Stability, convergence, and geometric properties of second-order-in-time space-time discretizations for linear and semilinear wave equations
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arXiv
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| Format: | Preprint |
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2026
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| author | Ferrari, Matteo Perugia, Ilaria Zampa, Enrico |
| author_facet | Ferrari, Matteo Perugia, Ilaria Zampa, Enrico |
| contents | We revisit second-order-in-time space-time discretizations of the linear and semilinear wave equations by establishing precise equivalences with first-order-in-time formulations. Focusing on schemes using continuous piecewise-polynomial trial functions in time, we analyze their stability, convergence, and geometric properties. We consider first a weak space-time formulation with test functions projected onto discontinuous polynomials of one degree lower in time, showing that it is equivalent to the scheme proposed in [French, Peterson 1996] in the linear case, and extended in [Karakashian, Makridakis 2005] to the semilinear case. In particular, this equivalence shows that this method conserves energy at mesh nodes but is not symplectic. We then introduce two symplectic variants, obtained through Gauss-Legendre and Gauss-Lobatto quadratures in time, and show that they correspond to specific Runge-Kutta time integrators. These connections clarify the geometric structure of the space-time methods considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03160 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stability, convergence, and geometric properties of second-order-in-time space-time discretizations for linear and semilinear wave equations Ferrari, Matteo Perugia, Ilaria Zampa, Enrico Numerical Analysis 65M60, 65M12, 65P10 We revisit second-order-in-time space-time discretizations of the linear and semilinear wave equations by establishing precise equivalences with first-order-in-time formulations. Focusing on schemes using continuous piecewise-polynomial trial functions in time, we analyze their stability, convergence, and geometric properties. We consider first a weak space-time formulation with test functions projected onto discontinuous polynomials of one degree lower in time, showing that it is equivalent to the scheme proposed in [French, Peterson 1996] in the linear case, and extended in [Karakashian, Makridakis 2005] to the semilinear case. In particular, this equivalence shows that this method conserves energy at mesh nodes but is not symplectic. We then introduce two symplectic variants, obtained through Gauss-Legendre and Gauss-Lobatto quadratures in time, and show that they correspond to specific Runge-Kutta time integrators. These connections clarify the geometric structure of the space-time methods considered. |
| title | Stability, convergence, and geometric properties of second-order-in-time space-time discretizations for linear and semilinear wave equations |
| topic | Numerical Analysis 65M60, 65M12, 65P10 |
| url | https://arxiv.org/abs/2601.03160 |