2025/03/17 by Maximilian Ruff, Ruff, Maximilian · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Convergence (economics) #Discretization #Discretization error #Error analysis #Numerical analysis #Numerical methods for differential equations #Quantum chaos and dynamical systems #Quartic function #Torus #Wave equation #cs.NA #math.AP #math.NA
paper · pdf · doi:10.1007/s00211-026-01549-z
published in Numerische Mathematik 158(4), 1565-1612 (Springer Science+Business Media)
openalex publication_date 2026/06/22 · openalex created_date 2026/06/23 · openalex updated_date 2026/08/05
Abstract We study a variant of the Strang splitting for the time integration of the semilinear wave equation under the finite-energy condition on the torus \mathbb T3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> . In the case of a cubic nonlinearity, we show almost second-order convergence in L2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> and almost first-order convergence in H1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> . If the nonlinearity has a quartic form instead, we show analogous convergence results, where the order is reduced by 1/2 in both cases. To our knowledge these are the best convergence results available for the 3D cubic and quartic wave equations under the finite-energy condition. Our approach relies on continuous- and discrete-time Strichartz estimates. We also make use of the integration and summation by parts formulas to exploit cancellations in the error terms. Moreover, error bounds for a full discretization using the Fourier pseudo-spectral method in space are given. Finally, we discuss a numerical example indicating the sharpness of our theoretical results.