An exponential-type integrator for the KdV equation

Abstract

We introduce an exponential-type time-integrator for the KdV equation and prove its first-order convergence in H 1 for initial data in H 3 . Furthermore, we outline the generalization of the presented technique to a second-order method. t 0 e −∂ 3 x (t−s) ∂ x (u(s)) 2 ds

Two exponential-type integrators for the “good” Boussinesq equation

“…The idea of twisting the variable is widely applied in the analysis of PDEs in low regularity spaces [3]. It was also widely applied in the context of numerical analysis for the Schrödinger equation [18,25], the KdV equation [15] and Klein-Gordon type equations [1,2]. For implementation issues, we impose periodic boundary conditions and refer to [11,17,23] for the corresponding well-posedness results.…”

A Lawson-type exponential integrator for the Korteweg–de Vries equation

“…More specifically, first-order convergence in time was shown in [25] for the fully discrete pseudospectral method under the stability condition τ = O(h 3 ) for explicit and τ = O(h 2 ) for implicit discretization of the nonlinear term, respectively. For the rigorous analysis of splitting methods, we refer to [17,20].Nowadays, exponential time integration methods are widely applied for parabolic and hyperbolic problems [3,15,16]. In particular, a distinguished exponential-type integrator was derived for the KdV equation [16] by using a "twisting" technique.…”

“…Such an integrator, however, can hardly be extended to more general equations, e.g., the fifth-order KdV equation, without additional regularity assumptions. Furthermore, the spatial error was not considered in [16].In the present paper, we propose a Fourier pseudospectral method based on a classical Lawson-type exponential integrator, which integrates the linear part exactly, and the Rusanov scheme for Burgers' nonlinearity with an added artificial viscosity. The method is explicit, implemented with FFT and efficient in practical computation.…”

“…Such an integrator, however, can hardly be extended to more general equations, e.g., the fifth-order KdV equation, without additional regularity assumptions. Furthermore, the spatial error was not considered in [16].…”

Asymptotic consistent exponential-type integrators for Klein-Gordon-Schrödinger systems from relativistic to non-relativistic regimes

“…In addition, they appear in numerical analysis, for instance in the context of the modulated Fourier expansion [9,17], adiabatic integrators [17,25] as well as Lawson-type Runge-Kutta methods [24]. Recently, this technique was also established in the numerical analysis of low-regularity problems [21,28] and introduced for the highly oscillatory Klein-Gordon equation in [5]. In the latter we could develop uniformly accurate exponential-type integrators for the classical Klein-Gordon equation up to order two for the first time.…”