Conservative compact finite difference scheme for the N‐coupled nonlinear Klein–Gordon equations

Abstract

In this article, a compact finite difference method is developed for the periodic initial value problem of the N‐coupled nonlinear Klein–Gordon equations. The present scheme is proved to preserve the total energy in the discrete sense. Due to the difficulty in obtaining the priori estimate from the discrete energy conservation law, the cut‐off function technique is employed to prove the convergence, which shows the new scheme possesses second order accuracy in time and fourth order accuracy in space, respectively. Additionally, several numerical results are reported to confirm our theoretical analysis. Lastly, we apply the reliable method to simulate and study the collisions of solitary waves numerically.

Higher‐order explicit schemes based on the method of characteristics for hyperbolic equations with crossing straight‐line characteristics

“…To summarize, Equations (14)- (18) and (21) define an MoC-RK3 scheme that is a counterpart of the ODE solver (13). The reason why this is a, and not the, counterpart scheme is that approximate solutions y ± (1) and y ± (2) could, in principle, be computed by different first-and second-order schemes; other "degrees of freedom" in obtaining (y ± ) n+1 m will be mentioned in Section 3.3.…”

“…1∕2 from the linearization of (21a) and (18); this is quite tedious but straightforward. Combined with a similar calculation for (̃− 2 ) n m+1 , this yields: .…”

“…) n+1 m being found with respective local errors O(h 2 ) and O(h 3 ) by (18). Once the stage derivatives 𝜿 ± 1,2 are found at time level t n − 1 , they are stored for one time step, to advance the solution from t n to t n + 1 ; see the last two terms in (44a).…”

“…Let us note that the Klein–Gordon equation, which serves as a simplified model for many dispersive wave problems: where c = const and g is some differentiable function of its arguments, can also be written in form (1b) [15] with N + = N − = 2. Numerical solution of various extensions of this well‐studied model has recently attracted attention [16, 17, 18]. The schemes that we propose below are applicable to all of those extensions.…”

Two regularized energy-preserving finite difference methods for the logarithmic Klein-Gordon equation

“…For the numerical part, different efficient and accurate numerical methods have been proposed and analyzed for computations of wave propagations in classic/relativistic physics, such as the standard finite difference time domain (FDTD) [18,19,20,21,22,23,24,25] , the conservative compact finite difference method [26], the multiscale time integrator Fourier pseudospectral (MWI-FP) method [27], the finite element method [28], the time-splitting spectral method (TSFP) [29], the exponential wave integrator Fourier pseudospectral (EWI-FP) method [18,30], the asymptotic preserving (AP) method [31], etc. Of course, each method has its advantages and disadvantages.…”