An implicit–explicit time discretization scheme for second-order semilinear wave equations with application to dynamic boundary conditions

Numerische Mathematik https://doi.org/10.1007/s00211-021-01184-w Numerische Mathematik An implicit–explicit time discretization scheme for second-order semilinear wave equations with application to dynamic boundary conditions Marlis Hochbuck1 · Jan Leibold1 Received: 27 July 2020 / Revised: 3 December 2020 / Accepted: 30 January 2021 © The Author(s) 2021 Abstract We construct and analyze a second-order implicit–explicit (IMEX) scheme for the time integration of semilinear second-order wave equations. The scheme treats the stiff linear part of the problem implicitly and the nonlinear part explicitly. This makes the scheme unconditionally stable and at the same time very efficient, since it only requires the solution of one linear system of equations per time step. For the combination of the IMEX scheme with a general, abstract, nonconforming space discretization we prove a full discretization error bound. We then apply the method to a nonconforming finite element discretization of an acoustic wave equation with a kinetic boundary condition. This yields a fully discrete scheme and a corresponding a-priori error estimate. Keywords Implicit–explicit time integration · IMEX · Dynamic boundary conditions · Semilinear wave equation · Nonconforming space discretization · Error analysis · A-priori error bounds · Semilinear evolution equations · Operator semigroups Mathematics Subject Classification Primary 65M12 · 65M15 · Secondary 65M60 · 65J08 Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project-ID 258734477—SFB 1173. B Jan Leibold Marlis Hochbuck 1 Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technologie, 76049 Karlsruhe, Germany 123 M. Hochbuck, J. Leibold 1 Introduction In this paper we construct and analyze an implicit–explicit (IMEX) time integration scheme for second-order semilinear wave equations of the form u  (t) + Bu  (t) + Au(t) = f (t, u(t)) in a suitable Hilbert space. Here, A and B are unbounded operators and f is a locally Lipschitz continuous nonlinearity. The IMEX scheme is constructed as a combination of the explicit leapfrog method and the implicit Crank–Nicolson scheme. It treats the unbounded linear part of the differential equation implicitly and the nonlinear part explicitly. We show that the scheme is unconditionally stable in the sense that the time-step size is only restricted by the Lipschitz constant of f but not by the linear operators A and B. We combine this IMEX scheme with an abstract, nonconforming space discretization within the framework of [11–13]. These papers provide a unified error analysis (UEA) which allows one to analyze nonconforming space discretizations of wave equations in a systematic way. Our main result is an error bound that is second order in time and contains abstract space-discretization errors. The error result can then be used to prove convergence rates for specific problems and discretizations by plugging in geometric and interpolation error results. The fully discrete scheme is very efficient. In fact, we show that one time step only requires the solution of one linear system and one application of discretizations of A, B, and f , respectively. Since the construction of the scheme is based on two second-order methods and our analysis makes use of the specific form of the method, the generalization to higher order is not straightforward and out of the scope of this paper. Higher-order IMEX schemes for second-order equations will be part of future work. There is a rich literature on IMEX schemes for first-order equations, in particular, there is a well-developed theory for IMEX Runge–Kutta schemes [3,6] or IMEX multistep schemes [1,4,8,15], for instance. In [6,15] an error analysis for ODEs is presented, while [1] contains discretization errors for IMEX schemes applied to conformal space discretizations of quasilinear parabolic evolution equations. IMEX schemes are used in applications, e.g., in structural dynamics and fluid-structure interaction [22], hydrodynamics [16], sea-ice dynamics [20], or atmospheric dynamics, see, e.g., [9], to mention just a few examples. There exists also a so-called Crank–Nicolson-leapfrog IMEX scheme which is obtained from a combination of the Crank–Nicolson and the leapfrog scheme for first-order equations, cf. [17,18], and references therein. However, this scheme is not equivalent to the scheme we construct here, since the leapfrog schemes for first- and second-order equations are not equivalent and indeed have completely different stability properties. More precisely, the Crank–Nicolson-leapfrog scheme is only stable, if the explicitly treated part is skew symmetric. To solve a second-order equation one can either reformulate it equivalently into a first-order equation and apply an IMEX scheme to it or one can design a scheme for the original second-order form. An example for the first option is the Crank– Nicolson-leapfrog IMEX scheme presented in [17,18]. In contrast, we decided to take the second option and present a new scheme which to the best of our knowledge was not 123 An implicit–explicit time discretization scheme… considered in literature so far. In fact, we are not aware of any IMEX scheme exploiting the special structure of second-order equations. The advantage of this approach is the efficiency of the scheme, as will be discussed in detail in Sect. 2.2. The main contribution of this paper is to provide a full discretization error analysis of semilinear wave equations in a quite general framework. So far, such an error analysis does not even exist for the Crank–Nicolson scheme, which is also covered as a byproduct of our analysis of the IMEX scheme. The challenge of such a rigorous analysis is that it applies to abstract, non-conforming space discretizations of semilinear wave-type equations. As an application of our abstract theory, we consider an acoustic wave equation with kinetic boundary conditions that fits into the abstract setting, cf., [13]. Kinetic boundary conditions are a special case of dynamic boundary conditions that contain tangential derivatives and are intrinsically posed on domains with (piecewise) smooth and therefore possibly curved boundaries. Hence, the spatial discretization has to be done on an approximated domain rendering the discretization nonconforming. The paper is organized as follows: in Sect. 2 we present the problem setting, introduce the IMEX scheme for second-order wave equations, and state a second-order error bound for the time discretization error. In Sect. 3 we briefly recall the UEA and present the fully discrete scheme as a combination of the IMEX scheme with a general space discretization. Afterwards we state and prove the main result, namely the abstract full discretization error bound. Finally, in Sect. 4, we consider a semilinear acoustic wave equation with a kinetic boundary conditions as an example fitting into the abstract setting. We present (...truncated)