Overcoming the order barrier two in splitting methods when applied to semilinear parabolic problems with non-periodic boundary conditions
This work addresses a specific computational bottleneck in numerical methods for partial differential equations, offering an incremental improvement for researchers in applied mathematics and scientific computing.
The authors tackled the problem of order reduction in high-order splitting methods for semilinear parabolic problems with non-periodic boundary conditions by introducing a third-order splitting method that avoids this issue, achieving third-order convergence as proven in a linear setting and confirmed numerically, with numerical evidence also showing persistence for a fourth-order variant and nonlinear terms.
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.