Optimal Error Estimate Of Linearized Numerical Methods For Some Evolution Equations

The movement of particles,and the transmission and diffusion of objects in different states are not single,but complex and changeable.Therefore,the application of nonlinear systems is becoming more and more extensive.Numerical methods in temporal direction for solving nonlinear evolution equations mainly include: implicit,explicit,semi-implicit and linearized implicit methods.In order to reduce the computation cost,many researchers used the linearized methods to handle the nonlinear term.The commonly used linearized methods are: linearized Crank-Nicolson methods,linearized backward differential formula(BDF)methods and so on.In this thesis,we estimate the convergence of the full discrete scheme unconditionally by using the time-space splitting methods.That is to say,the convergence results can be proved without time-space step ratio limit,while all previous theoretical analyses always required certain restrictions.Our research is mainly divided into:Part Ⅰ : we use Crank-Nicolson method in temporal direction and finite element methods in the spatial one to solve semi-linear parabolic equations,and obtain a one-setp linearized method with considering Newton linearized method.The unconditional convergence of the full discrete scheme is proved with time-space error splitting skills and Gr ¨onwall inequalities.The boundedness of the numerical solution is given without the constraint of time-space step ratio.Finally,theoretical results are verified by numerical experiments.Part Ⅱ: we propose a linearized numerical method for solving the nonlinear time fractional partial parabolic and Schr ¨odinger equations.The solution owns weak singularity at the initial time caused by fractional derivative.In order to reduce the order reduction caused by singularity and obtain a high-order numerical scheme,Alikhanov methods are used with non-uniform meshes to discrete the fractional derivate.Finite element methods are applied to discrete in spatial direction.The error estimate is carried out by using the fractional discrete Gr ¨onwall inequality.Applying the time-space splitting methods to prove the unconditional convergence as well as the boundedness of the numerical solution.In Chapter 5,the Alikhnov method in graded mesh is applied to solve Schr ¨odinger problem and extrapolation method is used to approximate the nonlinear term.Then,we can obtain the linearized scheme,and the first layer is approximated by a low-order method.Unconditional stability about numerical solution of the fully discrete scheme can also be obtained by means of the error splitting method.Part Ⅲ: we introduce exponential function transformations to get the analytical solution of the logarithmic Schr ¨odinger equation with a potential function.The regularity of the equation is low due to the logarithms.The transformations not only convert logarithmic terms into linear parts,but also keep the solution positive.When potential function is a spacial one,we can get the analytical solution by solving a nonlinear second-order ordinary partial equations.When it is a more general function,the analytical solution is difficult and the numerical methods is applied to study the property of the solution.At the same time,because of the logarithm and the characteristics of the equation itself,the corresponding standing wave solution can be obtained by solving the steady-state problem.Finally,Newton iteration is used to solve the steady-state equations.