An energy-preserving algorithm for nonlinear Hamiltonian wave equations with Neumann boundary conditions
As model problem the authors consider the nonlinear Klein-Gordon equation \[ \frac{\partial^2 u(x,t)}{\partial t^2} - \lambda \frac{\partial^2 u(x,t)}{\partial x^2} = - \nabla V(u) \, , \; (x,t) \in [x_L,x_R] \times [t_0,T], \] where \(V(u)\) is a smooth potential energy and \(\lambda\) is a positive real number. Additionally, initial conditions and Neumann boundary conditions are given. For the discretization in space the finite element method with standard continuous piecewise polynomial basis functions is used. This results in a system of ordinary differential equations. The stability of this semi-discrete finite element scheme is analysed. For the time discretization of the system of ordinary differential equations the average vector field approach is used. In this way the semi-discrete energy is preserved. The presented approach is illustrated in detail in the case of piecewise-linear basis functions in the finite element method. Finally, the conservative property of the presented algorithm is shown by two numerical examples.
scientific article; zbMATH DE number 6839403
scientific article