In general, high order splitting methods suffer from an order reduction phenomena when applied to the time integration of partial differential equations with non-periodic boundary conditions. In the last decade, there were introduced several modifications to prevent the second order Strang Splitting method from such a phenomena. In this article, inspired by these recent corrector techniques, we introduce a splitting method of order three for a class of semilinear parabolic problems that avoids order reduction in the context of non-periodic boundary conditions. We give a proof for the third order convergence of the method in a simplified linear setting and confirm the result by numerical experiments. Moreover, we show numerically that the result also persists with a nonlinear source term.

翻译：一般而言，高阶分裂方法在应用于非周期边界条件的偏微分方程时间积分时，会出现阶数降低现象。过去十年间，已有多种修正方案被提出，用以防止二阶Strang分裂方法出现此类降阶问题。受近期校正技术的启发，本文针对一类半线性抛物问题，提出了一种避免非周期边界条件下阶数降低的三阶分裂方法。我们在简化的线性设定下证明了该方法的第三阶收敛性，并通过数值实验验证了该结果。此外，数值结果表明，该方法在非线性源项存在时仍能保持其三阶收敛性。