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Mathematical modelling of ionic electrodiffusion and water movement isemerging as a powerful avenue of investigation to provide new physiologicalinsight into brain homeostasis. However, in order to provide solid answers andresolve controversies, the accuracy of the predictions is essential. Ionicelectrodiffusion models typically comprise non-trivial systems of non-linearand highly coupled partial and ordinary differential equations that governphenomena on disparate time scales. Here, we study numerical challenges relatedto approximating these systems. We consider a homogenized model forelectrodiffusion and osmosis in brain tissue and present and evaluate differentassociated finite element-based splitting schemes in terms of their numericalproperties, including accuracy, convergence, and computational efficiency forboth idealized scenarios and for the physiologically relevant setting ofcortical spreading depression (CSD). We find that the schemes display optimalconvergence rates in space for problems with smooth manufactured solutions.However, the physiological CSD setting is challenging: we find that theaccurate computation of CSD wave characteristics (wave speed and wave width)requires a very fine spatial and fine temporal resolution.

Accurate numerical simulation of electrodiffusion and water movement in brain tissue