To date, there have been a number of attempts in simulating surface discharge. These works generally fall into one of the two categories: stochastic or deterministic. Stochastic techniques are able to produce discharge figures that have fractal characteristics but do not take into consideration any of the physical processes involved in the discharge phenomenon. Deterministic approaches, on the other hand, take ionisation, attachment, recombination, photo-ionisation and photo-emission processes into account. This project, therefore, pursues a deterministic scheme which is characterised by a set of partial differential continuity equations coupled with the Poissonâs equation. These equations govern the evolution of charged particles along the surface streamer channel. Due to the non-linearity and sharp gradient nature of the problem at hand, an accurate numerical method is required. Various numerical techniques have been compared including the Eulerian differencing, Lax-Wendroff and the Flux-Corrected-Transport (FCT) algorithm. The FCT method has shown superiority owning to its accuracy, stability and nonânegativity. The principle of the method is based on the flux limitation process which makes the best out of the low and high order schemes without introducing either diffusion or spurious oscillation. Possionâs equation, on the other hand, can be solved using the disc method or 3-D finite-difference method depending on the accuracy desired. From simulated results, one can obtain the streamer velocity, charged particle densities and electric field along the streamer channel. In addition, the discharge figure can be achieved by considering the growth of the future streamer in the direction of the maximum field induced from the preâexisting streamers. By using a different set of swarm parameters, it is also possible to simulate surface discharge in SF6 gas.