Numerical methods are described and results are presented for a system of convection–diffusion dispersion–reaction equations. Discretization methods that were developed earlier by the author are used. The methods allow large time steps for simulating the transport–reaction model of a waste disposal. With higher-order discretization methods, based on finite-volume methods, one may use large time steps without loss of accuracy. A multi-physical multi-dimensional equation is broken down into simpler physical and one-dimensional equations. These simpler equations are handled with locally higher-order discretization methods and the results are coupled by operator-splitting methods. An improved explicit time-discretization method, with embedded analytical solutions, for the convection–reaction equation and an implicit time-discretization diffusion–dispersion equation is described. For the numerical experiments the underlying program-tool R ³ T is briefly introduced and the main concepts are presented. Benchmark problems for testing the discretization methods of higher order are described. Real-life problems for simulating radioactive-waste disposals with underlying flowing groundwater are presented and discussed.