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__Stability in the presence of transport terms__

A. Iserles

**Abstract**

Numerous PDEs are of the form $u_t+V(x)\cdot \nabla u = \mathcal{L}(u,x,t)$, where $\mathcal{L}$ is an operator, e.g. convection-diffusion, the forward Kolmogorov equation, the Fokker-Planck equation and the Bolzmann equation. Once this equation, on an arbitrary grid, is discretised using finite differences, in tandem with splitting (e.g. the Strang splitting), the discretisation of transport terms $-V\cdot \nabla u$ is typically unstable. In this talk we prove that such a discretisation is always stable as long as differentiation is approximated with a skew-symmetric matrix but that skew-symmetry on an arbitrary grid is generically consistent with just first-order approximation. We derive conditions that guarantee high-order skew-symmetric approximation of the first derivative and present a method to construct matrices of this kind which are, in addition, banded.