Capacitive deionization (CDI) is a desalination method that has been expanding substantially in recent years. As processes are getting more complex, corresponding developments in theory and software are necessary to keep up and drive future research. In this work, we derive a new CDI theory based on a tertiary current distribution, meaning each ionic species is resolved individually in a unified framework in 1D/2D. The results show that this approach is ideal for simulations with multiple ionic species and materials that affect cations and anions differently. Direct examples include such as intercalation materials and membranes. It is also effective for incorporating electrode replenishment in flow-electrode CDI (FCDI). By benchmarking with traditional methods, we demonstrate that numerical stability is a central limitation of traditional methods for these applications. The results identify physical processes involving rapid changes to cause major instabilities. This can thus be handled by introducing specific numerically stabilizing factors. Finally, the theory is compiled into comprehensive software that researchers can straightforwardly apply in future studies without having to reconstruct methods from scratch. A corresponding video tutorial has also been deposited. In conclusion, the work pushes the limits of the simulation capabilities in a wide range of CDI processes.