Exponential-type Integrators For Semilinear Delay Differential Equations And Nonlinear Partial Differential Equations

Delay differential equations and nonlinear partial differential equations are widely used for describing various phenomena in the field of natural science.This dissertation is concerned with the analysis of the exponential-type integrators for semilinear delay differential equations and two classes of nonlinear partial differential equations.The con-vergence and stability properties of exponential Runge–Kutta methods for semilinear de-lay differential equations are investigated.Meanwhile,the convergence of the operator splitting methods for the Gardner equation and the Camassa–Holm equation is analyzed.The main contents of the dissertation are as follow.Stability properties of explicit exponential integrators are analyzed for three kinds of delay differential equations.For linear autonomous delay differential equations,sufficient conditions for P-and GP-contractivity of explicit exponential Runge–Kutta methods are investigated.For linear nonautonomous delay differential equations,it is shown that Mag-nus integrator is PN-and GPN-stable and convergent of order two.For semilinear delay differential equations on C~N,sufficient conditions for RN-and GRN-stability of explicit exponential Runge–Kutta methods are derived.Some examples of P-and GP-contractive,RN-and GRN-stable explicit exponential Runge–Kutta methods are given.Numerical experiments are included to illustrate the theoretical results.We concern with the semilinear delay differential equations on complex Hilbert s-pace.D-convergence and conditional GDN-stability of exponential Runge–Kutta meth-ods are investigated.By introducing the concepts of exponential algebraic stability and conditional GDN-stability,we investigate the properties of methods with exponential al-gebraical stability,diagonal stability and stage order p.It is shown that exponentially algebraically stable and diagonally stable exponential Runge–Kutta methods with stage order p,together with a Lagrange interpolation of order q(q?p),are D-convergent of order p.It is also shown that exponentially algebraically stable and diagonally stable ex-ponential Runge–Kutta methods are conditionally GDN-stable.We construct exponential Runge–Kutta methods which are exponentially algebraically stable and diagonally stable.We also present the numerical results.We focus on semilinear parabolic delay differential equations.Stiff convergence and conditional DN-stability of explicit exponential Runge–Kutta methods are studied.Under the framework of analytic semigroup,we derive the stiff order conditions up to order four.We also give the methods that have stiff order one to four respectively.In particular,it is shown that all explicit exponential Runge–Kutta methods are condition-ally DN-stable.Comparing with some classical implicit Runge–Kutta methods,explicit exponential Runge–Kutta methods are shown to be more accurate and effective.Then we proceed to analyze the convergence property of the Strang splitting for the Gardner equation.We first build the regularity properties of the nonlinear subequation.Then we show that the Strang splitting converges of order one in H~2and the numerical solution is bounded in H~5.With the help of the boundedness of the numerical solution,it is shown that the Strang splitting converges of order two in L~2.Numerical experiments serve to compare the accuracy and efficiency of three kinds of classical time stepping methods.At the same time,the proposed method is applied to simulate the multi solitons collisions for the Gardner equation.The convergence analysis of Lie–Trotter and Strang splitting methods for the Camassa–Holm equation is provided.We assume that the exact solution is bounded in H~4.The analysis is built upon the regularity properties of the Camassa–Holm equation and the di-vided equations.Under the regularity properties,we show that the Lie–Trotter and Strang splitting converge of order one in H~2and the numerical solution is bounded in H~4.Fi-nally,we prove that the Strang splitting is convergent of order two in H~1with the help of boundedness of the numerical solution.Numerical experiments are presented to illustrate the theoretical result.