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Abstract: Two candidates for optimal numerical methods for highly nonlinear partial differential equations are compared. The chosen methods are characterized by their speed, accuracy and simplicity of formulation. They can be easily formulated in high level scripting languages like MATLAB or python and are suited for practical implementation. The setting is that of 1D general convection–diffusion–adsorption–reaction systems, a setting of high relevance in chemical and groundwater engineering. First, the mathematical model is numerically solved by the method of lines (MOL) using a space discretization with moving grid points (r-adaptivity). The advantage is that with minimal grid points one captures the sharp fronts of the solution which can arise due to a strong adsorption. A large variety of isotherms can be included in the adsorption model for both equilibrium and non-equilibrium modes. In the second method, the mathematical model is approximated using interface modelling. However, this method is only applicable for adsorption in equilibrium mode. The numerical efficiency of the methods is discussed and the obtained numerical results are compared to determine their optimal use. Both methods are very suitable for solving inverse problems in practical implementations, as they are robust and fast. [Copyright &y& Elsevier]