Highly Efficient Numerical Algorithms For A Class Of Complex Nonlinear Systems

Many modern scientific problems can be described by complex systems with energy dissipation or conservation properties.The constructed numerical schemes need to satisfy these properties,or other physical constraints.The main goal of this thesis is to construct effective schemes for nonlinear systems,and carry out their error analysis and numerical simulations.The schemes are based on the scalar auxiliary variable approach(SAV)in time,and spectral method in space.The outline of this thesis is as follows:In Chapter 1,we briefly review the current study on complex nonlinear systems,summarize the main contribution of this work,and list some relevant preliminaries.In Chapter 2,we develop efficient and accurate numerical schemes based on the SAV approach for the generalized Zakharov system and generalized vector Zakharov system.These schemes are second-order in time,linear,unconditionally stable,only require solving linear systems with constant coefficients at each time step.Moreover,the schemes preserve exactly a modified Hamiltonian.Ample numerical results are presented to demonstrate the accuracy and robustness of the schemes.In Chapter 3,we develop efficient numerical schemes for solving the micropolar Navier-Stokes equations by combining the SAV approach and pressure-correction method.Our first-and second-order semi-discrete schemes enjoy remarkable properties such as unconditional energy stable with a modified energy,and only a sequence of decoupled linear equations with constant coefficients need to be solved at each time step.We also construct fully discrete versions of these schemes with a special spectral discretization which preserve the essential properties of the semi-discrete schemes.Numerical experiments are presented to validate the proposed schemes.In Chapter 4,we construct several efficient SAV schemes based on the Fourierspectral method in space for the Cahn-Hilliard-Hele-Shaw system.The temporal discretizations are built upon the first-order Euler and second-order BDF method respectively.We derive the unconditional energy stability for both schemes and also establish the rigorous error estimates for the first-order SAV Fourier-spectral scheme.Finally various numerical experiments are presented to demonstrate the accuracy and performance of the constructed schemes.In Chapter 5,we present efficient implicit-explicit BDFk SAV schemes for solving the sixth-order Cahn-Hilliard-type equations.The new schemes enjoy the following advantages such as,it only requires solving one linear equation with constant coefficients at each time step,which is the same as the IMEX schemes;they are unconditional energy stable,and lead to a uniform bound of the numerical solution in the norm.We also present numerical experiments to validate the stability and accuracy of the proposed schemes.In Chapter 6,we make a conclusion of this thesis and discuss some future works.