Highly Efficient Numerical Methods For Energy Dissipative/Conservative Nonlinear Systems

Nonlinear partial differential equations have significant applications in many fields.For the dissipative/conservative nonlinear systems,how to construct an efficient and accurate numerical scheme is a challenge.The difficulties mainly include the needs to handle with the strong nonlinear term,realize long-time simulation,and construct numerical scheme to satisfy the inherent properties of the equation.In this thesis,we propose a series of efficient and accurate numerical algorithms for a large class of dissipative/conservative nonlinear systems,mainly including the following three parts:Firstly,we propose a relaxed generalized scalar auxiliary variable(R-GSAV)approach for general dissipation systems.The scalar auxiliary variable(SAV)approach[114]and its generalized version GSAV proposed in[76]are very popular methods to construct efficient and accurate energy stable schemes for nonlinear dissipative systems.However,the discrete value of the SAV is not directly linked to the free energy of the dissipative system,and may lead to inaccurate solutions if the time step is not sufficiently small.Inspired by the relaxed SAV method proposed in[78]for gradient flows,we propose a generalized SAV approach with relaxation(R-GSAV)for general dissipative systems.The R-GSAV approach preserves all the advantages of the GSAV approach,in addition,it dissipates a modified energy that is directly linked to the original free energy.We prove that the k-th order implicit-explicit(IMEX)schemes based on R-GSAV are unconditionally energy stable,and we carry out a rigorous error analysis for 1≤k≤5.We present ample numerical results to demonstrate the improved accuracy and effectiveness of the R-GSAV approach.Secondly,we construct three efficient and accurate numerical methods for solving the Klein-Gordon-Schrodinger(KGS)equations with/without damping terms.The first one is based on the original SAV approach,it preserves/dissipates a modified energy but does not preserve the wave energy.The second one is based on the Lagrange multiplier approach,it preserves both the original energy and wave energy,but requires solving a nonlinear algebraic system which may require smaller time steps to have real solutions.The third one is also based on the Lagrange multiplier approach and preserves/dissipates the energy and preserves wave energy in a slightly different form,but it leads to a nonlinear quadratic system for the Lagrange multiplier which can always be explicitly solved.We present ample numerical tests to validate the three schemes,and provide a comparison on the efficiency and accuracy of the three schemes for the KGS equations.Thirdly,we develop an innovative efficient and accurate numerical scheme for constrained general dissipative systems.Our approach are based on relaxed generalized SAV approach,the Lagrange multiplier approach and operator splitting approach which is called R-GSAV/LM-OS approach for short.The approach enjoys a lot of attractive advantages,which only requires solving one linear system at each time step;is easy to construct higher-order BDFk unconditionally energy stable scheme;preserves global constraints exactly;and satisfies energy law according to original energy in most cases.We apply the R-GSAV/LM-OS approach to imaginary time gradient flow of Bose-Einstein condensates,optimal partition problem and Klein-Gordon Schrondinger equations.And ample numerical results are presented to demonstrate the effectiveness and remarkable advantages of the scheme we proposed.In addition,all the numerical simulations we consider in this part satisfy energy dissipative law according to original energy.