Discontinuous Galerkin discretisation in time of the second-order hyperbolic PDEs

Thesis

A discontinuous Galerkin (DG) time-stepping method is presented for solving second-order hyperbolic partial differential equations (PDEs). The proposed numerical method combines the $hp$-version discontinuous Galerkin finite element method (hp-DGFEM) in the time direction with an H¹(Ω)-conforming finite element approximation for the spatial variables. We start with the construction and analysis of the discontinuous-in-time scheme to linear hyperbolic PDEs of second order in Chapter 2. Our analysis shows that this method is consistent and stable, with arbitrarily high-order convergence rates in the temporal domain for sufficiently smooth solutions. Error bounds in both energy and L² norms are derived. These error estimates show that this method allows for a broad range of hp-refinement strategies with varying time step sizes and polynomial degrees, thus having the potential to give exponential rates of convergence. Numerical experiments on linear wave equations and elastodynamics systems show significant gains in accuracy over existing time integration schemes.

We then extend this DG time-stepping method to approximate solutions of second-order quasilinear hyperbolic systems in Chapter 3. In particular, we study the nonlinear elastodynamics problem. The resulting numerical scheme is stable and convergent, and we derive a priori error bounds at nodal points in the L²-norm for sufficiently regular solutions. Numerical experiments on a nonlinear elastodynamics problem with smooth solutions demonstrate the convergence rates.

Chapter 4 further applies this discontinuous-in-time scheme to a nonlinear damped equation, which is derived from Maxwell's equations by assuming a linearly polarised wave propagating on an infinite cylindrical domain. Again, we show a priori error estimates of the solutions at the nodal points in this chapter. Numerical experiments on nonlinear wave equations illustrate and confirm these theoretical findings.

Authors: Shao, A

• DOI: 10.5287/ora-xqpvzj1vz
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Numerical analysis; second-order hyperbolic PDEs; Discontinuous time stepping; Finite element method; Discontinuous Galerkin method; Nonlinear PDEs
• Subjects: Mathematics