Splitting methods constitute an important class of time integration schemes for evolution equations: one decomposes the vector field into disjoint parts, integrates the resulting sub-problems separately (on an appropriate time interval), and finally combines the single flows in the right way to obtain the sought-after numerical approximation.
In this talk we will consider problems containing a Burgers-type nonlinearity. Typical examples comprise the Korteweg--de Vries (KdV) and the Kadomtsev--Petviashvili (KP) equation, which model shallow water in 1 and 2D, respectively. Whereas the linear part in both equations is efficiently integrated by fast Fourier techniques, the nonlinearity can be handled with the method of characteristics. For the Burgers-type nonlinearity we propose a semi-Lagrangian approach based on polynomial interpolation. It is shown that this interpolation procedure can be implemented efficiently. The resulting splitting schemes are highly competitive with previous integrators. In addition, the conservation properties of the numerical schemes under consideration are investigated.
As a second example we discuss reaction-diffusion equations, where the diffusion is modeled by the Laplacian and the reaction by a (locally acting) non-linearity, respectively. Separating the diffusion from the reaction gives, on the one hand, a free heat equation that can often by solved efficiently by fast Fourier techniques and, on the other hand, a set of ordinary differential equations that describe the local reaction at each grid point. However, the above approach requires some care if the problem is endowed with non-periodic boundary conditions. In this case, a straightforward application of splitting will often lead to a strong order reduction and consequently to computational inefficiency. In the talk, we will exemplify the problem of boundary conditions in splitting methods with the help of typical examples. Based on these observations and further theoretical investigations, we will present remedies to avoid order reduction.
The talk is based on joint work with Lukas Einkemmer.