An Asymptotic Preserving Method for the Weakly Nonlinear Klein-Gordon Equation

Abstract

In this paper, we propose an asymptotic preserving (AP) method for the weakly nonlinear Klein-Gordon equation (NKGE). Firstly, we apply a multi-scale expansion for the weakly NKGE and obtain the equation for the leading-order term, for which an error estimate has been provided. Secondly, by solving the equation for the leading-order term numerically, we construct an AP method for the weakly NKGE. Finally, numerical results in one spatial dimension are provided to show that:

(i) The method is asymptotic preserving, i.e., the error between the leading-order term and the solution of the weakly NKGE behaves as $\mathcal{O}(\varepsilon)$ as $\varepsilon \to 0$.

(ii) It is uniformly accurate since the numerical solution obtained by the method is independent of the small parameter $\varepsilon$.

(iii) It can make correct predictions about the solution of the original NKGE. Moreover, extension of the method to the two-dimensional weakly NKGE are also provided.