High-order Structure-preserving Difference Methods For BBMB Equations And Rosenau-burgers Equations

In recent years,scalar auxiliary variables(SAV)methods have received much attention due to their advantages in constructing unconditional energy-stable numerical algorithms.Some improved versions including the exponential scalar auxiliary variables(E-SAV)method have been proposed.Compared with the original SAV method,the E-SAV method retains the advantages of the original SAV method and simplifies the computation.The SAV and E-SAV were originally proposed in the context of the gradient flow problem.However,many mathematical models with strong application backgrounds do not have this form.For this reason,this thesis extends the E-SAV method to two representative non-gradient flow problems and constructs the corresponding structure-preserving algorithms.This extends the applicability of the SAV method to a certain degree.Specifically,an equivalent system is developed for the Benjamin-Bona-Mahony-Burgers(BBMB)equations by defining appropriate exponential scalar auxiliary variables.Based on the obtained equivalent system,three linear difference schemes with fourth-order accuracy in the spatial direction and second-order accuracy in the temporal direction are constructed using different discretization in time,and the structure-preserving performance of the proposed difference schemes is analyzed theoretically and verified numerically.Similarly,a corresponding equivalent form for the Rosenau-Burgers equation is constructed by the E-SAV method,and three linear difference schemes of fourth-order in space and second-order in time are constructed based on the resulting equivalent form.The structurepreserving properties of the proposed difference schemes are analyzed and verified numerically using the discrete energy method.