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Suitable discretizations through tensor product formulas of popularmultidimensional operators (diffusion or diffusion--advection, for instance)lead to matrices with $d$-dimensional Kronecker sum structure. For evolutionaryPartial Differential Equations containing such operators and integrated in timewith exponential integrators, it is then of paramount importance to efficientlyapproximate the actions of $\varphi$-functions of the arising matrices. In thiswork, we show how to produce directional split approximations of third orderwith respect to the time step size. They conveniently employ tensor-matrixproducts (the so-called $\mu$-mode product and related Tucker operator,realized in practice with high performance level 3 BLAS), and allow for theeffective usage of exponential Runge--Kutta integrators up to order three. Thetechnique can also be efficiently implemented on modern computer hardware suchas Graphic Processing Units. The approach has been successfully tested againststate-of-the-art techniques on two well-known physical models that lead toTuring patterns, namely the 2D Schnakenberg and the 3D FitzHugh--Nagumosystems, on different architectures.

Efficient third order tensor-oriented directional splitting for exponential integrators