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 high order convergence persists for an order four variant of a splitting method, and also for a nonlinear source term.

翻译：一般而言，高阶分裂方法在应用于具有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。过去十年中，已有多种修正方法被引入以防止二阶斯特朗分裂方法出现该现象。受近期修正技术启发，本文针对一类半线性抛物问题提出了一种三阶分裂方法，该方法在非周期边界条件下避免了阶数降低。我们在线性简化设定下证明了该方法的三阶收敛性，并通过数值实验验证了该结果。此外，数值结果表明，对于四阶变体分裂方法以及非线性源项，该方法的高阶收敛性依然保持。