliminate the excessive numerical dis-persions.Douglas and Russell[46]proposed modified method of characteristics methods for solving convection-diffusion problems.Combined with finite element methods,[7,15,54,109,113]developed the characteristic method.However,most of characteristic methods are only of time first-order accuracy.[87]developed a time second-order characteristic finite element scheme for solving advection e-quation with a nonlinear coagulation integration term,but the equation has no diffusion term.Hence,there is of importance to develop and analyze the time second-order characteristic finite element method to solve nonlinear advection-diffusion equations.We first transferred time derivative term and the advection term into the global derivative term and then discretized it by the central differ-ence operator along the characteristic curve.The diffusion term is approximated by the second-order average operator along the characteristic curve.For treating the nonlinear right side term,the second-order extrapolation along the charac-teristics is applied.Using the theory of variation and prior estimates,we prove that the developed scheme has second-order accuracy in time and can provide efficiently high accuracy solutions when using large time step sizes.Finally,nu-merical tests are given to verify theoretical results and simulate the single-species spatio-temporal population dynamic models.