High-temperature refractory ceramics can be produced in the combustion regime by using self-propagating, high-temperature synthesis (SHS) processes. The numerical simulation of the SHS process in a simplified diffusionreaction system is investigated. The SHS process is simplified by the one- and two-dimensional pseudo-homogeneous environment. The stiff equations of the SHS process are solved by using finite difference methods on twodimensional adaptive meshes. Travelling waves with constant patterns are observed for adiabatic and nonadiabatic systems. For higher values of heat of reaction and activation energy, the combustion front starts to oscillate. Single and complex oscillating waves are detected. In oscillating combustion fronts, the temperature can overshoot the adiabatic temperature to result in the complete conversion of solid reactants. In two dimensional systems, travelling, fingering, and rotating waves are detected in the combustion synthesis process.