Despite the fundamental role of movements of molecules and ions in and between cellular compartments for brain function, most computational models for excitable tissue assume constant ion concentrations. Although these models have provided valuable insight into how neurons function and communicate, they fail to describe vital processes related to signalling and brain homeostasis, and pathologies involving substantial changes in the extracellular ion composition such as epilepsy and spreading depression. Here, we discuss the emerging KNP-EMI framework, describing the coupling of ion concentration dynamics and electrical properties in excitable tissue with explicit representation of the cells, allowing for morphologically detailed descriptions of the neuropil. In contrast to classical homogenized models, the highly detailed KNP-EMI framework enables modelling of single cells and small collections of cells, uneven distributions of membrane mechanisms, and how the cellular morphology affects tissue dynamics. The system consists of non-linear, highly coupled, and mixed–dimensional partial differential equations (PDEs). Numerical strategies for approximating the system must as such be chosen with care. We present a novel numerical strategy with three key components: (i) a splitting scheme to decouple the PN equations governing ionic transport and the equation arising from the electro-neutrality assumption, (ii) a high-order Discontinuous Galerkin (DG) discretization of the decoupled system, and (iii) a robust and scalable solver. We further analyse and discuss stability and convergence properties of the proposed scheme, and show that our solution approach is robust and scalable allowing for large-scale, high-realism in-silico studies of electrodiffusive processes in excitable tissues.