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

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.

An Efficient Numerical Method for Time Domain Electromagnetic Wave Propagation in Co-axial Cables

“…In [9], [2], the objective was to treat in a particular way the boundary condition arising from the formulation of elastodynamics by way of potentials introduced for treatinf separately the two types of waves. In [14], the objective was to treat explicitly the linear part of the model and implicitly the nonlinear one. Here, these are the direction of space differentiation, longitudinal or transverse, which are treated in a different manner.…”

A second-order low-regularity correction of Lie splitting for the semilinear Klein--Gordon equation

“…• The recent second-order IMEX method for semilinear second-order wave equations from [16], that we refer to as hl21. This method computes approximations u n+1 , v n+1 to u(t n+1 ), u t (t n+1 ) at discrete times t n+1 = t 0 + (n + 1)τ with the time step size τ as…”

“…The numerical approximation of semilinear Klein-Gordon equations in the form of (1.1) has been extensively studied in computational mathematics. A large variety of numerical schemes for approximating the time dynamics of the semilinear Klein-Gordon equation has been proposed and analyzed, including trigonometric/exponential integrators that are based on the variation-of-constants formula (for example, see [5,11,13,17,29]), splitting methods (for example, see [1,2,5,10]), finite difference methods (such as the Crank-Nicolson and Runge-Kutta methods, see [6,16,19,[21][22][23]26]), and symplectic methods [7,8,14].…”

Splitting schemes for the semi-linear wave equation with dynamic boundary conditions

“…A slight change in the incorporation of the right-hand side, namely replacing the trapezoidal rule by a left/right rectangle rule, leads to a Crank-Nicolson scheme with an explicit implementation of the nonlinearity, cf. [HL21]. The resulting implicit-explicit scheme reads…”

“…For wave systems with dynamic boundary conditions, the spatial discretization was rigorously analyzed in [HHS18,HL20,HK20]. Moreover, for the temporal discretization, an implicit-explicit variant of the Crank-Nicolson scheme was introduced and analyzed in [HL21].…”