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| Home > Publications database > Investigation of Exponential Time Differencing Schemes for Advection-Diffusion-Reaction Problems in the Presence of Significant Advection |
| Master Thesis | FZJ-2023-01486 |
2022
Please use a persistent id in citations: http://hdl.handle.net/2128/34151
Abstract: Advection-diffusion-reaction equations are partial differential equations (PDEs)with various applications across the sciences. Exponential time differencing schemesare efficient methods of numerically solving PDEs of this type. We consider anexponential time differencing scheme called ETD-RDP-IF that approximates thearising matrix exponentials using a rational approximation with real distinct poles andemploys a dimensional splitting technique to improve computational performance.The scheme has originally been derived for systems without advection. We show thatthe derivation still holds in the presence of advection and prove new results on thesecond-order temporal accuracy of the scheme. In numerical experiments, weinvestigate the real-world performance of the scheme depending on the strength ofadvection as quantified by the Péclet and Courant numbers. We confirmsecond-order convergence in space and time for linear problems with smooth initialcondition and observe order reduction for non-smooth initial conditions. We furtherfind that upwind-biased discretizations of advection improve computational efficiency.A comparison with an ETD scheme that uses Krylov-subspace approximations of thematrix exponentials shows that the Krylov-subspace technique has a bettercomputational performance in low-advection regimes. Outside of these regimes,ETD-RDP-IF is more robust and therefore more widely applicable.
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