Splitting methods are a well-established tool for the numerical integration of time-dependent partial differential equations. The basic idea behind splitting is to decompose the vector field into disjoint parts, integrate them separately (with an appropriate time step), and finally combine the single flows in the right way to obtain the sought-after numerical approximation. For example, in diffusion-reaction problems, the linear diffusion equation is split from the nonlinear reaction. The diffusion part is the solved with the help of a fast Poisson solver (e.g., FFT, multigrid) whereas the nonlinearity, which is just a local ODE or a system of ODEs at each discretization point, is cheaply integrated.
In contrast to standard time integration schemes for diffusion--reaction equations (e.g., implicit Runge--Kutta or multistep methods), the splitting approach offers computational efficiency and better geometric properties. For example, positivity conservation for second-order integrators can be achieved. However, when the problem is equipped with nontrivial boundary conditions, splitting methods usually suffer from order reduction and some additional loss of accuracy.
After a brief introduction to splitting methods and their numerical analysis, the problem of order reduction for diffusion-reaction equations will be discussed. In particular, we will propose a modification of the classic splitting schemes, which resolves the problem of order reduction in the case of oblique boundary conditions: the modified Strang splitting scheme is second-order convergent, the modified Lie splitting has improved accuracy. The proposed modification only depends on the given boundary data and the computed numerical solution. Therefore, an efficient implementation is possible that makes the modified schemes superior to their classical versions.
The talk is based on joint work with Lukas Einkemmer.