High Efficient Numerical Methods For Higher Order Evolution Problems

Many problems in modern science,technology and engineering are related to time,and their mathematical models can be described by initial/boundary value problems of linear or nonlinear evolution equations(systems).These problems,especially those related to nonlin-ear evolution equations(systems),are generally very complex.It is very difficult to find their explicit solutions,and thus the numerical solution is imperative.The purpose of this disserta-tion is to construct efficient numerical algorithms for several kinds of important higher-order evolution equations(systems)and carry out systematic numerical simulation.Therefore,this research is of great importance both in theory and applications.First of all,we develop a fast compact time integrator(FCTI)method for numerically solving a family of general order semilinear evolution equations in regular domains.Con-cretely speaking,we first do the spatial discretization for the equation in terms of the fourth-order compact difference scheme,which combined with the spectral decomposition leads to the semi-discrete scheme described as a system of ordinary differential equations.Then the explicit time integral expression of the system's solution is obtained by the constant varia-tion formula.Based on this formulation,the desired numerical method is devised by further adopting the Lagrange polynomial interpolation to the nonlinear source term together with exact evaluation of the resulting integration.Since the spectral decompositions under the two boundary conditions correspond to the discrete sine transform and the discrete Fourier trans-form,respectively,the previous numerical method can be implemented efficiently using the FFT algorithm.Linear stability analysis is performed for second-order evolution equations.The numerical results are also presented to demonstrate efficiency,accuracy,and stability of the proposed method.Furthermore,the numerical experiments also show that the modi-fied FCTI method can effectively solve some nonstandard higher order semilinear evolution equation.Secondly,we devise a modified fast compact time integration method for solving n-order semilinear evolution equations based on Hermite interpolation.The ideas behind the method are very simple.concretely speaking,when the FCTI method is used to solve higher-order equations(n? 2),the values of the numerical solution and its derivative functions at the right end of each time step,i.e.U(t)(m+1),0 ?l?n-1,can be obtained by(3.10).However,only the function value U(0)(tm+1)is used for further computation in the next time step.If we make full use of these available values to construct interpolation polynomials,we can get a more compact and higher order scheme in temporal discretization.Therefore,we can derive an n-order accurate temporal discretization by only using the approximate solution in the preceding time subinterval.The numerical simulation results verify the effec-tiveness of the method.In addition,we propose a fast compact exponential time differencing method for solving the first and second order evolution equations with Neumann boundary conditions.It is mentioned in[106]that it is difficult to construct a high-order discrete scheme with fast solvers in this case.By making full use of the equation itself and Theorem 1 in[68],we construct a high-order discrete scheme for Neumann boundary conditions,which combined with the compact difference scheme(2.17)on the inner grid point,gives rise to a fourth-order compact scheme globally.The fast calculation is realized by using the fast implementation technique as shown in[54,100].A series of numerical experiments show that numerical results are satisfactory.Finally,we use the numerical methods developed above to solve several coupling prob-lems which are important in the subject of mathematical physics,including the coupling Schrodinger equations,Klein-Gordon-Schrodinger equations,and Klein-Gordon-Zakharov equations,respectively.Numerical results are satisfactory.