3. November 2023, 10:15
Roland Schnaubelt (KIT)
Error analysis of the Lie splitting for semilinear wave equations at H^1-regularity
We study the semilinear wave equation on R^3 for data in the energy space H^1\times L^2 with a power nonlinearity of order a between 3 and 5. For a=5, the problem is energy-critical, and this quintic nonlinearity is the borderline case for local wellposedness. Time integration schemes for this equation have been studied before only at higher regularity levels. The local wellposedness theory suggests to exploit the dispersive nature of the system by using Strichartz estimates. Motivated by related work on the nonlinear Schrödinger equation, we first show discrete-in-time Strichartz estimates for the linear wave equation on R^3. These require frequency cut-offs of the solution because of the discrete setting. We then analyze a Lie splitting scheme with frequency cut-offs for the semilinear problem with data in the energy space. Our main result yields first-order convergence in L^2\times H^{-1} of the time-discrete approximation to the solution (u,u_t) to the PDE. We also investigate a higher-order variant. This is joint work with Maximilian Ruff (Karlsruhe).