Energy-preserving Schemes For Semilinear Wave Equation Under Different Boundary Conditions

In recent years,in the area of numerical computation,structure-preserving algorithms have drawn much attention.Structure-preserving algorithm is also known as geometric numerical integration.It exactly preserves one or more physical /geometric properties of the system when numerically solving differential equations.Compared to the traditional numerical methods,structure-preserving algorithms are more effective and suitable for long-term numerical simulations.In the design of numerical algorithms,to make numerical methods correctly reproduce the qualitative behavior of the systems,one should try to ensure that the basic structures of the system is not to be altered by the numerical methods.In this thesis,energy-preserving schemes are established for semilinear wave equation.Semilinear wave equation is an important class of Hamiltonian partial differential equations,the energy of which has different evolutionary behaviors under different boundary conditions.This thesis proposes new energy-preserving schemes for semilinear wave equation under periodic boundary conditions,Dirichlet boundary conditions and Neumann boundary conditions.The main contents are as follows:1)The wave equation is semi-discretized in spatial direction by finite differences for different boundary conditions such that the Hamiltonian functions of the resulting semi-discrete Hamiltonian systems are the discrete analogue of the original energy.Meanwhile,the Hamiltonian functions of the resulting semi-discrete systems have similar evolutionary behaviours as the original energy.2)It is noted that the semi-discrete system exhibits an oscillatory structure.Subsequently,the semi-discrete approximation of the energy evolution as well as the oscillatory structure is passed to the fully-discrete numerical solutions by applying extended discrete gradient formula in temporal direction.In particular,under Dirichlet boundary conditions and Neumann boundary conditions,the semi-discrete Hamiltonian systems are non-autonomous.In these two cases,auxiliary conjugate variables and the augmented Hamiltonian are introduced to make the extended discrete gradient formula applicable.3)Numerical experiments are carried out to show the effectiveness of the new schemes in comparison with the traditional discrete gradient schemes.