A symmetric exponential wave integrator for the long-time dynamics of weakly nonlinear Klein–Gordon equation

We propose a symmetric midpoint-rule exponential wave integrator (sMEWI) for the weakly nonlinear Klein–Gordon equation over long-time intervals of order 𝑂 ⁡ ( 𝜀 − 2 ) . This scheme is derived from a centered two-sided Duhamel formula combined with integral midpoint approximation, and thus yields an explicit, symmetric scheme with second-order accuracy. Under the long-time regularity assumption on the exact solution, we employ the regularity compensation oscillation (RCO) technique to establish an improved uniform error bound O (ε 2 τ 2 ). This bound holds for the long-time scale 𝑇 𝜀 = 𝑇 / 𝜀 2 , where T is a fixed constant. Furthermore, for the fully discrete Fourier pseudospectral implementation, we prove an error estimate of 𝑂 ⁢ ( ℎ 𝑚 + 𝜀 2 ⁢ 𝜏 2 ) . Extensive numerical experiments are conducted to validate our theoretical error bounds and demonstrate the advantages of the proposed sMEWI. Comparative studies with the Strang splitting method, the Gautschi-type exponential wave integrator Fourier pseudospectral (EWI-FP) method, and the symmetric exponential wave integrator (sEWI) show that sMEWI exhibits stronger long-time stability and maintains near discrete energy conservation.