Efficient third-order tensor-oriented directional splitting for exponential integrators

Suitable discretizations of popular multidimensional operators (for instance of diffusion or diffusion-advection type) by tensor product formulas lead to matrices with d-dimensional Kronecker sum structure. For evolutionary partial differential equations containing such operators and when integrating in time with exponential integrators, it is then of paramount importance to efficiently approximate the actions of phi-functions of the arising matrices. In this work we show how to produce directional split approximations of third order with respect to the time step size. These approximations conveniently employ tensor-matrix products (the so-called mu-mode product and the related Tucker operator, realized in practice with high-performance level 3 BLAS operations) and allow for the effective usage of exponential Runge–Kutta integrators up to order three. The technique can also be efficiently implemented on modern computer hardware such as Graphic Processing Units. This approach is successfully tested against state-of-the-art techniques on two well-known physical models that lead to Turing patterns, namely the 2D Schnakenberg and the 3D FitzHugh–Nagumo systems, on different hardware and software architectures.